A370519 Intersection of A002061 and A016105.

21, 57, 133, 381, 553, 813, 993, 1057, 1333, 1561, 1641, 1893, 1981, 2653, 2757, 3193, 3661, 5257, 5853, 6973, 8373, 8557, 9121, 9313, 10713, 10921, 12657, 13341, 15253, 15501, 16257, 18633, 19741, 22053, 24493, 29413, 30801, 32221, 32581, 33673, 35157, 39801

Offset

1,1

Comments

If p is a cuban prime (A002407) and p == 3 (mod 4) (A002145), then m = 3*p is a term. Indeed, there is k for which p = 1 + 3*k*(k + 1) and m = 3*p = 3 + 9*k*(k + 1) = (3*k + 2)^2 - (3*k + 2) + 1, so m is a term.

The sequence also includes terms that do not have this form: 133 = 12^2 - 12 + 1 = 7*19, 553 = 24^2 - 24 + 1 = 7*79, 1057 = 33^2 - 33 + 1 = 7*151, 1333 = 37^2 - 37 + 1= 31*43 and others.

Links

Table of n, a(n) for n=1..42.

Example

A002061(5) = 21 = A016105(1), so 21 is a term.
A002061(8) = 57 = A016105(3), so 57 is a term.

Mathematica

 TR=40000; R1=Ceiling[(1+Sqrt[1-4(1-TR)])/2]; R2=TR/4; Intersection[Table[n^2-n+1, {n, 0, R1}], Select[4Range[5, R2]+1, PrimeNu[#]==2&&MoebiusMu[#]==1&&Mod[FactorInteger[#][[1, 1]], 4]!=1&]](* James C. McMahon, Feb 27 2024 *)

Prog

(Magma) pd:=PrimeDivisors; blum:=func<n|#Divisors(n) eq 4 and #pd(n) eq 2 and pd(n)[1] mod 4  eq 3 and pd(n)[2] mod 4 eq 3>; [n:n in [s^2-s+1:s in [2..2000]]|blum(n)];

Crossrefs

Cf. A002061, A002145, A002407, A016105.

Sequence in context: A118057 A327902 A020148 * A037305 A370109 A223467

Adjacent sequences: A370516 A370517 A370518 * A370520 A370521 A370522

Keyword

nonn

Author

Marius A. Burtea, Feb 27 2024

Status

approved