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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.
4
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
1,4
COMMENTS
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.
LINKS
FORMULA
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).
MATHEMATICA
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 *)
PROG
(PARI)
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));
KEYWORD
nonn
AUTHOR
Antti Karttunen, Nov 08 2021
STATUS
approved