A347131 -id:A347131 - cp's OEIS frontend

A347130 a(n) = Sum_{d|n} d * A003415(n/d), where A003415 is the arithmetic derivative.

0, 1, 1, 6, 1, 10, 1, 24, 9, 14, 1, 48, 1, 18, 16, 80, 1, 63, 1, 72, 20, 26, 1, 176, 15, 30, 54, 96, 1, 124, 1, 240, 28, 38, 24, 270, 1, 42, 32, 272, 1, 164, 1, 144, 117, 50, 1, 560, 21, 135, 40, 168, 1, 324, 32, 368, 44, 62, 1, 552, 1, 66, 153, 672, 36, 244, 1, 216, 52, 236, 1, 936, 1, 78, 165, 240, 36, 284, 1, 880

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of the identity function (A000027) with the arithmetic derivative of n (A003415).

Dirichlet convolution of Euler phi (A000010) with A319684.

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

Inverse Möbius transform of A347131.
Cf. A000010, A000027, A003415, A003557, A319684, A347129.
Cf. also A300251, A305809, A319684, A347132, A347133.

Table[DivisorSum[n, #*(If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &[n/#]) &], {n, 80}] (* 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]));
A347130(n) = sumdiv(n,d,d*A003415(n/d));

a(n) = Sum_{d|n} d * A003415(n/d).
a(n) = Sum_{d|n} A000010(n/d) * A319684(d).
a(n) = Sum_{d|n} A347131(d).
a(n) = A003557(n) * A347129(n).

A347235 Dirichlet convolution of Euler phi with A342001, where A342001(n) = A003415(n) / A003557(n).

Original entry on oeis.org

0, 1, 1, 3, 1, 8, 1, 7, 4, 12, 1, 21, 1, 16, 14, 15, 1, 27, 1, 33, 18, 24, 1, 47, 6, 28, 13, 45, 1, 87, 1, 31, 26, 36, 22, 69, 1, 40, 30, 75, 1, 119, 1, 69, 51, 48, 1, 99, 8, 63, 38, 81, 1, 84, 30, 103, 42, 60, 1, 219, 1, 64, 67, 63, 34, 183, 1, 105, 50, 183, 1, 153, 1, 76, 75, 117, 34, 215, 1, 159, 40, 84, 1, 303, 42

Offset: 1

Antti Karttunen, Aug 26 2021

nonn

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

Cf. A000010, A003415, A003557, A342001.

Cf. also A346485, A347131, A347233, A347234, A347395.

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]));
A342001(n) = (A003415(n) / A003557(n));
A347235(n) = sumdiv(n,d,eulerphi(d)*A342001(n/d));

A347235(n) = sum(k=1,n,A342001(gcd(n,k))); \\ (Slow) - Antti Karttunen, Sep 02 2021

a(n) = Sum_{d|n} A000010(n/d) * A342001(d).

a(n) = Sum_{k=1..n} A342001(gcd(n,k)). - Antti Karttunen, Sep 02 2021

A347132 a(n) = Sum_{d|n} A001615(n/d) * A003415(d), where A003415 is the arithmetic derivative and A001615 is Dedekind psi function.

Original entry on oeis.org

0, 1, 1, 7, 1, 12, 1, 30, 10, 16, 1, 65, 1, 20, 18, 104, 1, 83, 1, 93, 22, 28, 1, 254, 16, 32, 63, 121, 1, 167, 1, 320, 30, 40, 26, 391, 1, 44, 34, 374, 1, 215, 1, 177, 143, 52, 1, 840, 22, 165, 42, 205, 1, 450, 34, 494, 46, 64, 1, 827, 1, 68, 183, 912, 38, 311, 1, 261, 54, 295, 1, 1430, 1, 80, 197, 289, 38, 359

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of Dedekind psi function (A001615) with the arithmetic derivative (A003415).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001615, A003415.

Cf. also A305809, A322577, A347131, A347133, A347135.

Table[DivisorSum[n, DirichletConvolve[j, MoebiusMu[j]^2, j, n/#]*If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &], {n, 78}] (* Michael De Vlieger, Oct 19 2021, after Jan Mangaldan at A001615 *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A347132(n) = sumdiv(n,d,A001615(n/d)*A003415(d));

a(n) = Sum_{d|n} A001615(n/d) * A003415(d).

A348981 a(n) = Sum_{d|n} phi(n/d) * A322582(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, 4, 1, 7, 1, 12, 7, 11, 1, 24, 1, 15, 13, 32, 1, 35, 1, 40, 17, 23, 1, 68, 13, 27, 35, 56, 1, 71, 1, 80, 25, 35, 21, 112, 1, 39, 29, 116, 1, 99, 1, 88, 77, 47, 1, 176, 19, 91, 37, 104, 1, 151, 29, 164, 41, 59, 1, 232, 1, 63, 105, 192, 33, 155, 1, 136, 49, 159, 1, 308, 1, 75, 117, 152, 33, 183, 1, 304, 151

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of Euler phi (A000010) with A322582.

Möbius transform of A348980.

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

Cf. A000010, A003958, A008683, A018804, A322582, A348980 (Inverse Möbius transform), A348981, A348982, A348983, A349131.

Cf. also A347131, A349141.

f[p_, e_] := (p - 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (# - s[#]) * EulerPhi[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));
A348981(n) = sumdiv(n,d,A322582(n/d)*eulerphi(d));

a(n) = Sum_{d|n} A000010(n/d) * A322582(d).
a(n) = Sum_{d|n} A008683(n/d) * A348980(d).
a(n) = Sum_{k=1..n} A322582(gcd(n,k)).
For all n >= 1, a(n) <= A347131(n) <= A349141(n).
a(n) = A018804(n) - A349131(n). - Antti Karttunen, Nov 14 2021

A349141 a(n) = Sum_{d|n} phi(n/d) * A348507(d), where A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 6, 1, 9, 1, 26, 9, 13, 1, 44, 1, 17, 15, 98, 1, 57, 1, 68, 19, 25, 1, 176, 15, 29, 57, 92, 1, 105, 1, 342, 27, 37, 23, 252, 1, 41, 31, 280, 1, 141, 1, 140, 111, 49, 1, 636, 21, 125, 39, 164, 1, 309, 31, 384, 43, 61, 1, 480, 1, 65, 147, 1138, 35, 213, 1, 212, 51, 209, 1, 960, 1, 77, 155, 236, 35, 249, 1, 1028

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

Dirichlet convolution of Euler phi (A000010) with A348507.

Möbius transform of A349140.

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

Cf. A000010, A003959, A008683, A018804, A348507, A349140 (inverse Möbius transform), A349142, A349143, A349171.

Cf. also A347131, A348981.

f[p_, e_] := (p + 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, (s[#] - #) * EulerPhi[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);
A349141(n) = sumdiv(n,d,eulerphi(d)*A348507(n/d));

a(n) = Sum_{d|n} A000010(n/d) * A348507(d).
a(n) = Sum_{d|n} A008683(n/d) * A349140(d).
a(n) = Sum_{k=1..n} A348507(gcd(n,k)).
For all n >= 1, a(n) >= A347131(n) >= A348981(n).
a(n) = A349171(n) - A018804(n). - Antti Karttunen, Nov 14 2021

A347133 a(n) = Sum_{d|n} A003415(n/d) * A069359(d).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 6, 1, 2, 0, 16, 0, 2, 2, 24, 0, 19, 0, 20, 2, 2, 0, 72, 1, 2, 9, 24, 0, 40, 0, 80, 2, 2, 2, 111, 0, 2, 2, 96, 0, 48, 0, 32, 25, 2, 0, 256, 1, 29, 2, 36, 0, 117, 2, 120, 2, 2, 0, 244, 0, 2, 29, 240, 2, 64, 0, 44, 2, 56, 0, 446, 0, 2, 31, 48, 2, 72, 0, 352, 54, 2, 0, 308, 2, 2, 2, 168, 0, 304, 2, 56

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of A003415 (arithmetic derivative) with A069359.

Dirichlet convolution of A001221 (omega, number of distinct prime factors of n) with A347131.

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

Cf. A001221, A003415, A069359, A347131.

Cf. also A300251, A305809, A347132, A347134, A347135.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A347133(n) = sumdiv(n,d,A003415(n/d)*A069359(d));

a(n) = Sum_{d|n} A003415(n/d) * A069359(d).

a(n) = Sum_{d|n} A001221(n/d) * A347131(d).

A347134 a(n) = Sum_{d|n} phi(n/d) * A069359(d), where phi is Euler totient function.

Original entry on oeis.org

0, 1, 1, 3, 1, 8, 1, 8, 5, 12, 1, 23, 1, 16, 14, 20, 1, 36, 1, 35, 18, 24, 1, 60, 9, 28, 21, 47, 1, 87, 1, 48, 26, 36, 22, 103, 1, 40, 30, 92, 1, 119, 1, 71, 66, 48, 1, 148, 13, 92, 38, 83, 1, 144, 30, 124, 42, 60, 1, 247, 1, 64, 86, 112, 34, 183, 1, 107, 50, 183, 1, 268, 1, 76, 110, 119, 34, 215, 1, 228, 81, 84

Offset: 1

Antti Karttunen, Aug 23 2021

nonn

Dirichlet convolution of A000010 (Euler totient function phi) with A069359.

Dirichlet convolution of A001221 (omega) with A029935 (the convolution square of Euler phi).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000010, A001221, A069359, A029935.

Cf. also A347131, A347133, A347135.

Table[DivisorSum[n,EulerPhi[n/#]*#*DivisorSum[#,1/#&,PrimeQ]&],{n,82}] (* Giorgos Kalogeropoulos, Oct 28 2021 *)

A069359(n) = (n*sumdiv(n, d, isprime(d)/d)); \\ From A069359
A347134(n) = sumdiv(n,d,eulerphi(n/d)*A069359(d));

a(n) = Sum_{d|n} A000010(n/d) * A069359(d)
a(n) = Sum_{d|n} A001221(n/d) * A029935(d).
a(n) = Sum_{k=1..n} A069359(gcd(n,k)). - Antti Karttunen, Oct 17 2021

A349123 a(n) = Sum_{d|n} A038040(n/d) * A003415(d), where A038040(n) = n*tau(n), and A003415 is the arithmetic derivative of n.

Original entry on oeis.org

0, 1, 1, 8, 1, 15, 1, 40, 12, 21, 1, 96, 1, 27, 24, 160, 1, 126, 1, 144, 30, 39, 1, 440, 20, 45, 90, 192, 1, 279, 1, 560, 42, 57, 36, 720, 1, 63, 48, 680, 1, 369, 1, 288, 234, 75, 1, 1680, 28, 270, 60, 336, 1, 810, 48, 920, 66, 93, 1, 1656, 1, 99, 306, 1792, 54, 549, 1, 432, 78, 531, 1, 3120, 1, 117, 330, 480, 54, 639

Offset: 1

Antti Karttunen, Nov 08 2021

nonn

This sequence is the Dirichlet convolution of at least the following pairs of sequences:
  - A003415 (the arithmetic derivative) with A038040,
  - A000027 (the identity function) with A347130,
  - A000203 (sigma) with A347131,
  - A018804 with A319684,
  - A060640 with A300251.

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

Cf. A000027, A000203, A003415, A003557, A018804, A060640, A300251, A319684, A347130, A349124.

Cf. also A348983, A349143.

d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, d[#]*(n/#)*DivisorSigma[0, n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A038040(n) = (n*numdiv(n));
A349123(n) = sumdiv(n,d,A038040(d)*A003415(n/d));

a(n) = Sum_{d|n} A038040(d) * A003415(n/d).
a(n) = Sum_{d|n} d * A347130(n/d).
a(n) = Sum_{d|n} A000203(d) * A347131(n/d).
a(n) = Sum_{d|n} A018804(d) * A319684(n/d).
a(n) = Sum_{d|n} A060640(d) * A300251(n/d).
For all n >= 1, A348983(n) <= a(n) <= A349143(n).
a(n) = A003557(n) * A349124(n).

A348027 Dirichlet convolution of Euler phi with A324198.

Original entry on oeis.org

1, 2, 5, 4, 5, 8, 7, 8, 15, 14, 11, 16, 13, 14, 37, 16, 17, 24, 19, 28, 35, 22, 23, 32, 49, 26, 45, 28, 29, 60, 31, 32, 55, 34, 41, 48, 37, 38, 65, 56, 41, 62, 43, 44, 111, 46, 47, 64, 55, 114, 85, 52, 53, 72, 59, 62, 95, 58, 59, 120, 61, 62, 123, 64, 65, 88, 67, 68, 115, 134, 71, 96, 73, 74, 293, 76, 83, 104, 79

Offset: 1

Antti Karttunen, Sep 25 2021

Antti Karttunen, Table of n, a(n) for n = 1..11550
Index entries for sequences related to primorial base

Cf. A000010, A324198.

Cf. also A346242, A347131, A347235, A347958.

s[n_] := Module[{k = n, m = 1, p = 2}, While[k > 0, m *= (p^Min[Mod[k, p], IntegerExponent[n, p]]); k = Quotient[k, p]; p = NextPrime[p]]; m]; a[n_] := DivisorSum[n, s[#] * EulerPhi[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 27 2021 *)

A324198(n) = { my(m=1, p=2, orgn=n); while(n, m *= (p^min(n%p, valuation(orgn, p))); n = n\p; p = nextprime(1+p)); (m); };
A348027(n) = sumdiv(n,d,eulerphi(d)*A324198(n/d));

a(n) = Sum_{d|n} phi(n/d) * A324198(d).

a(n) = Sum_{k=1..n} A324198(gcd(n,k)).

cp's OEIS Frontend

A347130 a(n) = Sum_{d|n} d * A003415(n/d), where A003415 is the arithmetic derivative.

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A347235 Dirichlet convolution of Euler phi with A342001, where A342001(n) = A003415(n) / A003557(n).

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A347132 a(n) = Sum_{d|n} A001615(n/d) * A003415(d), where A003415 is the arithmetic derivative and A001615 is Dedekind psi function.

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A348981 a(n) = Sum_{d|n} phi(n/d) * A322582(d), where A322582(n) = n - A003958(n), and A003958 is fully multiplicative with a(p) = (p-1).

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A349141 a(n) = Sum_{d|n} phi(n/d) * A348507(d), where A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

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A347133 a(n) = Sum_{d|n} A003415(n/d) * A069359(d).

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A347134 a(n) = Sum_{d|n} phi(n/d) * A069359(d), where phi is Euler totient function.

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A349123 a(n) = Sum_{d|n} A038040(n/d) * A003415(d), where A038040(n) = n*tau(n), and A003415 is the arithmetic derivative of n.

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