cp's OEIS Frontend

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,...

First differs from A374590 and A375432 at n = 25: A374590(25) = A375432(25) = 216 is not a term of this sequence.

Numbers k such that A382291(k) = 2, i.e., numbers whose number of infinitary divisors is twice the number of their unitary divisors.

Numbers whose prime factorization has a single exponent that is a sum of two distinct powers of 2 (A018900) and all the other exponents, if they exist, are powers of 2. Equivalently, numbers of the form p^e * m, where p is a prime, e is a term in A018900, and m is a term in A138302 that is coprime to p.

If k is a term then k^2 is also a term. If m is a term in A138302 that is coprime to k then k * m is also a term. The primitive terms, i.e., the terms that cannot be generated from smaller terms using these rules, are the numbers of the form p^(2^i+1), where p is prime and i >= 1.

Analogous to A060687, which is the sequence of numbers k with prime excess A046660(k) = 2.

The asymptotic density of this sequence is A271727 * Sum_{p prime} (((1 - 1/p)/f(1/p)) * Sum_{k>=1} 1/p^A018900(k)) = 0.11919967112489084407..., where f(x) = 1 - x^3 + Sum_{k>=2} (x^(2^k)-x^(2^k+1)).

Contains all odd multiples of 2^5: Each second term of A174312 is in this sequence.

The asymptotic density of this sequence is 1 - Product_{p prime} (1 - 1/p^5 + 1/p^6) = 0.01863624892... . - Amiram Eldar, May 05 2023

First differs from A286708 at n = 25.

Number of the form p^i * q^j, where p != q are primes and i,j > 1.

Numbers k such that A001221(k) = 2 and A051904(k) >= 2.

The possible values of the number of the divisors (A000005) of terms in this sequence is any composite number that is not 8 or twice a prime (A264828 \ {1, 8}).

675 = 3^3*5^2 and 676 = 2^2*13^2 are 2 consecutive integers in this sequence. There are no other such pairs below 10^22 (the lesser members of such pairs are terms of A060355).

Number of the form A036966(m)/p, m >= 2, where p is a prime divisor of A036966(m).