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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 (list; graph; refs; listen; history; text; internal format)
 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 Antti Karttunen, Table of n, a(n) for n = 1..16384 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)); CROSSREFS Cf. A000027, A000203, A003415, A003557, A018804, A060640, A300251, A319684, A347130, A349124. Cf. also A348983, A349143. Sequence in context: A181762 A209684 A173988 * A158893 A342636 A332941 Adjacent sequences:  A349120 A349121 A349122 * A349124 A349125 A349126 KEYWORD nonn AUTHOR Antti Karttunen, Nov 08 2021 STATUS approved

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Last modified January 18 03:02 EST 2022. Contains 350410 sequences. (Running on oeis4.)