A344480
a(n) = Sum_{d|n} d * sigma_d(d), where sigma_k(n) is the sum of the k-th powers of the divisors of n.
Original entry on oeis.org
1, 11, 85, 1103, 15631, 284795, 5764809, 134745175, 3486961642, 100097682141, 3138428376733, 107019534806039, 3937376385699303, 155577590686826319, 6568408813691811835, 295152408847835466855, 14063084452067724991027, 708238048886862707907062, 37589973457545958193355621, 2097154000001929438984022793
Offset: 1
a(6) = Sum_{d|6} d * sigma_d(d) = 1*(1^1) + 2*(1^2 + 2^2) + 3*(1^3 + 3^3) + 6*(1^6 + 2^6 + 3^6 + 6^6) = 284795.
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Table[Sum[k*DivisorSigma[k, k] (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 30}]
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a(n) = sumdiv(n, d, d*sigma(d, d)); \\ Michel Marcus, May 21 2021
A344787
a(n) = n * Sum_{d|n} sigma_d(d) / d, where sigma_k(n) is the sum of the k-th powers of the divisors of n.
Original entry on oeis.org
1, 7, 31, 287, 3131, 47527, 823551, 16843583, 387440266, 10009772937, 285311670623, 8918294639219, 302875106592267, 11112685050294387, 437893920912795941, 18447025553014982271, 827240261886336764195, 39346558271492953948522, 1978419655660313589123999
Offset: 1
a(4) = 4 * Sum_{d|4} sigma_d(d) / d = 4 * ((1^1)/1 + (1^2 + 2^2)/2 + (1^4 + 2^4 + 4^4)/4) = 287.
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Table[n*Sum[DivisorSigma[k, k] (1 - Ceiling[n/k] + Floor[n/k])/k, {k, n}], {n, 20}]
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, k)*x^k/(1-x^k)^2)) \\ Seiichi Manyama, Dec 16 2022
A356076
a(n) = Sum_{k=1..n} sigma_k(k) * floor(n/k).
Original entry on oeis.org
1, 7, 36, 315, 3442, 50926, 874471, 17717759, 405157961, 10414927743, 295726598356, 9214021189459, 312089127781714, 11424774177252514, 449318695090042077, 18896344248088180470, 846136606134424944649, 40192694877626991357901
Offset: 1
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Table[Sum[DivisorSigma[k,k]Floor[n/k],{k,n}],{n,20}] (* Harvey P. Dale, Sep 08 2024 *)
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a(n) = sum(k=1, n, sigma(k, k)*(n\k));
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a(n) = sum(k=1, n, sumdiv(k, d, sigma(d, d)));
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, sigma(k, k)*x^k/(1-x^k))/(1-x))
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from sympy import divisor_sigma
def A356079(n): return n+sum(divisor_sigma(k,k)*(n//k) for k in range(2,n+1)) # Chai Wah Wu, Jul 25 2022
Showing 1-3 of 3 results.
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