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A039955
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Squarefree numbers congruent to 1 mod 4.
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8
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1, 5, 13, 17, 21, 29, 33, 37, 41, 53, 57, 61, 65, 69, 73, 77, 85, 89, 93, 97, 101, 105, 109, 113, 129, 133, 137, 141, 145, 149, 157, 161, 165, 173, 177, 181, 185, 193, 197, 201, 205, 209, 213, 217, 221, 229, 233, 237, 241, 249, 253, 257, 265, 269
(list;
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refs;
listen;
history;
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internal format)
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OFFSET
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1,2
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COMMENTS
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The subsequence of primes is A002144.
The subsequence of semiprimes (intersection with A001358) begins: 21, 33, 57, 65, 69, 77, 85, 93, 129, 133, 141, 145, 161, 177, 185, 201, 205, 209, 213, 217, 221, 237, 249, 253, 265.
The subsequence with more than two prime factors (intersection with A033942) begins: 105 = 3 * 5 * 7, 165 = 3 * 5 * 11, 273, 285, 345, 357, 385, 429, 465. [Jonathan Vos Post, Feb 19 2011].
Except for a(1) = 1 these are the squarefree members of A079896 (i.e., squarefree determinants D of indefinite binary quadratic forms). - Wolfdieter Lang, Jun 01 2013
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REFERENCES
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R. A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. M. Legendre, Diviseurs de la formule t^2+a*u^2, a étant de la forme 4 n + 1, Essai sur la Théorie des Nombres An VI, Table IV. See first column. [Paul Curtz, Aug 14 2019]
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MATHEMATICA
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fQ[n_] := Max[Last /@ FactorInteger@ n] == 1 && Mod[n, 4] == 1; Select[ Range@ 272, fQ] (* Robert G. Wilson v *)
Select[Range[1, 300, 4], SquareFreeQ[#]&] (* Harvey P. Dale, Mar 27 2020 *)
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PROG
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(MAGMA) [4*n+1: n in [0..67] | IsSquarefree(4*n+1)]; // Bruno Berselli, Mar 03 2011
(Haskell)
a039955 n = a039955_list !! (n-1)
a039955_list = filter ((== 1) . (`mod` 4)) a005117_list
-- Reinhard Zumkeller, Aug 15 2011
(PARI) list(lim)=my(v=List([1])); forfactored(n=5, lim\1, if(vecmax(n[2][, 2])==1 && n[1]%4==1, listput(v, n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017
(PARI) is(n)=n%4==1 && issquarefree(n) \\ Charles R Greathouse IV, Nov 05 2017
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CROSSREFS
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Cf. A002144, A039956, A039957.
Sequence in context: A226165 A166409 A077425 * A213340 A014539 A249034
Adjacent sequences: A039952 A039953 A039954 * A039956 A039957 A039958
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KEYWORD
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nonn,easy,nice
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AUTHOR
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R. K. Guy
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STATUS
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approved
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