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A354875 Dirichlet inverse of A344005, the smallest positive m such that n divides the oblong number m*(m+1).

%I A354875 #11 Jun 12 2022 15:38:57
%S A354875 1,-1,-2,-2,-4,2,-6,-2,-4,4,-10,7,-12,6,11,0,-16,4,-18,16,18,10,-22,4,
%T A354875 -8,12,-2,23,-28,-11,-30,4,29,16,34,12,-36,18,36,5,-40,-18,-42,39,27,
%U A354875 22,-46,-6,-12,8,47,48,-52,2,70,25,54,28,-58,-78,-60,30,21,8,71,-29,-66,64,65,-34,-70,24,-72,36,16,71,99
%N A354875 Dirichlet inverse of A344005, the smallest positive m such that n divides the oblong number m*(m+1).
%H A354875 Antti Karttunen, <a href="/A354875/b354875.txt">Table of n, a(n) for n = 1..20000</a>
%F A354875 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A344005(n/d) * a(d).
%F A354875 a(n) = A354876(n) - A344005(n).
%t A354875 f[n_] := Module[{m = 1}, While[! Divisible[m*(m + 1), n], m++]; m]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, a[#]*f[n/#] &, # < n &]; Array[a, 100] (* _Amiram Eldar_, Jun 12 2022 *)
%o A354875 (PARI)
%o A354875 A344005(n) = for(m=1, oo, if((m*(m+1))%n==0, return(m))); \\ From A344005
%o A354875 memoA354875 = Map();
%o A354875 A354875(n) = if(1==n,1,my(v); if(mapisdefined(memoA354875,n,&v), v, v = -sumdiv(n,d,if(d<n,A344005(n/d)*A354875(d),0)); mapput(memoA354875,n,v); (v)));
%Y A354875 Cf. A002378, A344005, A354876, A354877 (positions of 0's).
%Y A354875 Cf. also A345055.
%K A354875 sign
%O A354875 1,3
%A A354875 _Antti Karttunen_, Jun 12 2022