cp's OEIS Frontend

Numbers with exactly three factorizations: A001055(a(n)) = 3 (e.g., a(4) = 1*343 = 7*49 = 7*7*7). - Reinhard Zumkeller, Dec 29 2001

Intersection of A014612 and A000578. Intersection of A014612 and A030513. - Wesley Ivan Hurt, Sep 10 2013

Let r(n) = (a(n)-1)/(a(n)+1) if a(n) mod 4 = 1, (a(n)+1)/(a(n)-1) otherwise; then Product_{n>=1} r(n) = (9/7) * (28/26) * (124/126) * (344/342) * (1332/1330) * ... = 48/35. - Dimitris Valianatos, Mar 06 2020

There exist 5 groups of order p^3, when p prime, so this is a subsequence of A054397. Three of them are abelian: C_p^3, C_p^2 X C_p and C_p X C_p X C_p = (C_p)^3. For 8 = 2^3, the 2 nonabelian groups are D_8 and Q_8; for odd prime p, the 2 nonabelian groups are (C_p x C_p) : C_p, and C_p^2 : C_p (remark, for p = 2, these two semi-direct products are isomorphic to D_8). Here C, D, Q mean Cyclic, Dihedral, Quaternion groups of the stated order; the symbols X and : mean direct and semidirect products respectively. - Bernard Schott, Dec 11 2021

This sequence is a permutation of A355038.

This sequence is also a permutation of the exponentially odd numbers (A268335) multiplied by the square of their squarefree kernel (A007947).

a(n)/rad(a(n)) is a permutation of the squares.

a(n)/rad(a(n))^2 is a permutation of the exponentially odd numbers.

Numbers of the form p^11 A079395, p*q^5 A178740, p*q*r^2 A085987, or p^2*q^3 A143610, where p, q and r are distinct primes. - R. J. Mathar, Mar 01 2010, Mar 17 2010

If n = p - 1 where p is prime, then row n lists the numbers with p divisors.

The partial sums of column k give the column k of A319076.

Number of elements in the set {(x, y): x|n, y|n, x < y, gcd(x, y) > 1}.

Every element of the sequence is repeated indefinitely, for instance:

a(n)=0 if n prime;

a(n)=1 if n = p^2 for p prime (A001248);

a(n)=2 if n is a squarefree semiprime (A006881);

a(n)=3 if n = p^3 for p prime (A030078);

a(n)=6 if n = p^4 for p prime (A030514);

a(n)=8 if n is a number which is the product of a prime and the square of a different prime (A054753);

a(n)=10 if n = p^5 for p prime (A050997);

a(n)=15 if n is in the set {A007304} union {64} = {30, 42, 64, 66, 70,...} = {Sphenic numbers} union {64};

a(n)=18 if n is the product of the cube of a prime (A030078) and a different prime (see A065036);

a(n)=21 if n = p^7 for p prime (A092759);

a(n)=24 if n is square of a squarefree semiprime (A085986);

a(n)=32 if n is the product of the 4th power of a prime (A030514) and a different prime (see A178739);

a(n)=36 if n = p^9 for p prime (A179665);

a(n)=44 if n is the product of exactly four primes, three of which are distinct (A085987);

a(n)=45 if n is a number with 11 divisors (A030629);

a(n)=49 if n is of the form p^2*q^3, where p,q are distinct primes (A143610);

a(n)=50 if n is the product of the 5th power of a prime (A050997) and a different prime (see A178740);

a(n)=55 if n if n = p^11 for p prime(A079395);

a(n)=72 if n is a number with 14 divisors (A030632);

a(n)=80 if n is the product of four distinct primes (A046386);

a(n)=83 if n is a number with 15 divisors (A030633);

a(n)=89 if n is a number with prime factorization pqr^3 (A189975);

a(n)=96 if n is a number that are the cube of a product of two distinct primes (A162142);

a(n)=98 if n is the product of the 7th power of a prime and a distinct prime (p^7*q) (A179664);

a(n)=116 if n is the product of exactly 2 distinct squares of primes and a different prime (p^2*q^2*r) (A179643);

a(n)=126 if n is the product of the 5th power of a prime and different distinct prime of the 2nd power (p^5*q^2) (A179646);

a(n)=128 if n is the product of the 8th power of a prime and a distinct prime (p^8*q) (A179668);

a(n)=150 if n is the product of the 4th power of a prime and 2 different distinct primes (p^4*q*r) (A179644);

a(n)=159 if n is the product of the 4th power of a prime and a distinct prime of power 3 (p^4*q^3) (A179666).

It is possible to continue with a(n) = 162, 178, 209, 224, 227, 238, 239, 260, 289, 309, 320, 333,...

Subsequence of A301517 and A374459 and first differs from them at n = 21. A301517(21) = A374459(21) = 216 is not a term of this sequence.

Numbers having exactly one non-unitary prime factor and its multiplicity is odd.

Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m+1} with m >= 1, i.e., any number (including zero) of 1's and then a single odd number > 1.

The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/((p-1)*(p+1)^2) = 0.093382464285953613312...

Subsequence of A102834 and first differs from it at n = 14: A102834(14) = 432 = 2^4 * 3^3 is not a term of this sequence.

Powerful numbers k such that A051903(k) is odd.

Equivalently, numbers whose prime factorization exponents are all larger than 1 and their maximum is odd. The maximum exponent in the prime factorization of 1 is considered to be A051903(1) = 0, and therefore 1 is not a term of this sequence.

The numbers of terms that do not exceed the 10^k-powerful number (A376092(k)), for k = 1, 2, ..., are 3, 40, 416, 4255, 42829, 429393, 4299797, 43022803, ... . Apparently, the asymptotic density of this sequence within the powerful numbers (A001694) exists and approximately equals 0.43.

Subsequence of A059269 and first differs from it at n = 36: A059269(136) = 44 has 15 = 3 * 5 divisors and thus is not a term of this sequence.

Numbers k such that A000005(k) is in A065119.

Numbers k such that A071188(k) = 3.

Equals the complement of A354181, without the terms of A036537 (i.e., complement(A354181) \ A036537).

The asymptotic density of this sequence is Product_{p prime} (1-1/p) * (Sum_{k>=1} 1/p^(A003586(k)-1)) - A327839 = 0.26087647470200496716... .

First differs from A344653 and A345193 at n = 17: a(17) = 120 is not a term of these sequences.

Numbers whose prime signature (A118914) is of the form {1, 1, ..., m} with m >= 3, i.e., any number (including zero) of 1's and then a single number >= 3.

The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/(p*(p^2-1)) = A369632 / A013661 = 0.13463358553764438661... .