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A074172
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Smaller of two consecutive numbers of the form p^2*q where p and q are primes.
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10
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44, 75, 98, 116, 147, 171, 244, 332, 387, 507, 548, 603, 604, 724, 844, 908, 931, 963, 1075, 1083, 1251, 1324, 1412, 1467, 1556, 1587, 1675, 1772, 2523, 2524, 2636, 2644, 2763, 3283, 3356, 3411, 3508, 3788, 3987, 4075, 4203, 4204, 4418, 4491, 4804, 4868
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OFFSET
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1,1
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COMMENTS
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From Robert Israel, Dec 06 2018: (Start) There are four forms of terms, for odd primes p,q,r:
4*p where 4*p+1 = q^2*r, r == 1 (mod 4)
2*p^2 where 2*p^2+1 = q^2*r, r == 3 (mod 4)
p^2*q where p^2*q+1 = 2*r^2, q == 1 (mod 4)
p^2*q where p^2*q+1 = 4*r, q == 3 (mod 4).
Are there infinitely many terms of each type?
(End)
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LINKS
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Robert Israel, Table of n, a(n) for n = 1..10000
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EXAMPLE
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44 is a member as 44 = 2^2*11 and 45 = 3^2*5.
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MAPLE
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filter:= proc(n) local F;
F:= map(t -> t[2], ifactors(n)[2]);
F = [2, 1] or F = [1, 2]
end proc:
A054753:= select(filter, {$1..10000}):
sort(convert(A054753 intersect map(`-`, A054753, 1), list)); # Robert Israel, Dec 06 2018
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MATHEMATICA
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lst={}; Do[f1=FactorInteger[n]; If[Sort[Transpose[f1][[2]]]=={1, 2}, f2=FactorInteger[n+1]; If[Sort[Transpose[f2][[2]]]=={1, 2}, AppendTo[lst, n]]], {n, 3, 10000}]; lst
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PROG
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(PARI) isok1(n) = vecsort(factor(n)[, 2]) == [1, 2]~;
isok(n) = isok1(n) && isok1(n+1); \\ Michel Marcus, Sep 20 2017
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CROSSREFS
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Cf. A054753, A074173, A074174, A178032, A308683, A141621.
Sequence in context: A156812 A171665 A049103 * A039386 A043209 A043989
Adjacent sequences: A074169 A074170 A074171 * A074173 A074174 A074175
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KEYWORD
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nonn
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AUTHOR
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Amarnath Murthy, Aug 30 2002
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EXTENSIONS
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More terms from T. D. Noe, Oct 04 2004
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STATUS
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approved
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