A374464 Numbers k such that the odd parts of k and sigma(A003961(k)) are equal, where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

1, 2, 3, 6, 40, 120, 351, 702, 1000, 3000, 14040, 351000

Offset: 1

Author

Antti Karttunen, Jul 11 2024

Keywords

• nonn
• hard
• more

Comments

If x and y are included, and they are coprime (gcd(x,y) = 1), then x*y is also included.

A003961 applied to this sequence gives the odd terms of A374463, which after 1 is a subsequence of A347391.

Conjecture: the sequence is finite.

Examples

			351 = 3^3 * 13 is included as sigma(A003961(351)) = sigma(2125) = 2808, with A000265(2808) = A000265(351) = 351.
		

Crossrefs

Cf. A000203, A000265, A003961, A003973, A347391, A374463.

After the initial 1, a subsequence of A348738. Cf. also A326042.

Programs

PARI

A000265(n) = (n>>valuation(n,2));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA374464(n) = (A000265(sigma(A003961(n)))==A000265(n));

Formula

{k | A000265(k) = A000265(sigma(A003961(k)))}.