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A285931
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Number of primes q < p such that q^(p-1) == 1 (modulo p^2), where p = prime(n).
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0
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0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0
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OFFSET
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1,70
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COMMENTS
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Pairs of prime numbers (q, p) satisfying the conditions in the definition are sometimes called "Wieferich prime pairs" (cf. Mossinghoff, 2009).
a(n) > 0 iff p is a term of A222184.
First occurrence of k beginning at 0: 1, 5, 70, 1618, 2702, etc. - Robert G. Wilson v, May 10 2017
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LINKS
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EXAMPLE
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For n = 70: prime(70) = 349 and there are two primes q < 349 such that q^(349-1) == 1 (modulo 349^2), namely 223 and 317, so a(70) = 2.
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MATHEMATICA
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f[n_] := Block[{c = 0, p = Prime@ n, q = 2}, While[q < p, If[ PowerMod[q, p - 1, p^2] == 1, c++]; q = NextPrime@q]; c]; Array[f, 105] (* Robert G. Wilson v, May 10 2017 *)
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PROG
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(PARI) a(n) = my(p=prime(n), i=0); forprime(q=1, p-1, if(Mod(q, p^2)^(p-1)==1, i++)); i
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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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