A321141
a(n) = Sum_{d|n} sigma_n(d).
Original entry on oeis.org
1, 6, 29, 291, 3127, 48246, 823545, 16909060, 387459858, 10019533302, 285311670613, 8920489178073, 302875106592255, 11113363271736486, 437893951444713443, 18447307036548136965, 827240261886336764179, 39346708467688595378892, 1978419655660313589123981
Offset: 1
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[&+[DivisorSigma(n, d):d in Divisors(n)]:n in [1..20]]; // Vincenzo Librandi, Feb 16 2020
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with(numtheory): seq(coeff(series(add(sigma[n](k)*x^k/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 20); # Muniru A Asiru, Oct 28 2018
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Table[Sum[DivisorSigma[n, d], {d, Divisors[n]}] , {n, 19}]
Table[SeriesCoefficient[Sum[DivisorSigma[n, k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 19}]
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a(n) = sumdiv(n, d, sigma(d, n)); \\ Michel Marcus, Oct 28 2018
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from sympy import divisor_sigma, divisors
def A321141(n):
return sum(divisor_sigma(d,0)*(n//d)**n for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020
A321294
a(n) = Sum_{d|n} mu(n/d)*d*sigma_n(d).
Original entry on oeis.org
1, 9, 83, 1058, 15629, 282381, 5764807, 134480900, 3486902505, 100048836321, 3138428376731, 107006403495850, 3937376385699301, 155572843119518781, 6568408661060858767, 295150157013526773768, 14063084452067724991025, 708236697425777157039381
Offset: 1
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Table[Sum[MoebiusMu[n/d] d DivisorSigma[n, d], {d, Divisors[n]}], {n, 18}]
Table[Sum[EulerPhi[n/d] d^(n + 1), {d, Divisors[n]}], {n, 18}]
Table[Sum[GCD[n, k]^(n + 1), {k, n}], {n, 18}]
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a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, n)); \\ Michel Marcus, Nov 03 2018
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from sympy import totient, divisors
def A321294(n):
return sum(totient(d)*(n//d)**(n+1) for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020
A322104
Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n} d*sigma_k(d).
Original entry on oeis.org
1, 1, 5, 1, 7, 7, 1, 11, 13, 17, 1, 19, 31, 35, 11, 1, 35, 85, 95, 31, 35, 1, 67, 247, 311, 131, 91, 15, 1, 131, 733, 1127, 631, 341, 57, 49, 1, 259, 2191, 4295, 3131, 1615, 351, 155, 34, 1, 515, 6565, 16775, 15631, 8645, 2409, 775, 130, 55, 1, 1027, 19687, 66311, 78131, 49111, 16815, 4991, 850, 217, 23
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, ...
5, 7, 11, 19, 35, 67, ...
7, 13, 31, 85, 247, 733, ...
17, 35, 95, 311, 1127, 4295, ...
11, 31, 131, 631, 3131, 15631, ...
35, 91, 341, 1615, 8645, 49111, ...
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Table[Function[k, Sum[d DivisorSigma[k, d], {d, Divisors[n]}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
Table[Function[k, SeriesCoefficient[Sum[j DivisorSigma[k, j] x^j/(1 - x^j), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 11}, {n, 1, i}] // Flatten
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T(n,k)={sumdiv(n, d, d^(k+1)*sigma(n/d))}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018
Showing 1-3 of 3 results.