A343274 a(n) = Sum_{d|n} d^d * sopf(d).

0, 8, 81, 520, 15625, 233369, 5764801, 33554952, 1162261548, 70000015633, 3138428376721, 44580502475161, 3937376385699289, 100008061435786953, 3503151123046890706, 36893488147452658184, 14063084452067724991009, 196732040376483850371956, 37589973457545958193355601

Offset

1,2

Comments

If p is prime, a(p) = Sum_{d|p} d^d * sopf(d) = 1^1*0 + p^p*p = p^(p+1).

Inverse Möbius transform of n^n * sopf(n). - Wesley Ivan Hurt, Mar 31 2025

Links

Robert Israel, Table of n, a(n) for n = 1..384

Formula

a(p^k) = p * Sum_{i=1..k} (p^i)^(p^i) for p prime and k >= 0. - Wesley Ivan Hurt, May 23 2026

Example

a(4) = Sum_{d|4} d^d * sopf(d) = 1^1*sopf(1) + 2^2*sopf(2) + 4^4*sopf(4) = 0 + 8 + 512 = 520.

Maple

f:= proc(n) local d; add(d^d * convert(numtheory:-factorset(d), `+`), d=numtheory:-divisors(n)) end proc:
map(f, [$1..20]); # Robert Israel, Nov 25 2025

Mathematica

Table[Sum[i^i*Sum[k (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[i/k] + Floor[i/k]), {k, i}] (1 - Ceiling[n/i] + Floor[n/i]) , {i, n}], {n, 20}]

Crossrefs

Cf. A000312 (n^n), A008472 (sopf), A344578.

Sequence in context: A303515 A027768 A236749 * A240480 A274752 A240632

Adjacent sequences: A343271 A343272 A343273 * A343275 A343276 A343277

Keyword

nonn

Author

Wesley Ivan Hurt, May 24 2021

Status

approved