A004614 Numbers that are divisible only by primes congruent to 3 mod 4.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A004613.

Cf. A002145 (subsequence of primes).

Programs

Haskell

a004614 n = a004614_list !! (n-1)
a004614_list = filter (all (== 1) . map a079261 . a027748_row) [1..]
-- Reinhard Zumkeller, Jan 07 2013

Magma

[n: n in [1..300] | forall{d: d in PrimeDivisors(n) | d mod 4 eq 3}]; // Vincenzo Librandi, Aug 21 2012

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 *)

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

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)))
A004614_list = list(islice(A004614_gen(),30)) # Chai Wah Wu, Aug 21 2024

Formula

Product(A079261(A027748(a(n),k)): k=1..A001221(a(n))) = 1. - Reinhard Zumkeller, Jan 07 2013