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URL: https://oeis.org/A343274

⇱ A343274 - OEIS

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A343274
a(n) = Sum_{d|n} d^d * sopf(d).
2
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
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
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, May 24 2021
STATUS
approved