Multiplicative · 199 sequences
The sequences of the family “multiplicative”, by A-number, with the share of their decomposable terms in the level class (k > L).
- primes · 1717
- polynomial · 1232
- quadratic form · 430
- prime values · 423
- residue class · 357
- digit rule · 306
- multiplicative · 199
- Beatty · 86
- divisor functions · 58
- binary rule · 48
- self-referential · 35
- complement · 23
- powers · 22
- summatory · 21
- arithmetic progression · 10
- smooth · 10
- forced divisor · 9
- sieve · 8
- block · 5
- base case · 1
| A-number | Name | Level |
|---|---|---|
| A000028 | Let k = p_1^e_1 p_2^e_2 p_3^e_3 ... be the prime factorization of n. Sequence gives k such that the sum of the numbers of 1's in the binary expansions of e_1, e_2, e_3, ... is odd | 12.7 % |
| A000379 | Numbers where total number of 1-bits in the exponents of their prime factorization is even; a 2-way classification of integers: complement of A000028 | 12.7 % |
| A000415 | Numbers that are the sum of 2 but no fewer nonzero squares | 16.0 % |
| A000430 | Primes and squares of primes | 23.0 % |
| A000977 | Numbers that are divisible by at least three different primes | 13.8 % |
| A001358 | Semiprimes (or biprimes): products of two primes | 15.7 % |
| A001481 | Numbers that are the sum of 2 squares | 15.9 % |
| A001694 | Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squareful, square full, square-full or 2-powerful numbers) | 71.9 % |
| A002035 | Numbers that contain primes to odd powers only | 10.1 % |
| A004611 | Divisible only by primes congruent to 1 mod 3 | 32.6 % |
| A004614 | Numbers that are divisible only by primes congruent to 3 mod 4 | 23.6 % |
| A004709 | Cubefree numbers: numbers that are not divisible by any cube > 1 | 10.2 % |
| A005117 | Squarefree numbers: numbers that are not divisible by a square greater than 1 | 10.7 % |
| A005238 | Numbers k such that k, k+1 and k+2 have the same number of divisors | 36.7 % |
| A006049 | Numbers k such that k and k+1 have the same number of distinct prime divisors | 16.7 % |
| A006073 | Numbers k such that k, k+1 and k+2 all have the same number of distinct prime divisors | 22.0 % |
| A006881 | Squarefree semiprimes: Numbers that are the product of two distinct primes | 15.7 % |
| A007304 | Sphenic numbers: products of 3 distinct primes | 17.0 % |
| A007674 | Numbers m such that m and m+1 are squarefree | 14.6 % |
| A007675 | Numbers m such that m, m+1 and m+2 are squarefree | 28.7 % |
| A007774 | Numbers that are divisible by exactly 2 different primes; numbers n with omega(n) = A001221(n) = 2 | 13.2 % |
| A008846 | Hypotenuses of primitive Pythagorean triangles | 23.8 % |
| A013929 | Numbers that are not squarefree. Numbers that are divisible by a square greater than 1. The complement of A005117 | 13.0 % |
| A014567 | Numbers k such that k and sigma(k) are relatively prime, where sigma(k) = sum of divisors of k (A000203) | 13.5 % |
| A014612 | Numbers that are the product of exactly three (not necessarily distinct) primes | 15.8 % |
| A014613 | Numbers that are products of 4 primes | 16.9 % |
| A014614 | Numbers that are products of 5 primes (or 5-almost primes, a generalization of semiprimes) | 18.0 % |
| A014657 | Numbers m that divide 2^k + 1 for some nonnegative k | 23.6 % |
| A014661 | Numbers that do not divide 2^k + 1 for any k>0 | 11.2 % |
| A016105 | Blum integers: numbers of the form p * q where p and q are distinct primes congruent to 3 (mod 4) | 29.9 % |
| A025583 | Composite numbers that are not the sum of 2 primes | 5.4 % |
| A026424 | Number of prime divisors (counted with multiplicity) is odd; Liouville function lambda(n) (A008836) is negative | 12.8 % |
| A028260 | Numbers with an even number of prime divisors (counted with multiplicity); numbers k such that the Liouville function lambda(k) (A008836) is positive | 12.8 % |
| A030059 | Numbers that are the product of an odd number of distinct primes | 14.6 % |
| A030229 | Numbers that are the product of an even number of distinct primes | 14.7 % |
| A030230 | Numbers that have an odd number of distinct prime divisors | 12.7 % |
| A030231 | Numbers with an even number of distinct prime factors | 12.7 % |
| A030632 | Numbers with 14 divisors | 28.9 % |
| A030636 | Numbers with 18 divisors | 33.9 % |
| A031363 | Positive numbers of the form x^2 + xy - y^2; or, of the form 5x^2 - y^2 | 23.4 % |
| A033948 | Numbers that have a primitive root (k such that the multiplicative group modulo k is cyclic) | 15.3 % |
| A033949 | Positive integers that do not have a primitive root | 11.6 % |
| A033950 | Refactorable numbers: number of divisors of k divides k. Also known as tau numbers | 16.9 % |
| A033992 | Numbers that are divisible by exactly three different primes | 13.6 % |
| A033993 | Numbers that are divisible by exactly four different primes | 17.2 % |
| A036668 | Hati numbers: of form 2^i*3^j*k, i+j even, (k,6)=1 | 13.1 % |
| A036785 | Numbers divisible by the squares of two distinct primes | 19.2 % |
| A037020 | Numbers whose sum of proper (or aliquot) divisors is a prime | 28.4 % |
| A037144 | Numbers with at most 3 prime factors (counted with multiplicity) | 9.6 % |
| A038509 | Composite numbers congruent to +-1 mod 6 | 15.1 % |
| A039955 | Squarefree numbers congruent to 1 (mod 4) | 22.9 % |
| A039956 | Even squarefree numbers | 17.5 % |
| A039957 | Squarefree numbers congruent to 3 mod 4 | 23.0 % |
| A045920 | Numbers m such that the factorizations of m..m+1 have the same number of primes (including multiplicities) | 20.1 % |
| A045939 | Numbers m such that the factorizations of m..m+2 have the same number of primes (including multiplicities) | 30.4 % |
| A046099 | Numbers that are not cubefree. Numbers divisible by a cube greater than 1. Complement of A004709 | 14.1 % |
| A046100 | Biquadratefree numbers: numbers that are not divisible by any 4th power greater than 1 | 9.9 % |
| A046101 | Biquadrateful numbers | 13.7 % |
| A046306 | Numbers that are divisible by exactly 6 primes with multiplicity | 19.0 % |
| A046308 | Numbers that are divisible by exactly 7 primes counting multiplicity | 19.7 % |
| A046310 | Numbers that are divisible by exactly 8 primes counting multiplicity | 20.5 % |
| A046312 | Numbers that are divisible by exactly 9 primes with multiplicity | 21.2 % |
| A046314 | Numbers that are divisible by exactly 10 primes with multiplicity | 21.9 % |
| A046315 | Odd semiprimes: odd numbers divisible by exactly 2 primes (counted with multiplicity) | 21.0 % |
| A046316 | Numbers of the form p*q*r where p,q,r are (not necessarily distinct) odd primes | 25.7 % |
| A046386 | Products of exactly four distinct primes | 21.0 % |
| A046387 | Products of exactly 5 distinct primes | 25.6 % |
| A046388 | Odd numbers of the form p*q where p and q are distinct primes | 21.0 % |
| A048103 | Numbers not divisible by p^p for any prime p | 11.8 % |
| A050384 | Nonprimes such that n and phi(n) are relatively prime | 19.5 % |
| A050931 | Numbers having a prime factor congruent to 1 mod 6 | 11.6 % |
| A051270 | Numbers that are divisible by exactly 5 different primes | 20.2 % |
| A051283 | Numbers k such that if one writes k = Product p_i^e_i (p_i primes) and P = max p_i^e_i, then k/P > P | 17.9 % |
| A052214 | Numbers n with prime signature(n) = prime signature(n+1) = prime signature(n+2) | 42.4 % |
| A052485 | Weak numbers (i.e., not powerful (1)): there is a prime p where p|n is true but p^2|n is not true | 9.6 % |
| A054753 | Numbers which are the product of a prime and the square of a different prime (p^2 * q) | 27.4 % |
| A056809 | Numbers k such that k, k+1 and k+2 are products of two primes | 54.5 % |
| A056867 | Nilpotent numbers: n such that every group of order n is nilpotent | 14.5 % |
| A056868 | Numbers that are not nilpotent numbers | 12.5 % |
| A059404 | Numbers with different exponents in their prime factorizations | 13.0 % |
| A061346 | Odd numbers that are neither primes nor prime powers | 21.0 % |
| A062503 | Squarefree numbers squared | 100.0 % |
| A062721 | Numbers k such that k is a product of two primes and k-2 is prime | 40.2 % |
| A062832 | Numbers k such that k and k+2 have the same number of divisors | 22.1 % |
| A063464 | Numbers k such that omega(k) = omega(k+2), where omega(k) is the number of distinct prime divisors of k | 14.7 % |
| A063465 | Number k such that omega(k) = omega(k+3), where omega(k) is the number of distinct prime divisors of k | 16.6 % |
| A066031 | Composite numbers n the sum of whose prime factors divides n, but which are not themselves powers of primes | 36.1 % |
| A067259 | Cubefree numbers which are not squarefree | 17.9 % |
| A067885 | Products of exactly 6 distinct primes | 29.5 % |
| A068781 | Lesser of two consecutive numbers each divisible by a square | 24.9 % |
| A069272 | 11-almost primes (generalization of semiprimes) | 22.5 % |
| A069273 | 12-almost primes (generalization of semiprimes) | 23.3 % |
| A069274 | 13-almost primes (generalization of semiprimes) | 24.6 % |
| A069977 | Numbers k such that k and k+2 are squarefree | 19.6 % |
| A070552 | Semiprimes k such that k+1 is also a semiprime | 31.9 % |
| A071139 | Numbers k such that the sum of distinct primes dividing k is divisible by the largest prime dividing k | 23.1 % |
| A072202 | Same numbers of prime factors of forms 4*k+1 and 4*k+3, counted with multiplicity | 16.0 % |
| A072437 | Numbers with no prime factors of form 4*k+3 | 15.7 % |
| A072587 | Numbers having at least one prime factor with an even exponent | 15.8 % |
| A072978 | Numbers of the form m * 2^bigomega(m), where m>1 is odd and bigomega(m) = A001222(m), the number of prime factors of m | 14.0 % |
| A073247 | Squarefree numbers k such that k-1 and k+1 are not squarefree | 20.4 % |
| A073492 | Numbers having at least one prime gap in their factorization | 10.5 % |
| A073493 | Numbers having exactly one prime gap in their factorization | 14.2 % |
| A074969 | Numbers with six distinct prime divisors | 22.8 % |
| A075592 | Numbers n such that number of distinct prime divisors of n is a divisor of n | 13.5 % |
| A078972 | Brilliant numbers: semiprimes (products of two primes, A001358) whose prime factors have the same number of decimal digits | 35.1 % |
| A084969 | Numbers whose smallest prime factor is 11 | 14.8 % |
| A084970 | Numbers whose smallest prime factor is 13 | 15.6 % |
| A085722 | Numbers k such that k^2 + 1 is a semiprime | 16.7 % |
| A085746 | Numbers n such that n^2 + n + 1 is a semiprime | 17.2 % |
| A086005 | Semiprimes sandwiched between semiprimes | 50.0 % |
| A089352 | Numbers that are divisible by the sum of their distinct prime factors (A008472) | 23.4 % |
| A092192 | Semiprimes that are the sum of two successive semiprimes | 34.9 % |
| A092207 | Semiprimes k such that k+2 is also a semiprime | 29.5 % |
| A100493 | a(n) = n + n-th semiprime | 19.8 % |
| A105441 | Numbers with at least two odd prime factors (not necessarily distinct) | 12.9 % |
| A105571 | Numbers m such that m - 2 and m + 2 are semiprimes | 27.5 % |
| A108181 | Semiprimes of the form 4n + 1 | 26.2 % |
| A108769 | Numbers m such that m^2 + (m+1)^2 is a semiprime | 15.8 % |
| A109373 | Semiprimes of the form semiprime + 1 | 30.3 % |
| A112771 | Semiprimes of the form 6n + 1 | 33.5 % |
| A112772 | Semiprimes of the form 6n+2 | 35.3 % |
| A112774 | Semiprimes of the form 6n+4 | 35.2 % |
| A112775 | Numbers k such that 6k+1 is semiprime | 14.2 % |
| A112776 | Numbers k such that 6k+5 is semiprime | 13.3 % |
| A112777 | Numbers k such that 2*k^2 + 1 is a semiprime | 17.9 % |
| A120944 | Composite squarefree numbers | 11.9 % |
| A121495 | Numbers k such that k and k+1 are composite and squarefree | 16.2 % |
| A122488 | Numbers k such that 1 + 2k + 3k^2 is semiprime | 17.7 % |
| A123017 | Semiprimes k such that k+3 is also a semiprime | 23.4 % |
| A124269 | 3-almost primes indexed by primes | 33.6 % |
| A124283 | 4-almost primes indexed by primes | 33.7 % |
| A124940 | Numbers k such that k and k+3 are 3-almost primes | 23.0 % |
| A124941 | Numbers k such that k and k+4 are 4-almost primes | 26.2 % |
| A130091 | Numbers having in their canonical prime factorization mutually distinct exponents | 17.0 % |
| A134333 | Numbers n whose number of prime factors (counted with multiplicity) is a prime factor of n | 17.3 % |
| A134334 | Numbers which are not divisible by the number of their prime factors (counted with multiplicity) | 10.8 % |
| A134344 | Composite numbers such that the arithmetic mean of their prime factors (counted with multiplicity) is prime | 30.3 % |
| A134376 | Numbers whose sum of prime factors (counted with multiplicity) is not prime | 10.6 % |
| A134616 | Numbers such that the sum of squares of their prime factors (taken with multiplicity) is a prime | 24.7 % |
| A134617 | Numbers such that the arithmetic mean of the squares of their prime factors (taken with multiplicity) is a prime | 29.3 % |
| A134618 | Numbers such that the sum of cubes of their prime factors (taken with multiplicity) is a prime | 28.3 % |
| A134619 | Numbers such that the arithmetic mean of the cubes of their prime factors (taken with multiplicity) is a prime | 39.5 % |
| A137487 | Numbers with 24 divisors | 20.9 % |
| A137491 | Numbers with 28 divisors | 25.5 % |
| A137493 | Numbers with 30 divisors | 36.0 % |
| A138511 | Semiprimes where the larger prime factor is greater than the square of the smaller prime factor, short: semiprimes p*q, p^2 < q | 18.2 % |
| A144255 | Semiprimes of the form k^2+1 | 100.0 % |
| A157352 | Products (semiprimes) of two distinct safe primes | 50.0 % |
| A157483 | Numbers k such that k-1 and k+1 are divisible by exactly 3 primes, counted with multiplicity | 22.1 % |
| A157931 | Numbers that are both the sum and the product of two primes | 21.8 % |
| A172443 | Numbers with exactly 64 divisors | 27.1 % |
| A175461 | Semiprimes of form 8n+5 | 30.9 % |
| A175463 | Numbers k such that 8*k + 5 is semiprime | 15.9 % |
| A175648 | Semiprimes m such that m+4 is also semiprime | 26.8 % |
| A175742 | Numbers with 32 divisors | 23.8 % |
| A175746 | Numbers with 36 divisors | 27.9 % |
| A175749 | Numbers with 40 divisors | 27.9 % |
| A175750 | Numbers with 42 divisors | 40.9 % |
| A175754 | Numbers with 48 divisors | 22.5 % |
| A180748 | Numbers k such that k^2 - k + 1 is semiprime | 16.7 % |
| A186525 | Semiprimes of the form 7k+1 | 32.4 % |
| A195086 | Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 2 | 18.7 % |
| A195087 | Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 3 | 19.4 % |
| A195088 | Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 4 | 19.9 % |
| A195089 | Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 5 | 20.5 % |
| A195090 | Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 6 | 20.7 % |
| A195091 | Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 7 | 21.2 % |
| A195092 | Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 8 | 21.8 % |
| A195093 | Numbers k such that (number of prime factors of k counted with multiplicity) less (number of distinct prime factors of k) = 9 | 22.8 % |
| A209061 | Exponentially squarefree numbers | 9.7 % |
| A212164 | Numbers k such that the maximum exponent in its prime factorization is greater than the number of positive exponents (A051903(k) > A001221(k)) | 13.9 % |
| A212165 | Numbers k such that the maximum exponent in its prime factorization is not less than the number of positive exponents (A051903(k) >= A001221(k)) | 14.8 % |
| A212166 | Numbers k such that the maximum exponent in its prime factorization equals the number of positive exponents (A051903(k) = A001221(k)) | 18.1 % |
| A212168 | Numbers n such that the maximal exponent in its prime factorization is less than the number of positive exponents (A051903(n) < A001221(n)) | 11.9 % |
| A212707 | Semiprimes of the form 5*n^2 + 1 | 100.0 % |
| A228184 | Numbers k such that k^2 + k + 41 is semiprime | 13.1 % |
| A242330 | Numbers k such that k^2 + 2 is a semiprime | 22.1 % |
| A242331 | Numbers k such that k^2 + 3 is a semiprime | 14.7 % |
| A242332 | Numbers k such that k^2 + 4 is a semiprime | 25.2 % |
| A242333 | Numbers k such that k^2 + 5 is a semiprime | 17.0 % |
| A243436 | Numbers n such that n^2-n-1 is semiprime | 14.2 % |
| A243937 | Even numbers n>=6 for which lpf(n-1) > lpf(n-3), where lpf = least prime factor | 15.1 % |
| A246281 | Numbers k for which A003961(k) < 2*k; Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then product_{k >= 1} (p_{k+1})^(c_k) < 2*n, where p_k indicates the k-th prime, A000040(k) | 9.2 % |
| A246282 | Numbers k for which A003961(k) > 2*k; numbers n such that if n = Product_{k >= 1} (p_k)^(c_k), then Product_{k >= 1} (p_{k+1})^(c_k) > 2*n, where p_k indicates the k-th prime, A000040(k) | 14.2 % |
| A251728 | Semiprimes p*q for which p <= q < p^2 | 26.4 % |
| A261034 | Numbers m such that 3*m is squarefree | 5.3 % |
| A274546 | Numbers m such that 5*m is squarefree | 11.1 % |
| A276378 | Numbers k such that 6*k is squarefree | 11.4 % |
| A276713 | Numbers n such that n and n+3 have the same number of divisors (A000005) | 20.3 % |
| A332797 | Numbers whose smallest prime factor is 23 | 17.3 % |
| A332798 | Numbers whose smallest prime factor is 19 | 16.8 % |
| A332799 | Numbers whose smallest prime factor is 17 | 16.3 % |
| A344872 | Semiprimes of the form 3m+2 | 27.5 % |
| A353004 | Numbers k such that 2*k^2 + 29 is semiprime | 13.2 % |
| A360739 | Semiprimes of the form k^2 + 2 | 100.0 % |
| A360740 | Semiprimes of the form k^2 + 3 | 100.0 % |
| A360741 | Semiprimes of the form k^2 + 4 | 100.0 % |
| A361696 | Semiprimes of the form k^2 + 5 | 100.0 % |