A348498 -id:A348498 - cp's OEIS frontend

A348502 a(n) = A348498(A276086(n)).

0, 1, 1, 5, 1, 7, 1, 7, 8, 31, 13, 41, 1, 9, 11, 37, 8, 47, 3, 11, 14, 43, 19, 53, 2, 13, 17, 49, 11, 59, 1, 9, 10, 41, 17, 55, 12, 59, 71, 247, 106, 317, 19, 73, 92, 289, 127, 359, 26, 87, 113, 331, 148, 401, 33, 101, 134, 373, 169, 443, 1, 11, 13, 47, 10, 61, 17, 69, 86, 277, 121, 347, 12, 83, 107, 319, 71, 389, 31

Offset: 0

Antti Karttunen, Oct 23 2021

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

Cf. A276086, A348498, A348500, A348501.

Cf. also A342002 for a similar scatter plot.

b = MixedRadix[Reverse@ Prime@ Range@ 12]; f[n_] := If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Map[Function[n, GCD[f[n], DivisorSum[n, # f[n/#] &]]*Apply[Times, FactorInteger[n][[All, 1]]]/n], Array[Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ #, Reverse@ #} &@ IntegerDigits[#, b] &, 79, 0]] (* Michael De Vlieger, Oct 25 2021 *)

\\ Needs also code from A348498:
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A348502(n) = A348498(A276086(n));

A348496 a(n) = gcd(A018804(n), A347130(n)) / A003557(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 4, 1, 3, 1, 2, 1, 21, 1, 36, 5, 1, 1, 1, 1, 15, 3, 4, 1, 1, 1, 1, 7, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 12, 3, 5, 1, 10, 1, 3, 5, 4, 1, 9, 1, 1, 1, 1, 1, 12, 1, 3, 1, 1, 9, 1, 1, 12, 1, 1, 1, 1, 1, 3, 1, 4, 3, 1, 1, 2, 1, 1, 1, 4, 11, 15, 1, 35, 1, 3, 5, 36, 1, 1, 3, 1, 1, 3, 3, 1, 1

Offset: 1

Antti Karttunen, Oct 21 2021

nonn

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

Cf. A003557, A018804, A347128, A347129, A347130, A348495, A348501 [= a(A276086(n))].

Cf. also A348494, A348498.

Table[GCD[Total@ GCD[n, Range[n]], DivisorSum[n, #*(If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &[n/#]) &]]/Apply[Times, Map[#1^(#2 - 1) & @@ # &, FactorInteger[n]]], {n, 101}] (* Michael De Vlieger, Oct 21 2021 *)

\\ Needs also code from A348495:
A003557(n) = (n/factorback(factorint(n)[, 1]));
A348496(n) = (A348495(n)/A003557(n));

a(n) = A348495(n) / A003557(n).

a(n) = gcd(A347128(n), A347129(n)).

A348497 a(n) = gcd(A003415(n), A347130(n)), where A003415 is the arithmetic derivative and A347130 is its Dirichlet convolution with the identity function.

Original entry on oeis.org

0, 1, 1, 2, 1, 5, 1, 12, 3, 7, 1, 16, 1, 9, 8, 16, 1, 21, 1, 24, 10, 13, 1, 44, 5, 15, 27, 32, 1, 31, 1, 80, 14, 19, 12, 30, 1, 21, 16, 68, 1, 41, 1, 48, 39, 25, 1, 112, 7, 45, 20, 56, 1, 81, 16, 92, 22, 31, 1, 92, 1, 33, 51, 96, 18, 61, 1, 72, 26, 59, 1, 156, 1, 39, 55, 80, 18, 71, 1, 176, 54, 43, 1, 124, 22, 45

Offset: 1

Antti Karttunen, Oct 23 2021

nonn

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

Cf. A003415, A003557, A347130, A348498.

Cf. also A348492, A348495.

f[n_] := If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Table[GCD[f[n], DivisorSum[n, # f[n/#] &]], {n, 86}] (* 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]));
A347130(n) = sumdiv(n,d,d*A003415(n/d));
A348497(n) = gcd(A003415(n), A347130(n));

a(n) = gcd(A003415(n), A347130(n)).

a(n) = A003557(n) * A348498(n).

cp's OEIS Frontend

A348502 a(n) = A348498(A276086(n)).

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A348496 a(n) = gcd(A018804(n), A347130(n)) / A003557(n).

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A348497 a(n) = gcd(A003415(n), A347130(n)), where A003415 is the arithmetic derivative and A347130 is its Dirichlet convolution with the identity function.

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