A322702 a(n) is the product of primes p such that p+1 divides n.

1, 1, 2, 3, 1, 10, 1, 21, 2, 1, 1, 330, 1, 13, 2, 21, 1, 170, 1, 57, 2, 1, 1, 53130, 1, 1, 2, 39, 1, 290, 1, 651, 2, 1, 1, 5610, 1, 37, 2, 399, 1, 5330, 1, 129, 2, 1, 1, 2497110, 1, 1, 2, 3, 1, 9010, 1, 273, 2, 1, 1, 10727970, 1, 61, 2, 651, 1, 10, 1, 201, 2

Offset: 1

Author

Daniel Suteu, Dec 23 2018

Keywords

• nonn

Comments

In general, a(n) is the product of A072627(n) distinct prime factors, with a(n) = 1 iff A072627(n) = 0.

Examples

			For n=12, the divisors of 12 are {1, 2, 3, 4, 6, 12}. The prime numbers p, such that p+1 is a divisor of 12, are {2, 3, 5, 11}, therefore a(12) = 2 * 3 * 5 * 11 = 330.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10080

Crossrefs

Cf. A072627, A027760, A322356, A323156.

Programs

Maple

a:= n-> mul(`if`(isprime(d-1), d-1, 1), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 29 2018

Mathematica

Array[Apply[Times, Select[Divisors@ #, PrimeQ[# - 1] &] - 1 /. {} -> {1}] &, 69] (* Michael De Vlieger, Jan 07 2019 *)

PARI

a(n) = my(d=divisors(n)); prod(k=1, #d, if(isprime(d[k]-1), d[k]-1, 1));

Formula

a(n) = Product_{p prime, p+1 divides n} p.

a(n) = denominator of Sum_{p prime, p+1 divides n} 1/p.

a(n) = Product_{d|n, d-1 is prime} (d-1), where d runs over the divisors of n.

a(2*n + 1) = 2, iff n == 1 (mod 3), else a(2*n + 1) = 1.

A001221(a(n)) = A072627(n). - Antti Karttunen, Jan 12 2019