A295878 -id:A295878 - cp's OEIS frontend

A363825 The number of infinitary divisors of n that are exponentially odd numbers (A268335).

1, 2, 2, 1, 2, 4, 2, 3, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 6, 1, 4, 3, 2, 2, 8, 2, 3, 4, 4, 4, 1, 2, 4, 4, 6, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 6, 4, 6, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 3, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4

Offset: 1

Amiram Eldar, Oct 19 2023

First differs from A295878 at n = 32.

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

Cf. A000005, A000120, A005117, A037445, A077609, A268335, A295878, A358260.

f[p_, e_] := 1 + If[OddQ[e], 2^DigitCount[e-1, 2, 1], 0]; 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]%2, 2^hammingweight(f[i, 2]-1) + 1, 1))};

Multiplicative with a(p^e) = 1 if e is even, and a(p^e) = 2^A000120(e-1) + 1 if e is odd.

a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).

A295879 Multiplicative with a(p) = 1, a(p^e) = prime(e-1) if e > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 2, 3, 2, 1, 13

Offset: 1

Antti Karttunen, Nov 29 2017

This sequence can be used as a filter. It matches at least to the following sequences related to the counting of various non-unitary prime divisors:
For all i, j:
  a(i) = a(j) => A056170(i) = A056170(j), as A056170(n) = A001222(a(n)).
  a(i) = a(j) => A162641(i) = A162641(j).
  a(i) = a(j) => A295659(i) = A295659(j).
  a(i) = a(j) => A295662(i) = A295662(j).
  a(i) = a(j) => A295883(i) = A295883(j), as A295883(n) = A007949(a(n)).
  a(i) = a(j) => A295884(i) = A295884(j).
An encoding of the prime signature of A057521(n), the powerful part of n. - Peter Munn, Apr 06 2024

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A003557, A008578, A057521, A064989, A181819.
Cf. also A293515, A294875, A294895, A294897, A295878, A351944.
Differs from A000688 for the first time at n=128, where a(128) = 13, while A000688(128) = 15.

Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> Which[p == 1, 1, e == 1, 1, True, Prime[e - 1]]] &, 128] (* Michael De Vlieger, Nov 29 2017 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 1, prime(f[i,2]-1)));} \\ Amiram Eldar, Nov 18 2022

a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product A008578(e(i)).
a(n) = A064989(A181819(n)).
a(n) = A181819(A003557(n)). - Antti Karttunen, Apr 03 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^2  + Sum_{k>=1} (prime(k+1)-prime(k))/p^(k+2)) = 2.208... . - Amiram Eldar, Nov 18 2022

cp's OEIS Frontend

A363825 The number of infinitary divisors of n that are exponentially odd numbers (A268335).

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