A318470 -id:A318470 - cp's OEIS frontend

A318307 Multiplicative with a(p^e) = 2^A002487(e).

1, 2, 2, 2, 2, 4, 2, 4, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 8, 2, 4, 4, 4, 2, 8, 2, 8, 4, 4, 4, 4, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 4, 8, 2, 8, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4, 8, 2, 8, 4, 4, 4, 4, 4, 16, 2, 4, 4, 4, 2, 8, 2, 8, 8

Offset: 1

Antti Karttunen, Aug 29 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A318306, A318316, A318470, A318667.

Differs from A037445 for the first time at n=32, where a(32) = 8, while A037445(32) = 4.

f[m_] := Module[{a = 1, b = 0, n = m}, While[n > 0, If[OddQ[n], b += a, a += b]; n = Floor[n/2]]; b]; Array[Times @@ Map[2^f@ # &, FactorInteger[#][[All, -1]] ] - Boole[# == 1] &, 105] (* after Jean-François Alcover at A002487 *)

A002487(n) = { my(a=1, b=0); while(n>0, if(bitand(n, 1), b+=a, a+=b); n>>=1); (b); }; \\ From A002487
A318307(n) = factorback(apply(e -> 2^A002487(e),factor(n)[,2]));

from functools import reduce
from sympy import factorint
def A318307(n): return 1<Chai Wah Wu, May 18 2023

a(n) = 2^A318306(n).
a(n) = A061142(A318470(n)).
a(n^2) = a(n).
a(A003557(n^2)) = A318316(n).
Dirichlet convolution square of A318667(n)/A317934(n).

A318469 Multiplicative with a(p^e) = A019565(A003714(e)).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 10, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 7, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 20, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 14, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 20, 10, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 14, 2, 6, 6, 9, 2, 8, 2, 10, 8

Offset: 1

Antti Karttunen, Aug 30 2018

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

Cf. A003714, A019565, A072649, A318464, A318465.

Cf. also A293442, A293443, A300834, A318470.

A003714(n) = { my(s=0,w); while(n>2, w = A072649(n); s += 2^(w-1); n -= fibonacci(w+1)); (s+n); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A318469(n) = factorback(apply(e -> A019565(A003714(e)),factor(n)[,2]));

For all n >= 1, A001222(a(n)) = A318464(n).

cp's OEIS Frontend

A318307 Multiplicative with a(p^e) = 2^A002487(e).

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