A348502 -id:A348502 - cp's OEIS frontend

A348498 a(n) = gcd(A003415(n), A347130(n)) / A003557(n), where A003415 is the arithmetic derivative and A347130 is its Dirichlet convolution with the identity function.

0, 1, 1, 1, 1, 5, 1, 3, 1, 7, 1, 8, 1, 9, 8, 2, 1, 7, 1, 12, 10, 13, 1, 11, 1, 15, 3, 16, 1, 31, 1, 5, 14, 19, 12, 5, 1, 21, 16, 17, 1, 41, 1, 24, 13, 25, 1, 14, 1, 9, 20, 28, 1, 9, 16, 23, 22, 31, 1, 46, 1, 33, 17, 3, 18, 61, 1, 36, 26, 59, 1, 13, 1, 39, 11, 40, 18, 71, 1, 22, 2, 43, 1, 62, 22, 45, 32, 35, 1, 41

Offset: 1

Antti Karttunen, Oct 23 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A003415, A003557, A342001, A347129, A347130, A348497, A348502 [= a(A276086(n))].

Cf. also A348494, A348496.

f[n_] := If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Table[GCD[f[n], DivisorSum[n, # f[n/#] &]]*Apply[Times, FactorInteger[n][[All, 1]]]/n, {n, 90}] (* Michael De Vlieger, Oct 25 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003557(n) = (n/factorback(factorint(n)[, 1]));
A347130(n) = sumdiv(n,d,d*A003415(n/d));
A348498(n) = (gcd(A003415(n), A347130(n))/A003557(n));

a(n) = A348497(n) / A003557(n).

a(n) = gcd(A342001(n), A347129(n)).

A348501 a(n) = A348496(A276086(n)), where A348496(n) = A348495(n) / A003557(n).

Original entry on oeis.org

1, 1, 1, 5, 1, 21, 1, 1, 1, 1, 3, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 5, 35, 3, 3, 1, 3, 5, 1, 1, 3, 3, 1, 1, 13, 3, 3, 1, 3, 1, 3, 1, 3, 13, 3, 1, 1, 1, 3, 3, 1, 5, 5, 39, 3, 1, 3, 1, 3, 1, 3, 3, 9, 3, 3, 9, 9, 1, 3, 1, 3, 1, 3, 1, 3, 1, 57, 1, 3, 57, 9, 15, 15, 3, 9, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3

Offset: 0

Antti Karttunen, Oct 21 2021

Antti Karttunen, Table of n, a(n) for n = 0..11550
Index entries for sequences related to primorial base

Cf. A003557, A276086, A348495, A348496.

Cf. also A348500, A348502.

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12]}, Map[Function[n, GCD[Total@ GCD[n, Range[n]], DivisorSum[n, #*(If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &[n/#]) &]]/Apply[Times, Map[#1^(#2 - 1) & @@ # &, FactorInteger[n]]]], Table[Function[k, Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]@ IntegerDigits[n, b], {n, 0, 101}] ] ] (* Michael De Vlieger, Oct 21 2021 *)

\\ Needs also code from A348496:
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A348501(n) = A348496(A276086(n));

a(n) = A348496(A276086(n)).

cp's OEIS Frontend

A348498 a(n) = gcd(A003415(n), A347130(n)) / A003557(n), where A003415 is the arithmetic derivative and A347130 is its Dirichlet convolution with the identity function.

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