OFFSET
1,2
COMMENTS
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
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
Gareth A. Jones and Alexander K. Zvonkin, A number-theoretic problem concerning pseudo-real Riemann surfaces, arXiv:2401.00270 [math.NT], 2023. See page 4.
FORMULA
MAPLE
q:= n-> andmap(i-> irem(i[1], 4)=3, ifactors(n)[2]):
select(q, [$1..500])[]; # Alois P. Heinz, Jan 13 2024
MATHEMATICA
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 *)
PROG
(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..]
-- Reinhard Zumkeller, Jan 07 2013
(Python)
from itertools import count, islice
from sympy import primefactors
def A004614_gen(startvalue=1): # generator of terms >= startvalue
return filter(lambda n: n&1 and all(p&2 for p in primefactors(n>>(~n & n-1).bit_length())), count(max(startvalue, 1)))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved