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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.
3

%I #12 Jul 12 2024 14:23:46

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

%N 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.

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

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

%C Conjecture: the sequence is finite.

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

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

%o (PARI)

%o A000265(n) = (n>>valuation(n,2));

%o A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o isA374464(n) = (A000265(sigma(A003961(n)))==A000265(n));

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

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

%K nonn,hard,more

%O 1,2

%A _Antti Karttunen_, Jul 11 2024