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A121706
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a(n) = Sum_{k=1..n-1} k^n.
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9
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0, 1, 9, 98, 1300, 20515, 376761, 7907396, 186884496, 4914341925, 142364319625, 4505856912854, 154718778284148, 5729082486784839, 227584583172284625, 9654782997596059912, 435659030617933827136, 20836030169620907691465
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OFFSET
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1,3
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COMMENTS
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It appears that p^(3k-1) divides a(p^k) for all integer k > 1 and prime p > 2:
for prime p > 2, p^2 divides a(p), p^5 divides a(p^2) and p^8 divides a(p^3).
Additionally, p^3 divides a(3p) for prime p > 2.
For prime p > 3, p divides a(p+1) and p^3 divides a(2p+1);
for prime p > 5, p divides a(3p+1) and p^3 divides a(4p+1);
for prime p > 7, p divides a(5p+1) and p^3 divides a(6p+1):
It appears that p divides a((2k+1)p+1) for integer k >= 0 and prime p > 2k+3, and p^3 divides a(2kp+1) for integer k > 0 and prime p > 2k+2.
p divides a((p+1)/2) for primes in A002145: primes of the form 4n+3, n >= 1.
p^2 divides a((p+1)/2) for primes in A007522: primes of the form 8n+7, n >= 0.
n*(2*n+1) divides a(2*n+1) for n >= 1. - Franz Vrabec, Dec 20 2020
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LINKS
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FORMULA
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a(n) = zeta(-n) - zeta(-n, n).
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MAPLE
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(bernoulli(n+1, n)-bernoulli(n+1))/(n+1) ;
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MATHEMATICA
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Table[Sum[k^n, {k, 1, n-1}], {n, 1, 35}]
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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