A347130 -id:A347130 - cp's OEIS frontend

A347129 a(n) = A347130(n) / A003557(n), where A347130 is the Dirichlet convolution of the identity function with the arithmetic derivative of n.

0, 1, 1, 3, 1, 10, 1, 6, 3, 14, 1, 24, 1, 18, 16, 10, 1, 21, 1, 36, 20, 26, 1, 44, 3, 30, 6, 48, 1, 124, 1, 15, 28, 38, 24, 45, 1, 42, 32, 68, 1, 164, 1, 72, 39, 50, 1, 70, 3, 27, 40, 84, 1, 36, 32, 92, 44, 62, 1, 276, 1, 66, 51, 21, 36, 244, 1, 108, 52, 236, 1, 78, 1, 78, 33, 120, 36, 284, 1, 110, 10, 86, 1, 372, 44

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A003415, A003557, A347126 [= a(A276086(n))], A347130.

Cf. also A342001, A347127, A347128.

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));
A347129(n) = (A347130(n) / A003557(n));

a(n) = A347130(n) / A003557(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)).

A348495 a(n) = gcd(A018804(n), A347130(n)), where A347130 is the Dirichlet convolution of the identity function with the arithmetic derivative of n (A003415), and A018804 is Pillai's arithmetical function.

Original entry on oeis.org

1, 1, 1, 2, 1, 5, 1, 4, 3, 1, 1, 8, 1, 3, 1, 16, 1, 63, 1, 72, 5, 1, 1, 4, 5, 15, 27, 8, 1, 1, 1, 16, 7, 1, 3, 6, 1, 3, 1, 4, 1, 1, 1, 24, 9, 5, 1, 80, 7, 15, 5, 8, 1, 81, 1, 4, 1, 1, 1, 24, 1, 3, 3, 32, 9, 1, 1, 24, 1, 1, 1, 12, 1, 3, 5, 8, 3, 1, 1, 16, 27, 1, 1, 8, 11, 15, 1, 140, 1, 9, 5, 72, 1, 1, 3, 16, 1

Offset: 1

Antti Karttunen, Oct 21 2021

nonn

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

Cf. A003415, A003557, A018804, A347130, A348496.

Cf. also A348492, A348497.

Table[GCD[Total@ GCD[n, Range[n]], DivisorSum[n, #*(If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &[n/#]) &]], {n, 97}] (* 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
A347130(n) = sumdiv(n,d,d*A003415(n/d));
A348495(n) = gcd(A018804(n), A347130(n));

a(n) = gcd(A018804(n), A347130(n)).

a(n) = A003557(n) * A348496(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)).

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).

A347131 a(n) = Sum_{d|n} phi(n/d) * A003415(d), where A003415 is the arithmetic derivative and phi is Euler totient function.

Original entry on oeis.org

0, 1, 1, 5, 1, 8, 1, 18, 8, 12, 1, 33, 1, 16, 14, 56, 1, 45, 1, 53, 18, 24, 1, 110, 14, 28, 45, 73, 1, 87, 1, 160, 26, 36, 22, 169, 1, 40, 30, 182, 1, 119, 1, 113, 93, 48, 1, 328, 20, 107, 38, 133, 1, 216, 30, 254, 42, 60, 1, 337, 1, 64, 125, 432, 34, 183, 1, 173, 50, 183, 1, 538, 1, 76, 135, 193, 34, 215, 1, 552, 216

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of A000010 with A003415.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Möbius transform of A347130.
Cf. A000010, A003415.
Cf. also A300251, A305809, A322577, A347132, A347133, A347134, A347235.

f[p_, e_] := e/p; d[1] = 0; d[n_] := n * Plus @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, d[#] * EulerPhi[n/#] &]; Array[a, 100] (* Amiram Eldar, Sep 03 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A347131(n) = sumdiv(n,d,A003415(n/d)*eulerphi(d));

A347131(n) = sum(k=1,n,A003415(gcd(n,k))); \\ (Slow) - Antti Karttunen, Sep 02 2021

a(n) = Sum_{d|n} A000010(n/d) * A003415(d).
a(n) = Sum_{d|n} A008683(n/d) * A347130(d).
a(n) = Sum_{k=1..n} A003415(gcd(n,k)). - Antti Karttunen, Sep 02 2021

A348980 a(n) = Sum_{d|n} d * A322582(n/d), where A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

Original entry on oeis.org

0, 1, 1, 5, 1, 9, 1, 17, 8, 13, 1, 37, 1, 17, 15, 49, 1, 51, 1, 57, 19, 25, 1, 117, 14, 29, 43, 77, 1, 105, 1, 129, 27, 37, 23, 191, 1, 41, 31, 185, 1, 141, 1, 117, 99, 49, 1, 325, 20, 117, 39, 137, 1, 237, 31, 253, 43, 61, 1, 405, 1, 65, 131, 321, 35, 213, 1, 177, 51, 209, 1, 579, 1, 77, 145, 197, 35, 249, 1, 521

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A322582 with the identity function, A000027.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000027, A003958, A038040, A322582, A348981 (Möbius transform), A348982, A348983, A349130.

Cf. also A347130, A349140.

f[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, #*(n/# - s[n/#]) &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
A322582(n) = (n-A003958(n));
A348980(n) = sumdiv(n,d,d*A322582(n/d));

a(n) = Sum_{d|n} d * A322582(n/d).
For all n >= 1, a(n) <= A347130(n) <= A349140(n).
a(n) = A038040(n) - A349130(n). - Antti Karttunen, Nov 14 2021

A349140 a(n) = Sum_{d|n} d * A348507(n/d), where A348507(n) = A003959(n) - n, where A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 7, 1, 11, 1, 33, 10, 15, 1, 61, 1, 19, 17, 131, 1, 77, 1, 89, 21, 27, 1, 263, 16, 31, 67, 117, 1, 145, 1, 473, 29, 39, 25, 379, 1, 43, 33, 395, 1, 189, 1, 173, 137, 51, 1, 997, 22, 155, 41, 201, 1, 443, 33, 527, 45, 63, 1, 743, 1, 67, 177, 1611, 37, 277, 1, 257, 53, 265, 1, 1541, 1, 79, 187, 285, 37, 321

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of A348507 with the identity function, A000027.

Dirichlet convolution of sigma with A348971.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000027, A000203, A003959, A038040, A348507, A348971, A349141 (Möbius transform), A349142, A349143, A349170.

Cf. also A347130, A348980.

f[p_, e_] := (p + 1)^e; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; a[n_] := DivisorSum[n, #*s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349140(n) = sumdiv(n,d,d*A348507(n/d));

a(n) = Sum_{d|n} d * A348507(n/d).
a(n) = Sum_{d|n} A000203(d) * A348971(n/d).
a(n) = Sum_{d|n} A349141(d).
For all n >= 1, a(n) >= A347130(n) >= A348980(n).
a(n) = A349170(n) - A038040(n). - Antti Karttunen, Nov 15 2021

A347126 a(n) = A347129(A276086(n)).

Original entry on oeis.org

0, 1, 1, 10, 3, 21, 1, 14, 16, 124, 39, 246, 3, 27, 33, 222, 72, 423, 6, 44, 56, 344, 114, 636, 10, 65, 85, 490, 165, 885, 1, 18, 20, 164, 51, 330, 24, 236, 284, 1976, 636, 3804, 57, 438, 552, 3468, 1143, 6462, 104, 696, 904, 5296, 1776, 9624, 165, 1010, 1340, 7460, 2535, 13290, 3, 33, 39, 282, 90, 549, 51, 414, 516

Offset: 0

Antti Karttunen, Aug 25 2021

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

Cf. A003415, A003557, A276086, A347129, A347130.

Cf. also A342002, A346471, A347232.

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]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A347130(n) = sumdiv(n,d,d*A003415(n/d));
A347129(n) = (A347130(n) / A003557(n));
A347126(n) = A347129(A276086(n));

a(n) = A347129(A276086(n)).

A359425 Dirichlet convolution of the arithmetic derivative with the primorial base exp-function.

Original entry on oeis.org

0, 2, 2, 11, 2, 19, 2, 45, 18, 35, 2, 85, 2, 31, 40, 151, 2, 125, 2, 195, 36, 119, 2, 313, 38, 83, 120, 215, 2, 418, 2, 649, 124, 491, 52, 628, 2, 295, 88, 1057, 2, 1046, 2, 1629, 414, 2303, 2, 1777, 38, 1541, 496, 2241, 2, 4424, 140, 6421, 300, 11315, 2, 2048, 2, 83, 1002, 2013, 104, 1864, 2, 2073

Offset: 1

Antti Karttunen, Jan 02 2023

Antti Karttunen, Table of n, a(n) for n = 1..2309
Index entries for sequences related to primorial base

Cf. A003415, A276086, A359425, A347130, A383286.

Cf. also A347389, A347959.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A359425(n) = sumdiv(n,d,A003415(n/d)*A276086(d));

a(n) = Sum_{d|n} A003415(n/d) * A276086(d).

a(n) = Sum_{d|n} A347130(n/d) * A383286(d). - Antti Karttunen, May 14 2025

cp's OEIS Frontend

A347129 a(n) = A347130(n) / A003557(n), where A347130 is the Dirichlet convolution of the identity function with the arithmetic derivative of n.

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

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A348495 a(n) = gcd(A018804(n), A347130(n)), where A347130 is the Dirichlet convolution of the identity function with the arithmetic derivative of n (A003415), and A018804 is Pillai's arithmetical function.

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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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Formula

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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A347131 a(n) = Sum_{d|n} phi(n/d) * A003415(d), where A003415 is the arithmetic derivative and phi is Euler totient function.

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A348980 a(n) = Sum_{d|n} d * A322582(n/d), where A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

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A349140 a(n) = Sum_{d|n} d * A348507(n/d), where A348507(n) = A003959(n) - n, where A003959 is fully multiplicative with a(p) = (p+1).

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