A356985 Numbers k that divide the concatenation of sigma(k-1) and sigma(k+1).

2, 55, 56, 93, 170, 944, 1904, 2839, 3104, 4213, 4793, 4953, 5664, 35120, 48456, 145256, 213416, 224288, 267084, 276736, 310897, 366624, 627850, 802024, 837801, 1093671, 1451332, 2036312, 2909925, 3888750, 4373000, 4746840, 6754320, 6819840, 13327155, 18807624

Offset

1,1

Comments

The only known prime terms are 2 and 4793; there are no others < 1.5*10^10.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..53

Carlos Rivera, Puzzle 1101. Primes p that divide the concatenation of sigma(p-1) and sigma(p+1), The Prime Puzzles & Problems Connection, September 2022.

Formula

A000203(k-1).A000203(k+1) == 0 (mod a(k)), where . is concatenation of digits.

Example

2 is a term since sigma(1) = 1, sigma(3) = 4 and 2 divides 14.

Mathematica

Select[Range[2, 10^6], Divisible[FromDigits[Join @@ IntegerDigits @ DivisorSigma[1, # + {-1, 1}]], #] &] (* Amiram Eldar, Sep 09 2022 *)

Prog

(Python)
from sympy import divisor_sigma
def sigma(n): return divisor_sigma(n, 1)
def ok(n): return n > 1 and int(str(sigma(n-1)) + str(sigma(n+1)))%n == 0
print([k for k in range(10**5) if ok(k)])
(PARI) isok(k) = (eval(concat(Str(sigma(k-1)), Str(sigma(k+1)))) % k) == 0; \\ Michel Marcus, Sep 09 2022

Crossrefs

Cf. A000203.

Sequence in context: A089180 A034013 A340211 * A280209 A157262 A007975

Adjacent sequences: A356982 A356983 A356984 * A356986 A356987 A356988

Keyword

nonn,base

Author

Michael S. Branicky and Giorgos Kalogeropoulos, Sep 08 2022

Status

approved