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A212738
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a(n) = (7^p - 6^p - 1)/(1806p) where p is the n-th prime.
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1
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1, 43, 81271, 3570505, 7025726485, 314435374639, 639872336584027, 60775577624897675065, 2794429652350970000851, 276858360603194024261113585, 600808083611945729624598396925, 28083738921571587634894783049047, 61728002094732427074308383210511683
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OFFSET
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3,2
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COMMENTS
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7^p - 6^p - 1 is divisible by 1806p = 6*7*43*p where p prime > 3 (see the proof with the general case).
The sequence is generalizable with the form a(n) = ((k^p - (k-1)^p - 1)) /(k*(k-1)*p*q) where p = prime(n), k integer such that q = k*(k-1) + 1 prime (q = A002383(n) with k = A055494(n)).
k*(k-1)*p*q divides k^p - (k-1)^p - 1, proof :
(1) p divides k^p - (k-1)^p - 1 (Fermat’s theorem)
(2) k*(k-1) divides k^p - (k-1)^p - 1
(3) q = k*(k-1) + 1 divides k^p - (k-1)^p - 1. Suppose k^p - (k-1)^p - 1 ==r (mod q). Then ((k-1)^p)*k^p - ((k-1)^p)*(k-1)^p - (k-1)^p ==r*(k-1)^p (mod q). But the first term is congruent to -1 (mod q), the second term is congruent to k^p (mod q) and the last term is congruent to (k-1)^p (mod q). We obtain r (mod q) = r*(k-1)^p (mod q) => r = 0.
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LINKS
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MAPLE
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with(numtheory): for n from 3 to 25 do:p:=ithprime(n):x:=(7^p - 6^p - 1)/(1806*p): printf(`%d, `, x):od:
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PROG
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(PARI) a(n)={my(p=prime(n)); (7^p - 6^p - 1)/(1806*p)} \\ Andrew Howroyd, Feb 25 2018
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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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