A348494 -id:A348494 - cp's OEIS frontend

A348500 a(n) = A348494(A276086(n)).

1, 1, 1, 5, 1, 7, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 3, 5, 1, 1, 1, 3, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 13, 3, 1, 1, 1, 1, 3, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19, 1, 1, 19, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1

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, A348492, A348494.

Cf. also A348501.

Block[{b = MixedRadix[Reverse@ Prime@ Range@ 12]}, Map[GCD[Total@GCD[#1, Range[#1]], #1 Total[#2/#1 & @@@ #2]]/Apply[Times, Map[#1^(#2 - 1) & @@ # &, #2]] & @@ {#, FactorInteger[#]} &, Table[Function[k, Times @@ Power @@@ # &@ Transpose@ {Prime@ Range@ Length@ k, Reverse@ k}]@ IntegerDigits[n, b], {n, 0, 105}] ] ] (* Michael De Vlieger, Oct 21 2021 *)

\\ Needs also code from A348494:
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A348500(n) = A348494(A276086(n));

a(n) = A348494(A276086(n)), where A348494(n) = A348492(n) / A003557(n).

A348492 Greatest common divisor of the arithmetic derivative (A003415) and Pillai's arithmetical function (A018804).

Original entry on oeis.org

1, 1, 1, 4, 1, 5, 1, 4, 3, 1, 1, 8, 1, 3, 1, 16, 1, 21, 1, 24, 5, 1, 1, 4, 5, 15, 27, 8, 1, 1, 1, 16, 7, 1, 3, 12, 1, 3, 1, 4, 1, 1, 1, 24, 3, 5, 1, 16, 7, 15, 5, 8, 1, 81, 1, 4, 1, 1, 1, 4, 1, 3, 3, 64, 9, 1, 1, 24, 1, 1, 1, 12, 1, 3, 5, 8, 3, 1, 1, 16, 27, 1, 1, 4, 11, 15, 1, 140, 1, 3, 5, 24, 1, 1, 3, 16, 1, 7

Offset: 1

Antti Karttunen, Oct 21 2021

nonn

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

Cf. A003415, A003557, A018804, A348493, A348494.

Cf. also A342413, A342458, A348495.

Array[GCD[Total@ GCD[#, Range[#]], # Total[#2/#1 & @@@ FactorInteger[#]]] &, 98] (* Michael De Vlieger, Oct 21 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
A348492(n) = gcd(A003415(n), A018804(n));

a(n) = gcd(A003415(n), A018804(n)).
For n > 1, a(n) = A003415(n) / A348493(n).
a(n) = A003557(n) * A348494(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)).

A347128 a(n) = A018804(n) / A003557(n), where A018804 is Pillai's arithmetical function.

Original entry on oeis.org

1, 3, 5, 4, 9, 15, 13, 5, 7, 27, 21, 20, 25, 39, 45, 6, 33, 21, 37, 36, 65, 63, 45, 25, 13, 75, 9, 52, 57, 135, 61, 7, 105, 99, 117, 28, 73, 111, 125, 45, 81, 195, 85, 84, 63, 135, 93, 30, 19, 39, 165, 100, 105, 27, 189, 65, 185, 171, 117, 180, 121, 183, 91, 8, 225, 315, 133, 132, 225, 351, 141, 35, 145, 219, 65, 148

Offset: 1

Antti Karttunen, Aug 23 2021

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

Cf. A003557, A018804, A348494 [= gcd(a(n), A342001(n))], A348496 [= gcd(a(n), A347129(n))].

Cf. also A173557, A347127.

f[p_, e_] := (e*(p - 1)/p + 1)*p^e; A347128[n_] := (Times @@ (f @@@ FactorInteger[n]))/(n/Times @@ (First[Transpose[FactorInteger[n]]]));Table[A347128[n], {n, 1, 76}] (* Robert P. P. McKone, Aug 23 2021, after Amiram Eldar *)

A347128(n) = { my(f=factor(n)); prod(i=1, #f~, ((f[i, 1]-1)*f[i, 2] + f[i, 1])); };

Multiplicative with a(p^e) = ((p-1)*e + p).

a(n) = A018804(n) / A003557(n).

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.

Original entry on oeis.org

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)).

A348493 a(n) = A003415(n) / gcd(A003415(n), A018804(n)), where A003415 is the arithmetic derivative and A018804 is Pillai's arithmetical function.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 3, 2, 7, 1, 2, 1, 3, 8, 2, 1, 1, 1, 1, 2, 13, 1, 11, 2, 1, 1, 4, 1, 31, 1, 5, 2, 19, 4, 5, 1, 7, 16, 17, 1, 41, 1, 2, 13, 5, 1, 7, 2, 3, 4, 7, 1, 1, 16, 23, 22, 31, 1, 23, 1, 11, 17, 3, 2, 61, 1, 3, 26, 59, 1, 13, 1, 13, 11, 10, 6, 71, 1, 11, 4, 43, 1, 31, 2, 3, 32, 1, 1, 41, 4, 4, 34, 49, 8

Offset: 1

Antti Karttunen, Oct 21 2021

nonn

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

Cf. A003415, A018804, A348492, A348494.

Array[#2/GCD[#1, #2] & @@ {Total@ GCD[#, Range[#]], If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]]} &, 95] (* Michael De Vlieger, Oct 21 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A018804(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ From A018804
A348493(n) = { my(u=A003415(n)); (u/gcd(u,A018804(n))); };

a(n) = A003415(n) / A348492(n).

cp's OEIS Frontend

A348500 a(n) = A348494(A276086(n)).

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A348492 Greatest common divisor of the arithmetic derivative (A003415) and Pillai's arithmetical function (A018804).

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

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Formula

A347128 a(n) = A018804(n) / A003557(n), where A018804 is Pillai's arithmetical function.

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Mathematica

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Formula

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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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A348493 a(n) = A003415(n) / gcd(A003415(n), A018804(n)), where A003415 is the arithmetic derivative and A018804 is Pillai's arithmetical function.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula