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A091888
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Irregularity index of prime(n): number of numbers k, 1 <= k <= (p-3)/2, such that p = prime(n) divides the numerator of the Bernoulli number B(2k).
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3
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 2, 0, 0, 3, 0, 0, 0, 0, 1, 1, 2, 1, 0, 0, 0, 1
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OFFSET
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2,36
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COMMENTS
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Note offset is 2: only odd primes are considered.
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LINKS
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FORMULA
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0 if p is a regular prime; > 0 if p is an irregular prime.
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MATHEMATICA
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irregPrimeIndex[n_] := Block[{p = Prime[n], cnt = 0, k = 1}, While[ 2k + 2 < p, If[ Mod[ Numerator[ BernoulliB[ 2k]], p] == 0, cnt++]; k++]; cnt]; Array[ irregPrimeIndex, 105, 2] (* Robert G. Wilson v, Sep 20 2012 *)
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PROG
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(PARI) a(n)=sum(i=1, (prime(n)-1)/2, if(numerator(bernfrac(2*i))%prime(n), 0, 1)) \\ corrected by Amiram Eldar, May 10 2022
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CROSSREFS
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Cf. A073277 (primes having irregularity index 2), A060975 (primes having irregularity index 3), A061576 (least prime having irregularity index n), A091887 (irregularity index of irregular prime A000928(n)).
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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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