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A191218
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Odd numbers n such that sigma(n) is congruent to 2 modulo 4.
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26
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5, 13, 17, 29, 37, 41, 45, 53, 61, 73, 89, 97, 101, 109, 113, 117, 137, 149, 153, 157, 173, 181, 193, 197, 229, 233, 241, 245, 257, 261, 269, 277, 281, 293, 313, 317, 325, 333, 337, 349, 353, 369, 373, 389, 397, 401, 405, 409, 421, 425, 433, 449, 457, 461, 477
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OFFSET
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1,1
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COMMENTS
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Exactly the numbers of the form p^{4k+1}*m^2 with p a prime congruent to 1 modulo 4 and m a positive integer coprime with p. The odd perfect numbers are all of this form.
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LINKS
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EXAMPLE
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For n=3 one has a(3)=17 since sigma(17) = 18 = 4*4 +2 is congruent to 2 modulo 4
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MAPLE
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with(numtheory): genodd := proc(b) local n, s, d; for n from 1 to b by 2 do s := sigma(n);
if modp(s, 4)=2 then print(n); fi; od; end;
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MATHEMATICA
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Select[Range[1, 501, 2], Mod[DivisorSigma[1, #], 4]==2&] (* Harvey P. Dale, Nov 12 2017 *)
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PROG
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(PARI) forstep(n=1, 10^3, 2, if(2==(sigma(n)%4), print1(n, ", "))) \\ Joerg Arndt, May 27 2011
(PARI) list(lim)=my(v=List()); forstep(e=1, logint(lim\=1, 5), 4, forprimestep(p=5, sqrtnint(lim, e), 4, my(pe=p^e); forstep(m=1, sqrtint(lim\pe), 2, if(m%p, listput(v, pe*m^2))))); Set(v) \\ Charles R Greathouse IV, Feb 16 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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