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%I A004614 #61 Aug 21 2024 15:01:48
%S A004614 1,3,7,9,11,19,21,23,27,31,33,43,47,49,57,59,63,67,69,71,77,79,81,83,
%T A004614 93,99,103,107,121,127,129,131,133,139,141,147,151,161,163,167,171,
%U A004614 177,179,189,191,199,201,207,209,211,213,217,223,227,231,237,239,243,249,251
%N A004614 Numbers that are divisible only by primes congruent to 3 mod 4.
%C A004614 Numbers whose factorization as Gaussian integers is the same as their factorization as integers. - _Franklin T. Adams-Watters_, Oct 14 2005
%C A004614 Closed under multiplication. Primitive elements are the primes of form 4*k+3. - _Gerry Martens_, Jun 17 2020
%H A004614 T. D. Noe, <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL0EwMDQ2MTQvYjAwNDYxNC50eHQ">Table of n, a(n) for n = 1..10000</a>
%H A004614 Gareth A. Jones and Alexander K. Zvonkin, <a href="/load/view.php?a=aHR0cHM6Ly9hcnhpdi5vcmcvYWJzLzI0MDEuMDAyNzA">A number-theoretic problem concerning pseudo-real Riemann surfaces</a>, arXiv:2401.00270 [math.NT], 2023. See page 4.
%F A004614 Product(A079261(A027748(a(n),k)): k=1..A001221(a(n))) = 1. - _Reinhard Zumkeller_, Jan 07 2013
%p A004614 q:= n-> andmap(i-> irem(i[1], 4)=3, ifactors(n)[2]):
%p A004614 select(q, [$1..500])[];  # _Alois P. Heinz_, Jan 13 2024
%t A004614 ok[1] = True; ok[n_] := And @@ (Mod[#, 4] == 3 &) /@ FactorInteger[n][[All, 1]]; Select[Range[251], ok] (* _Jean-François Alcover_, May 05 2011 *)
%t A004614 A004614 = Select[Range[251],Length@Reduce[s^2 + t^2 == s # && s # > t > 0, Integers] == 0 &] (* _Gerry Martens_, Jun 05 2020 *)
%o A004614 (PARI) for(n=1,1000,if(sumdiv(n,d,isprime(d)*if((d-3)%4,1,0))==0, print1(n,",")))
%o A004614 (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
%o A004614 (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
%o A004614 (Magma) [n: n in [1..300] | forall{d: d in PrimeDivisors(n) | d mod 4 eq 3}]; // _Vincenzo Librandi_, Aug 21 2012
%o A004614 (Haskell)
%o A004614 a004614 n = a004614_list !! (n-1)
%o A004614 a004614_list = filter (all (== 1) . map a079261 . a027748_row) [1..]
%o A004614 -- _Reinhard Zumkeller_, Jan 07 2013
%o A004614 (Python)
%o A004614 from itertools import count, islice
%o A004614 from sympy import primefactors
%o A004614 def A004614_gen(startvalue=1): # generator of terms >= startvalue
%o A004614     return filter(lambda n: n&1 and all(p&2 for p in primefactors(n>>(~n & n-1).bit_length())), count(max(startvalue,1)))
%o A004614 A004614_list = list(islice(A004614_gen(),30)) # _Chai Wah Wu_, Aug 21 2024
%Y A004614 Cf. A004613.
%Y A004614 Cf. A002145 (subsequence of primes).
%K A004614 nonn,easy
%O A004614 1,2
%A A004614 _N. J. A. Sloane_

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