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 A295279 Number of strict tree-factorizations of n. 12
 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 10, 1, 2, 2, 4, 1, 8, 1, 6, 2, 2, 2, 12, 1, 2, 2, 10, 1, 8, 1, 4, 4, 2, 1, 26, 1, 4, 2, 4, 1, 10, 2, 10, 2, 2, 1, 28, 1, 2, 4, 12, 2, 8, 1, 4, 2, 8, 1, 44, 1, 2, 4, 4, 2, 8, 1, 26, 3, 2, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS A strict tree-factorization of n is either (case 1) the number n itself or (case 2) a set of two or more strict tree-factorizations, one of each factor in a factorization of n into distinct factors greater than one. a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..10000 FORMULA a(product of n distinct primes) = A005804(n). a(prime^n) = A273873(n). Dirichlet g.f.: (Zeta(s) + Product_{n >= 2}(1 + a(n)/n^s))/2. EXAMPLE The a(30) = 8 strict tree-factorizations are: 30, (2*3*5), (2*15), (2*(3*5)), (3*10), (3*(2*5)), (5*6), (5*(2*3)). The a(36) = 12 strict tree-factorizations are: 36, (2*3*6), (2*3*(2*3)), (2*18), (2*(2*9)), (2*(3*6)), (2*(3*(2*3))), (3*12), (3*(2*6)), (3*(2*(2*3))), (3*(3*4)), (4*9). MATHEMATICA sfs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[sfs[n/d], Min@@#>d&]], {d, Rest[Divisors[n]]}]]; sft[n_]:=1+Total[Function[fac, Times@@sft/@fac]/@Select[sfs[n], Length[#]>1&]]; Array[sft, 100] PROG (PARI) seq(n)={my(v=vector(n), w=vector(n)); w[1]=v[1]=1; for(k=2, n, w[k]=v[k]+1; forstep(j=n\k*k, k, -k, v[j]+=w[k]*v[j/k])); w} \\ Andrew Howroyd, Nov 18 2018 CROSSREFS Cf. A005804, A045778, A273873, A281113 A281118, A292504, A294786, A295281. Sequence in context: A303386 A295636 A050334 * A316784 A284974 A293222 Adjacent sequences:  A295276 A295277 A295278 * A295280 A295281 A295282 KEYWORD nonn AUTHOR Gus Wiseman, Nov 19 2017 STATUS approved

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Last modified September 20 20:02 EDT 2020. Contains 337265 sequences. (Running on oeis4.)