A231754 Products of distinct primes congruent to 1 modulo 4 (A002144).

1, 5, 13, 17, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 137, 145, 149, 157, 173, 181, 185, 193, 197, 205, 221, 229, 233, 241, 257, 265, 269, 277, 281, 293, 305, 313, 317, 337, 349, 353, 365, 373, 377, 389, 397, 401, 409, 421, 433, 445, 449

Offset

1,2

Comments

Contains A002144 as a subsequence, and is a subsequence of A016813 and of A005117.

Also, these numbers satisfy A231589(n) = floor(n*(n-1)/4) (A011848).

Links

R. J. Mathar, Table of n, a(n) for n = 1..4999

Rafael Jakimczuk, Generalizations of Mertens's Formula and k-Free and s-Full Numbers with Prime Divisors in Arithmetic Progression, ResearchGate, 2024.

Shailesh A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag., Vol. 70, No. 4 (Oct., 1997), pp. 263-272.

Formula

The number of terms that do not exceed x is ~ c * x / sqrt(log(x)), where c = A088539 * sqrt(A175647) / Pi = 0.3097281805... (Jakimczuk, 2024, Theorem 3.10, p. 26). - Amiram Eldar, Mar 08 2024

Example

65 = 5*13 is in the sequence since both 5 and 13 are congruent to 1 modulo 4.

Maple

isA231754 := proc(n)
    local d;
    for d in ifactors(n)[2] do
        if op(2, d) > 1 then
            return false;
        elif modp(op(1, d), 4) <> 1 then
            return false;
        end if;
    end do:
    true ;
end proc:
for n from 1 to 500 do
    if isA231754(n) then
        printf("%d, ", n) ;
    end if;
end do: # R. J. Mathar, Mar 16 2016

Mathematica

Select[Range[500], # == 1 || AllTrue[FactorInteger[#], Last[#1] == 1 && Mod[First[#1], 4] == 1 &] &] (* Amiram Eldar, Mar 08 2024 *)

Prog

(PARI) isok(n) = if (! issquarefree(n), return (0)); if (n > 1, f = factor(n); for (i=1, #f~, if (f[i, 1] % 4 != 1, return (0)))); 1

Crossrefs

Intersection of A005117 and A004613.

Cf. A002144, A011848, A016813, A167181, A231589.

Cf. A088539, A175647.

Sequence in context: A191218 A279857 A077426 * A175768 A351535 A002144

Adjacent sequences: A231751 A231752 A231753 * A231755 A231756 A231757

Keyword

nonn,easy

Author

Michel Marcus, Nov 13 2013

Status

approved