A347130 -id:A347130 - cp's OEIS frontend

A347965 Dirichlet convolution of A003415 (arithmetic derivative) with A003961 (prime shift towards larger primes).

0, 1, 1, 7, 1, 13, 1, 33, 11, 17, 1, 75, 1, 23, 20, 131, 1, 104, 1, 103, 26, 29, 1, 329, 17, 35, 82, 145, 1, 196, 1, 473, 32, 41, 30, 552, 1, 47, 38, 461, 1, 274, 1, 187, 181, 57, 1, 1259, 25, 194, 44, 229, 1, 682, 36, 659, 50, 65, 1, 1052, 1, 73, 247, 1611, 42, 352, 1, 271, 60, 366, 1, 2332, 1, 83, 245, 313, 42

Offset: 1

Antti Karttunen, Sep 25 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A003415, A003961.

Cf. also A347130, A347964.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A347965(n) = sumdiv(n,d,A003415(n/d)*A003961(d));

a(n) = Sum_{d|n} A003415(d) * A003961(n/d).

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).

A348279 a(n) = Sum_{d|n} d*d', where d' is the arithmetic derivative of d (A003415).

Original entry on oeis.org

0, 2, 3, 18, 5, 35, 7, 114, 57, 77, 11, 243, 13, 135, 128, 626, 17, 467, 19, 573, 220, 299, 23, 1395, 255, 405, 786, 1047, 29, 1160, 31, 3186, 476, 665, 432, 2835, 37, 819, 640, 3389, 41, 2100, 43, 2427, 1937, 1175, 47, 7283, 693, 2577, 1040, 3333, 53, 5570, 896, 6295, 1276

Offset: 1

Wesley Ivan Hurt, Oct 09 2021

nonn

			a(4) = 18; a(4) = 1*1' + 2*2' + 4*4' = 1*0 + 2*1 + 4*4 = 18.

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

Cf. A003415 (arithmetic derivative).
Inverse Möbius transform of A190116.
Cf. also A347130.

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = sumdiv(n, d, d*ad(d)); \\ Michel Marcus, Oct 10 2021

a(p) = p for primes p since we have a(p) = 1*1' + p*p' = 1*0 + p*1 = p.

a(n) = Sum_{d|n} A190116(d). - Antti Karttunen, Dec 07 2021

cp's OEIS Frontend

A347965 Dirichlet convolution of A003415 (arithmetic derivative) with A003961 (prime shift towards larger primes).

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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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A348279 a(n) = Sum_{d|n} d*d', where d' is the arithmetic derivative of d (A003415).

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