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A275741
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Sum of Wilson and Lerch remainders of n-th prime.
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1
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1, 3, 10, 6, 6, 17, 15, 11, 25, 38, 9, 37, 47, 39, 86, 58, 107, 50, 101, 36, 98, 45, 123, 92, 170, 57, 80, 72, 158, 194, 194, 67, 78, 133, 120, 302, 144, 158, 128, 97, 91, 303, 76, 191, 139, 178, 302, 117, 242, 179, 335, 390, 362, 197, 290, 314, 327, 227, 429
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OFFSET
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2,2
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COMMENTS
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a(n) = 0 if and only if prime(n) is in both A007540 and A197632, i.e., prime(n) is simultaneously a Wilson prime and a Lerch prime.
For n > 2, a(n) = 0 if and only if A027641(3*p-3) / A027642(3*p-3)-1 + 1/p == 0 (mod p^2), where p = prime(n) (cf. Dobson, 2016, theorem 2).
René Gy (see links) has shown that a number is simultaneously a Lerch prime and a Wilson prime if and only if it satisfies the congruence (p - 1)! + 1 == 0 (mod p^3). - John Blythe Dobson, Feb 23 2018
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LINKS
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FORMULA
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MATHEMATICA
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a[n_] := Module[{p = Prime[n]}, Mod[((p-1)!+1)/p, p] + Mod[(Sum[(k^(p-1)-1)/p, {k, 1, p-1}] - ((p-1)!+1)/p)/p, p]];
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PROG
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(PARI) a002068(n) = my(p=prime(n)); ((p-1)!+1)/p % p
a197631(n) = my(p=prime(n), m=p-1); sum(k=1, m, k^m, -p-m!)/p^2 % p
a(n) = a002068(n) + a197631(n)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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