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%I #35 Jun 25 2019 01:46:37
%S 781,1541,5461,13021,15751,25351,29539,38081,40501,79381,100651,
%T 121463,133141,195313,216457,315121,318551,319507,326929,341531,
%U 353827,375601,416641,432821,453331,464881,498451,555397,556421,753667,764941,863329,872101,886411
%N Overpseudoprimes to base 5.
%C If h_5(n) is the multiplicative order of 5 modulo n, r_5(n) is the number of cyclotomic cosets of 5 modulo n then, by the definition, n is an overpseudoprime of base 5 if h_5(n)*r_5(n)+1=n. These numbers are in A020231. In particular, if n is squarefree such that its prime factorization is n=p_1*...*p_k, then n is overpseudoprime to base 5 iff h_5(p_1)=...=h_5(p_k). E.g., since h_5(101)=h_5(251)=h_5(401)=25, the number 101*251*401=10165751 is in the sequence.
%H Amiram Eldar, <a href="/A141390/b141390.txt">Table of n, a(n) for n = 1..327</a> (calculated from the b-file at A020231)
%H V. Shevelev, <a href="http://arxiv.org/abs/0806.3412">Overpseudoprimes, Mersenne Numbers and Wieferich Primes</a>, arXiv:0806.3412 [math.NT], 2008-2012.
%H V. Shevelev, G. Garcia-Pulgarin, J. M. Velasquez and J. H. Castillo, <a href="http://arxiv.org/abs/1206.0606">Overpseudoprimes, and Mersenne and Fermat numbers as primover numbers</a>, arXiv preprint arXiv:1206:0606, 2012. - From _N. J. A. Sloane_, Oct 28 2012
%H V. Shevelev, G. Garcia-Pulgarin, J. M. Velasquez and J. H. Castillo, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Shevelev/shevelev19.html">Overpseudoprimes, and Mersenne and Fermat Numbers as Primover Numbers</a>, J. Integer Seq. 15 (2012) Article 12.7.7.
%t ops5Q[n_] := CompositeQ[n] && GCD[n, 5] == 1 && MultiplicativeOrder[5, n]*(DivisorSum[n, EulerPhi[#]/MultiplicativeOrder[5, #] &] - 1) + 1 == n; Select[Range[6, 10^6], ops5Q] (* _Amiram Eldar_, Jun 24 2019 *)
%o (PARI) isok(n) = (n>5) && !isprime(n) && (gcd(n,5)==1) && (znorder(Mod(5,n)) * (sumdiv(n, d, eulerphi(d)/znorder(Mod(5, d))) - 1) + 1 == n); \\ _Michel Marcus_, Oct 25 2018
%Y Cf. A141232, A141350, A020231, A020229.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Jun 29 2008
%E Inserted a(2) and a(8) and extended at the suggestion of Gilberto Garcia-Pulgarin by _Vladimir Shevelev_, Feb 06 2012
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