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A004614
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Numbers that are divisible only by primes congruent to 3 mod 4.
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34
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1, 3, 7, 9, 11, 19, 21, 23, 27, 31, 33, 43, 47, 49, 57, 59, 63, 67, 69, 71, 77, 79, 81, 83, 93, 99, 103, 107, 121, 127, 129, 131, 133, 139, 141, 147, 151, 161, 163, 167, 171, 177, 179, 189, 191, 199, 201, 207, 209, 211, 213, 217, 223, 227, 231, 237, 239, 243, 249, 251
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OFFSET
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1,2
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COMMENTS
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Numbers whose factorization as Gaussian integers is the same as their factorization as integers. - Franklin T. Adams-Watters, Oct 14 2005
Closed under multiplication. Primitive elements are the primes of form 4*k+3. - Gerry Martens, Jun 17 2020
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LINKS
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FORMULA
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MAPLE
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q:= n-> andmap(i-> irem(i[1], 4)=3, ifactors(n)[2]):
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MATHEMATICA
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ok[1] = True; ok[n_] := And @@ (Mod[#, 4] == 3 &) /@ FactorInteger[n][[All, 1]]; Select[Range[251], ok] (* Jean-François Alcover, May 05 2011 *)
A004614 = Select[Range[251], Length@Reduce[s^2 + t^2 == s # && s # > t > 0, Integers] == 0 &] (* Gerry Martens, Jun 05 2020 *)
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PROG
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(PARI) for(n=1, 1000, if(sumdiv(n, d, isprime(d)*if((d-3)%4, 1, 0))==0, print1(n, ", ")))
(PARI) forstep(n=1, 999, 2, for(j=1, #t=factor(n)[, 1], t[j]%4==1 && next(2)); print1(n", ")) \\ M. F. Hasler, Feb 26 2008
(PARI) list(lim)=my(v=List([1]), cur, idx, newIdx); forprime(p=3, lim, if(p%4>1, listput(v, p))); for(i=2, #v, cur=v[i]; idx=1; while(v[idx]*cur <= lim, my(newidx=#v+1, t); for(j=idx, #v, t=cur*v[j]; if(t<=lim, listput(v, t))); idx=newidx)); Set(v) \\ Charles R Greathouse IV, Feb 06 2018
(Magma) [n: n in [1..300] | forall{d: d in PrimeDivisors(n) | d mod 4 eq 3}]; // Vincenzo Librandi, Aug 21 2012
(Haskell)
a004614 n = a004614_list !! (n-1)
a004614_list = filter (all (== 1) . map a079261 . a027748_row) [1..]
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CROSSREFS
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Cf. A002145 (subsequence of primes).
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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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