A366989 -id:A366989 - cp's OEIS frontend

A366988 The number of prime powers of prime numbers (A053810) that divide n.

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 1, 0, 0, 0, 2, 2, 0, 0, 1, 0, 0, 0

Offset: 1

Amiram Eldar, Oct 31 2023

a(n) depends only on the prime signature of n.

Every nonnegative number appears in the sequence of record values. k >= 1 first occurs at n = 2^prime(k) (A034785).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720, A005117, A034785, A053810, A095691, A366989, A366990, A366993.

f[p_, e_] := PrimePi[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); sum(i = 1, #f~, primepi(f[i, 2]));}

Additive with a(p^e) = A000720(e).
a(n) = 0 if and only if n is squarefree (A005117).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} P(p) = 0.67167522222173297323..., where P(s) is the prime zeta function.

A366991 The number of divisors of n that are not terms of A322448.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 5, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 12, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4

Offset: 1

Amiram Eldar, Oct 31 2023

First differs from A365680 at n = 64.
The number of divisors of n whose prime factorization has exponents that are all either 1 or primes.
The sum of these divisors is A366992(n) and the largest of them is A366994(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000720, A046100, A365680, A366989, A366992, A366994.

f[p_, e_] := PrimePi[e] + 2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, primepi(f[i, 2]) + 2);}

Multiplicative with a(p^e) = A000720(e) + 2.

a(n) <= A000005(n), with equality if and only if n is a biquadratefree number (A046100).

A366994 The largest divisor of n that is not a term of A322448.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 8, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 24, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 32, 65, 66, 67, 68

Offset: 1

Amiram Eldar, Oct 31 2023

First differs from A365683 at n = 64.
The largest divisor of n whose prime factorization has exponents that are all either 1 or primes.
The number of these divisors is A366991(n) and their sum is A366992(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007917, A322448, A365683, A366989, A366991, A366992.

f[p_, e_] := p^If[e == 1, 1, NextPrime[e+1, -1]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, f[i, 1], f[i, 1]^precprime(f[i, 2])));}

Multiplicative with a(p) = p and a(p^e) = p^A007917(e) for e >= 2.
a(n) <= n, with equality if and only if n is not in A322448.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} f(1/p) = 0.48535795387619596052..., where f(x) = (1 - x) * (1 + Sum_{k>=1} x^(2*k-s(k))), s(k) = A007917(k) for k >= 2, and s(1) = 1.

cp's OEIS Frontend

A366988 The number of prime powers of prime numbers (A053810) that divide n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A366991 The number of divisors of n that are not terms of A322448.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A366994 The largest divisor of n that is not a term of A322448.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula