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

Author

Marius A. Burtea, Feb 27 2024

Keywords

• nonn

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.

Examples

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

Links

• Robert Israel, Table of n, a(n) for n = 1..572

Crossrefs

Cf. A002061, A002145, A002407, A016105.

Programs

Magma

pd:=PrimeDivisors; blum:=func; [n:n in [s^2-s+1:s in [2..2000]]|blum(n)];

Maple

N:= 10^5: # for terms <= N
P:= select(isprime, [seq(i,i=3..N/3,4)]):
sort(select(t -> t <= N and issqr(4*t-3), [seq(seq(P[i]*P[j],i=1..j-1),j=1..nops(P))])); # Robert Israel, Feb 27 2025

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 *)