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A350115
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Numbers p^2*q, p<q primes such that p^2 divides q-1.
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7
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20, 52, 68, 116, 148, 164, 171, 212, 244, 292, 333, 356, 388, 404, 436, 452, 548, 596, 628, 657, 692, 724, 772, 788, 916, 932, 964, 981, 1028, 1076, 1108, 1124, 1143, 1172, 1252, 1268, 1348, 1396, 1412, 1467, 1492, 1556, 1588, 1604, 1629, 1636, 1684, 1732, 1791, 1796, 1828, 1844
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OFFSET
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1,1
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COMMENTS
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For these terms m, there are precisely 5 groups of order m, so this is a subsequence of A054397.
Two of them are abelian: C_{p^2*q}, C_q X C_p X C_p = C_q X (C_p)^2, and the three others that are nonabelian are C_q : (C_p x C_p), and two nonisomorphic semi-direct products C_q : C_p^2. Here C means cyclic groups of the stated order, the symbols X and : mean direct and semidirect products respectively.
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REFERENCES
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Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.
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LINKS
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EXAMPLE
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20 = 2^2*5 and 2^2 divides 5-1, hence 20 is a term.
171 = 3^2*19 and 3^2 divides 19-1, hence 171 is another term.
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MATHEMATICA
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q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; e == {2, 1} && Divisible[p[[2]] - 1, p[[1]]^2]]; Select[Range[2000], q] (* Amiram Eldar, Dec 14 2021 *)
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PROG
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(PARI) isok(m) = {my(f=factor(m)); if (f[, 2] == [2, 1]~, my(p=f[1, 1], q=f[2, 1]); ((q-1) % p^2) == 0; ); } \\ Michel Marcus, Dec 14 2021
(Python)
from sympy import integer_nthroot, isprime, primerange
def aupto(limit):
aset, maxp = set(), integer_nthroot(limit, 4)[0]
for p in primerange(1, maxp+1):
m = p**2
for t in range(m, limit//m, m):
if isprime(t+1):
aset.add(p**2*(t+1))
return sorted(aset)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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