A345699 Multiplicative with a(p) = a(p-1) and a(p^e) = a(p) + a(e) if e>1.
1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 3, 3, 2, 2, 4, 1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 4, 4, 2, 2, 4, 4, 1, 1, 4, 4, 2, 2, 3, 2, 3, 3, 4, 4, 2, 4, 2, 2, 2, 2, 4, 4, 2, 2, 2, 4, 2, 2, 6, 2, 2, 2, 4, 4, 4, 3, 4, 2, 2, 2, 6, 3, 4, 4, 2, 6, 1, 2
Offset: 1
Examples
a(11)=a(10)=a(5)*a(2); a(2)=1; a(5)=a(4)=a(2)+a(2)=2; so a(11)=2.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Maple
a:= proc(n) option remember; mul(`if`(i[2]=1, a(i[1]-1), a(i[1])+a(i[2])), i=ifactors(n)[2]) end: seq(a(n), n=1..100); # Alois P. Heinz, Jun 28 2021 -
Mathematica
a[1]=1; a[p_,1]:= a[p-1]; a[p_, s_] := a[p, s] = a[p] + a[s]; a[n_]:=a[n]=Module[{aux=FactorInteger[n]},Product[a[aux[[i, 1]], aux[[i, 2]]], {i, Length[aux]}]]; -
PARI
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = a(f[k,1]-1); if (f[k,2] > 1, f[k,1] += a(f[k,2])); f[k,2] = 1); factorback(f); \\ Michel Marcus, Jun 26 2021