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

1,2

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

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

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

Crossrefs

Cf. A004613.

Cf. A002145 (subsequence of primes).

Sequence in context: A129747 A354570 A354039 * A112398 A197504 A167800

Adjacent sequences: A004611 A004612 A004613 * A004615 A004616 A004617

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved