A366988 -id:A366988 - cp's OEIS frontend

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).

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.

A366990 The sum of divisors of n that are terms of A056166.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 13, 10, 1, 1, 5, 1, 1, 1, 13, 1, 10, 1, 5, 1, 1, 1, 13, 26, 1, 37, 5, 1, 1, 1, 45, 1, 1, 1, 50, 1, 1, 1, 13, 1, 1, 1, 5, 10, 1, 1, 13, 50, 26, 1, 5, 1, 37, 1, 13, 1, 1, 1, 5, 1, 1, 10, 45, 1, 1, 1, 5, 1, 1, 1, 130, 1, 1, 26, 5, 1, 1, 1, 13

Offset: 1

Amiram Eldar, Oct 31 2023

The number of these divisors is A095691(n) and the largest of them is A366993(n).

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

Cf. A005117, A056166, A095691, A366988, A366993.

f[p_, e_] := 1 + 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 + sum(j = 1, f[i, 2], if(isprime(j), f[i, 1]^j)));}

Multiplicative with a(p^e) = 1 + Sum_{primes q <= e} p^q.

a(n) >= 1, with equality if and only if n is squarefree (A005117).

A366993 The largest divisor of n that is a term of A056166.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 8, 9, 1, 1, 4, 1, 1, 1, 8, 1, 9, 1, 4, 1, 1, 1, 8, 25, 1, 27, 4, 1, 1, 1, 32, 1, 1, 1, 36, 1, 1, 1, 8, 1, 1, 1, 4, 9, 1, 1, 8, 49, 25, 1, 4, 1, 27, 1, 8, 1, 1, 1, 4, 1, 1, 9, 32, 1, 1, 1, 4, 1, 1, 1, 72, 1, 1, 25, 4, 1, 1, 1, 8, 27, 1, 1, 4

Offset: 1

Amiram Eldar, Oct 31 2023

The number of these divisors is A095691(n) and their sum is A366990(n).

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

Cf. A005117, A007917, A056166, A095691, A366988, A366990.

f[p_, e_] := p^If[e == 1, 0, 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~, f[i, 1]^precprime(f[i, 2]));}

Multiplicative with a(p) = 1 and a(p^e) = p^A007917(e).
a(n) <= n, with equality if and only if n is in A056166.
a(n) >= 1, with equality if and only if n is squarefree (A005117).

A383104 Inverse Möbius transform of A382883.

Original entry on oeis.org

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

Offset: 1

Peter Luschny, Apr 16 2025

nonn

Cf. A382883, A366988.

# Seen as a special case of a transformation:
A382883Transform := (b, len) -> local n, d; seq(add(A382883(n/d)*b(d), d in numtheory:-divisors(n)), n = 1..len): A382883Transform(n -> 1, 99);

V[n_, e_] := If[e == 1, 1, IntegerExponent[n, e]]; f[n_] := f[n] = -DivisorSum[n, V[n, #] * f[#] &, # < n &]; f[1] = 1; a[n_] := DivisorSum[n, f[#] &]; Array[a, 100] (* Amiram Eldar, Apr 29 2025 *)

def a(n): return sum(A382883(n/d) for d in divisors(n))
print([a(n) for n in range(1, 90)])
# More general:
def A382883Transform(n: int, b: Callable[[int], int]) -> int:
    return sum(A382883(n/d)*b(d) for d in divisors(n))
def a(n) -> int: return A382883Transform(n, lambda x: 1)

a(n) = Sum_{d|n} A382883(d).

A384812 If n = Product prime(i)^e(i) then a(n) = Sum prime(e(i)).

Original entry on oeis.org

0, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 5, 2, 4, 4, 7, 2, 5, 2, 5, 4, 4, 2, 7, 3, 4, 5, 5, 2, 6, 2, 11, 4, 4, 4, 6, 2, 4, 4, 7, 2, 6, 2, 5, 5, 4, 2, 9, 3, 5, 4, 5, 2, 7, 4, 7, 4, 4, 2, 7, 2, 4, 5, 13, 4, 6, 2, 5, 4, 6, 2, 8, 2, 4, 5, 5, 4, 6, 2, 9, 7, 4, 2, 7, 4, 4, 4, 7, 2, 7

Offset: 1

Ilya Gutkovskiy, Jun 10 2025

nonn

Cf. A001222, A001414, A008472, A066328, A181819, A366988.

f:= proc(n) local t;
  add(ithprime(t[2]),t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Jul 25 2025

Table[Plus @@ (Prime[#[[2]]] & /@ FactorInteger[n]), {n, 1, 90}]

a(n) = my(f=factor(n)); sum(k=1, #f~, prime(f[k,2])); \\ Michel Marcus, Jun 10 2025

cp's OEIS Frontend

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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A366990 The sum of divisors of n that are terms of A056166.

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A366993 The largest divisor of n that is a term of A056166.

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A383104 Inverse Möbius transform of A382883.

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A384812 If n = Product prime(i)^e(i) then a(n) = Sum prime(e(i)).

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