A395563 Indices of 2 in A395473: Number k such that Q(chi) takes 2 different fields for chi running through the Dirichlet characters modulo k, where Q(chi) is the field generated by the values of chi.

5, 7, 9, 10, 11, 14, 15, 16, 18, 20, 21, 22, 23, 28, 30, 33, 36, 40, 42, 44, 46, 47, 48, 56, 59, 60, 63, 66, 69, 72, 80, 83, 84, 88, 92, 94, 107, 118, 120, 126, 132, 138, 141, 166, 167, 168, 177, 179, 184, 188, 214, 227, 236, 240, 249, 252, 263, 264, 276, 282, 321, 332, 334

Offset

1,1

Comments

Numbers k such that psi(k) = 2*p for some prime p.

Numbers that divide 240 or 504 but not 24, and numbers of the form d*q for some safe prime q and d | 24.

Occurrences of 4 consecutive numbers other than (20, 21, 22, 23) are congruent to (716, 717, 718, 719) (mod 720), generated by some prime p == 89 (mod 90).

Proof: Suppose that (d1*p1, d2*p2, d3*p3, d4*p4) are 4 consecutive members in A395563, i.e., d1, d2, d3, d4 are divisors of 24, and p1, p2, p3, p4 are safe primes != 7. We note that pi == 5 (mod 6), so none of the four di*pi can be congruent to 1 modulo 6. As a result, these four numbers are congruent to either (2, 3, 4, 5) or (3, 4, 5, 0) modulo 6.

If (d1*p1, d2*p2, d3*p3, d4*p4) == (3, 4, 5, 0) (mod 6), then d1 = 3, d2 = 2 (mod 6), d3 = 1, d4 == 0 (mod 6). It follows that d1*p1 == 15 (mod 18), so d4*p4 == 0 (mod 18), but this is impossible since it would imply d4 == 0 (mod 18).

If (d1*p1, d2*p2, d3*p3, d4*p4) == (2, 3, 4, 5) (mod 6), then d1 == 4 (mod 6), d2 = 3, d3 == 2 (mod 6), d4 = 1. Of course only one (and exactly one) of d1 and d3 is divisible by 4, so (d1, d2, d3, d4) = (4, 3, 2, 1). Write p1 = 2*p + 1 for some prime p, then p2 = (8*p + 5)/3, (p2 - 1)/2 = (4*p + 1)/3, p3 = 4*p + 3, p4 = 8*p + 7, and these must be all primes. Taking modulo 3 and 5 yields p == 89 (mod 90) if p > 2.

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Wikipedia, Dirichlet character.

Mathematica

PositionIndex[DivisorSigma[0, CarmichaelLambda[Range[1000]]/2]][2] (* Paolo Xausa, Jun 03 2026 *)
(* Alternative, after PARI: *)
A395563Q[k_] := ((Divisible[240, k] || Divisible[504, k]) && !Divisible[24, k]) || (PrimeQ[#] && PrimeQ[(#-1)/2] & [k/GCD[k, 24]]);
Select[Range[1000], A395563Q] (* Paolo Xausa, Jun 03 2026 *)

Prog

(PARI) isA395563(n) = if((240%n==0 || 504%n==0) && 24%n!=0, return(1)); n = n/gcd(24, n); return(isprime(n) && isprime((n-1)/2))

Crossrefs

Cf. A395473, A018253, A395464, A005385 (safe primes).

Cf. A396495 (primes p such that (8*p+4, 8*p+5, 8*p+6, 8*p+7) are four consecutive terms), A396497 (k such that 720*k-4, 720*k-3, 720*k-2, 720*k-1 are four consecutive trems).

Sequence in context: A126949 A079696 A275718 * A129270 A153031 A334303

Adjacent sequences: A395560 A395561 A395562 * A395564 A395565 A395566

Keyword

nonn,easy

Author

Jianing Song, May 28 2026

Status

approved