A381887 a(n) = 1 if n != p^m*(p-1) for any prime p and any m >= 0, otherwise Product_{p in W} p, where W are the primes such that n = p^m*(p-1) for some m >= 0.

2, 6, 1, 10, 1, 21, 1, 2, 1, 11, 1, 13, 1, 1, 1, 34, 1, 57, 1, 5, 1, 23, 1, 1, 1, 1, 1, 29, 1, 31, 1, 2, 1, 1, 1, 37, 1, 1, 1, 41, 1, 301, 1, 1, 1, 47, 1, 1, 1, 1, 1, 53, 1, 3, 1, 1, 1, 59, 1, 61, 1, 1, 1, 2, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1

Offset

1,1

Comments

a(n) = the product of the radicals of all prime powers whose totient equals n. The terms are successive quotients of the terms of A392539. - Hal M. Switkay, Jan 25 2026

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Manjul Bhargava, The factorial function and generalizations, Amer. Math. Monthly, 107 (Nov. 2000), 783-799, (page 793).

Angelo B. Mingarelli, Abstract factorials, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 4, 43-76, (page 54).

Formula

a(n) = 1 for n > 1 and odd.

a(n) > 1 is the product of at most two primes.

Example

a(100) = 505 = 5 * 101 because 100 = (5^2 * (5 - 1)) = 100^0 * (101 - 1).

Prog

(SageMath)
Max = 100 # Adjust for a larger range.
def a(n):
    f = 1
    for m in range(Max):
        for p in prime_range(2, Max):
            if p^m*(p-1) == n: f *= p
    return f
print([a(n) for n in range(1, 80)])
(PARI) a(n) = {my(f=factor(n), r=1); if(isprime(n+1), r=n+1); for(i=1, #f~, my([p, e]=f[i, ]); if(n/p^e==p-1, r*=p)); r} \\ Andrew Howroyd, Jan 25 2026

Crossrefs

Cf. A155457.

Cf. A000010, A002202, A007614, A025487, A032447, A378636, A378638.

Sequence in context: A100831 A136763 A109530 * A264789 A111519 A184941

Adjacent sequences: A381884 A381885 A381886 * A381888 A381889 A381890

Keyword

nonn

Author

Peter Luschny, Apr 05 2025

Status

approved