A323550 Numbers that can be expressed as (p - 1)*(q - 1) + 1, where p < q are primes.

3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 29, 31, 33, 37, 41, 43, 45, 47, 49, 53, 57, 59, 61, 65, 67, 71, 73, 79, 81, 83, 85, 89, 93, 97, 101, 103, 105, 107, 109, 113, 117, 121, 127, 131, 133, 137, 139, 141, 145, 149, 151, 157, 161, 163, 165, 167, 169, 173, 177, 179, 181, 185, 191, 193, 197, 199, 201, 205, 209

Offset

1,1

Comments

If p < q are primes and a(n) = (p - 1)*(q - 1) + 1, then x^a(n) == x (mod p*q) for every integer x.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

A. Bogomolny, Euler Function and Theorem

Example

181 is a term because 181 = (11 - 1)*(19 - 1) + 1. - Bernard Schott, Jan 19 2019

Maple

N:= 1000: # for terms <= N
S:= {}:
P:= select(isprime, [2, seq(i, i=3..N, 2)]): nP:= nops(P):
for i from 1 to nP do
  for j from i+1 to nP do
     v:= (P[i]-1)*(P[j]-1)+1;
     if v > N then break fi;
     S:= S union {v}
od od:
sort(convert(S, list)); # Robert Israel, May 22 2025

Mathematica

nmax = 100;
pairs = Table[Table[(Prime[i] - 1)*(Prime[j] - 1) + 1, {i, 1, j - 1}], {j, 2, Prime[nmax]}];
(DeleteDuplicates@Sort@Flatten@pairs)[[1 ;; nmax]]

Prog

(PARI) isok(n) = {if (n % 2, forprime(p = 2, n, forprime(q = p+1, n, if (n == (p - 1)*(q - 1) + 1, return (1)); ); ); ); } \\ Michel Marcus, Feb 25 2019

Crossrefs

Cf. A065091.

Sequence in context: A363691 A349174 A349177 * A165468 A353124 A354149

Adjacent sequences: A323547 A323548 A323549 * A323551 A323552 A323553

Keyword

nonn

Author

Andres Cicuttin, Jan 17 2019

Status

approved