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A320779
Inverse Euler transform of the number of divisors function A000005.
12
1, 1, 0, 0, -1, 1, -1, 0, 1, -1, 0, 1, -1, -1, 2, 1, -2, -2, 2, 3, -4, 0, 3, -3, 3, -2, -2, 2, 1, 7, -15, 0, 17, -11, -1, 0, 9, -4, -18, 26, -10, -10, 24, -17, -15, 21, 27, -42, -37, 69, 43, -113, -11, 149, -98, -24, 67, -57, 24, -53, 213, -243, -193, 704
OFFSET
1,15
COMMENTS
The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n).
LINKS
MAPLE
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-SumOfDivisors(n, 0))):
seq(a(n), n = 1..64); # Peter Luschny, Nov 21 2022
MATHEMATICA
EulerInvTransform[{}]={}; EulerInvTransform[seq_]:=Module[{final={}}, For[i=1, i<=Length[seq], i++, AppendTo[final, i*seq[[i]]-Sum[final[[d]]*seq[[i-d]], {d, i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]], {d, Divisors[i]}]/i, {i, Length[seq]}]];
EulerInvTransform[Table[DivisorSigma[0, n], {n, 100}]]
PROG
(Python)
from functools import lru_cache
from sympy import mobius, divisors, divisor_count
def A320779(n):
@lru_cache(maxsize=None)
def b(n): return divisor_count(n)
@lru_cache(maxsize=None)
def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1, n))
return sum(mobius(d)*c(n//d) for d in divisors(n, generator=True))//n # Chai Wah Wu, Jul 15 2024
CROSSREFS
Cf. A000005.
Sequence in context: A323088 A052954 A123505 * A356876 A114920 A283190
KEYWORD
sign
AUTHOR
Gus Wiseman, Oct 22 2018
STATUS
approved