A336841 Prime-shifted analog of A094471: a(n) = A336845(n) - A003973(n).

0, 2, 4, 14, 6, 36, 10, 68, 44, 52, 12, 192, 16, 84, 92, 284, 18, 326, 22, 274, 148, 100, 28, 840, 90, 132, 344, 438, 30, 648, 36, 1094, 176, 148, 212, 1622, 40, 180, 232, 1192, 42, 1032, 46, 520, 802, 228, 52, 3324, 230, 654, 260, 684, 58, 2376, 252, 1896, 316, 244, 60, 3156, 66, 292, 1278, 4010, 332, 1224, 70, 766

Offset: 1

Author

Antti Karttunen, Aug 06 2020

Keywords

• nonn

Comments

All terms are even because A003973 and A336845 match parity-wise. Also in the sum formulas, only even terms are summed (only one of which is zero).

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000005, A000203, A003961, A048673, A094471, A336837, A336839, A336840, A336845, A336856.

Cf. A336846 [= gcd(a(n), A003973(n))].

Twice the terms of A336854.

Programs

PARI

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336841(n) = ((numdiv(n)*A003961(n)) - sigma(A003961(n)));

PARI

A336841(n) = sumdiv(n,d,(A003961(n)-A003961(d)));

PARI

A336841(n) = sumdiv(A003961(n),d,(A003961(n)-d));

Formula

a(n) = A336845(n) - A003973(n) = (A000005(n)*A003961(n)) - A000203(A003961(n)).

a(n) = A094471(A003961(n)).

a(n) = Sum_{d|n} (A003961(n)-A003961(d)) = Sum_{d|A003961(n)} (A003961(n)-d).

a(n) = 2*A336854(n) = 2*Sum_{d|n} (A048673(n)-A048673(d)).

a(n) = ((A003961(n)+1)*A000005(n)) - 2*A336840(n).

a(n) = 2 * ((A000005(n)*A048673(n)) - A336840(n)).

a(n) = A000005(n) * (A336837(n)/A336839(n)) = A336837(n) * A336856(n).