A366994 -id:A366994 - cp's OEIS frontend

A366992 The sum of divisors of n that are not terms of A322448.

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 15, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 47, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 60, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 47, 84, 144

Offset: 1

Amiram Eldar, Oct 31 2023

First differs from A365682 at n = 64.
The sum of divisors of n whose prime factorization has exponents that are all either 1 or primes.
The number of these divisors is A366991(n) and the largest of them is A366994(n).

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

Cf. A000203, A046100, A365682, A366991, A366994.

f[p_, e_] := 1 + p + Total[p^Select[Range[e], PrimeQ]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

Multiplicative with a(p^e) = 1 + p + Sum_{primes q <= e} p^q.
a(n) <= A000203(n), with equality if and only if n is a biquadratefree number (A046100).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} f(1/p) = 0.77864544487983775708..., where f(x) = (1-x) * (1 + Sum_{k>=1} (1 + 1/x + Sum_{primes q <= k} 1/x^q) * x^(2*k)).

A366989 The number of prime powers p^q dividing n, where p is prime and q is either 1 or prime (A334393 without the first term 1).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 3, 3, 1, 3, 1, 4, 2, 2, 2, 4, 1, 2, 2, 4, 1, 3, 1, 3, 3, 2, 1, 4, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 4, 1, 2, 3, 4, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 3, 3, 2, 3, 1, 4, 3, 2, 1, 4, 2, 2, 2

Offset: 1

Amiram Eldar, Oct 31 2023

First differs from A122810 at n = 48, and from A318322 at n = 64.

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

Cf. A000720, A001221, A005117, A077761, A334393, A366988, A366991, A366992, A366994.

Cf. A122810, A318322.

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

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

Additive with a(p^e) = A000720(e) + 1.
a(n) = 1 is and only if n is squarefree (A005117) > 1.
a(n) = A366988(n) + A001221(n).
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{p prime} P(p) = 0.67167522222173297323..., and 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).

cp's OEIS Frontend

A366992 The sum of divisors of n that are not terms of A322448.

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A366989 The number of prime powers p^q dividing n, where p is prime and q is either 1 or prime (A334393 without the first term 1).

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

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