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 A045778 Number of factorizations of n into distinct factors greater than 1. 155
 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 5, 1, 3, 2, 2, 2, 5, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 4, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 2, 2, 1, 9, 2, 2, 2, 5, 1, 9, 2, 3, 2, 2, 2, 10, 1, 3, 3, 5, 1, 5, 1, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS This sequence depends only on the prime signature of n and not on the actual value of n. Also the number of strict multiset partitions (sets of multisets) of the prime factors of n. - Gus Wiseman, Dec 03 2016 LINKS David W. Wilson, Table of n, a(n) for n = 1..10000 Philippe A. J. G. Chevalier, On the discrete geometry of physical quantities, Preprint, 2012. P. A. J. G. Chevalier, On a Mathematical Method for Discovering Relations Between Physical Quantities: a Photonics Case Study, Slides from a talk presented at ICOL2014. P. A. J. G. Chevalier, A "table of Mendeleev" for physical quantities?, Slides from a talk, May 14 2014, Leuven, Belgium. A. Knopfmacher, M. Mays, Ordered and Unordered Factorizations of Integers: Unordered Factorizations with Distinct Parts, The Mathematica Journal 10(1), 2006. R. J. Mathar, Factorizations of n =1..1500 Eric Weisstein's World of Mathematics, Unordered Factorization FORMULA Dirichlet g.f.: Product_{n>=2}(1 + 1/n^s). Let p and q be two distinct prime numbers and k a natural number. Then a(p^k) = A000009(k) and a(p^k*q) = A036469(k). - Alexander Adam, Dec 28 2012 Let p_i with 1<=i<=k k distinct prime numbers. Then a(Product_{i=1..k} p_i) = A000110(k). - Alexander Adam, Dec 28 2012 EXAMPLE 24 can be factored as 24, 2*12, 3*8, 4*6, or 2*3*4, so a(24) = 5. The factorization 2*2*6 is not permitted because the factor 2 is present twice. a(1) = 1 represents the empty factorization. MAPLE with(numtheory): b:= proc(n, k) option remember;       `if`(n>k, 0, 1) +`if`(isprime(n), 0,       add(`if`(d>k, 0, b(n/d, d-1)), d=divisors(n) minus {1, n}))     end: a:= n-> b(n\$2): seq(a(n), n=1..120);  # Alois P. Heinz, May 26 2013 MATHEMATICA gd[m_, 1] := 1; gd[1, n_] := 0; gd[1, 1] := 1; gd[0, n_] := 0; gd[m_, n_] := gd[m, n] = Total[gd[# - 1, n/#] & /@ Select[Divisors[n], # <= m &]]; Array[ gd[#, #] &, 100]  (* Alexander Adam, Dec 28 2012 *) PROG (PARI) v=vector(100, k, k==1); for(n=2, #v, v+=dirmul(v, vector(#v, k, k==n)) ); v /* Max Alekseyev, Jul 16 2014 */ (PARI) A045778aux(n, k) = ((n<=k) + sumdiv(n, d, if(d > 1 && d <= k && d < n, A045778aux(n/d, d-1)))); A045778(n) = A045778aux(n, n); \\ After Alois P. Heinz's Maple-code by Antti Karttunen, Jul 23 2017 (Python) from sympy.core.cache import cacheit from sympy import divisors, isprime @cacheit def b(n, k): return (0 if n>k else 1) + (0 if isprime(n) else sum(0 if d>k else b(n//d, d - 1) for d in divisors(n)[1:-1])) def a(n): return b(n, n) print([a(n) for n in range(1, 121)]) # Indranil Ghosh, Aug 19 2017, after Maple code CROSSREFS Cf. A001055, A045779, A045780, A050323. a(p^k)=A000009. a(A002110) = A000110. Cf. A036469, A114591, A114592, A316441 (Dirichlet inverse). Sequence in context: A316364 A318357 A323091 * A320889 A296133 A033103 Adjacent sequences:  A045775 A045776 A045777 * A045779 A045780 A045781 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Edited by Franklin T. Adams-Watters, Jun 04 2009 STATUS approved

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Last modified September 21 03:13 EDT 2020. Contains 337266 sequences. (Running on oeis4.)