# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a124794 Showing 1-1 of 1 %I A124794 #34 Oct 16 2023 14:39:17 %S A124794 1,1,1,1,1,3,1,1,3,4,1,6,1,5,10,1,1,15,1,10,15,6,1,10,10,7,15,15,1,60, %T A124794 1,1,21,8,35,45,1,9,28,20,1,105,1,21,105,10,1,15,35,70,36,28,1,105,56, %U A124794 35,45,11,1,210,1,12,210,1,84,168,1,36,55,280,1,105,1,13,280,45,126,252,1 %N A124794 Coefficients of incomplete Bell polynomials in the prime factorization order. %C A124794 Coefficients of (D^k f)(g(t))*(D g(t))^k1*(D^2 g(t))^k2*... in the Faa di Bruno formula for D^m(f(g(t))) where k = k1 + k2 + ..., m = 1*k1 + 2*k2 + .... %C A124794 Number of set partitions whose block sizes are the prime indices of n (i.e., the integer partition with Heinz number n). - _Gus Wiseman_, Sep 12 2018 %H A124794 Alois P. Heinz, Table of n, a(n) for n = 1..20000 %H A124794 Eric Weisstein's World of Mathematics, Bell Polynomial %H A124794 Eric Weisstein's World of Mathematics, Faà di Bruno's Formula %F A124794 For n = p1^k1*p2^k2*... where 2 = p1 < p2 < ... are the sequence of all primes, a(n) = a([k1,k2,...]) = (k1+2*k2+...)!/((k1!*k2!*...)*(1!^k1*2!^k2*...)). %F A124794 a(2*prime(n)) = n + 1, for n > 1. See A065475. - _Bill McEachen_, Oct 11 2023 %e A124794 The a(6) = 3 set partitions of type (2,1) are {{1},{2,3}}, {{1,3},{2}}, {{1,2},{3}}. - _Gus Wiseman_, Sep 12 2018 %p A124794 with(numtheory): %p A124794 a:= n-> (l-> add(i*l[i], i=1..nops(l))!/mul(l[i]!*i!^l[i], %p A124794 i=1..nops(l)))([seq(padic[ordp](n, ithprime(i)), %p A124794 i=1..pi(max(1, factorset(n))))]): %p A124794 seq(a(n), n=1..100); # _Alois P. Heinz_, Feb 14 2020 %t A124794 numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn]; %t A124794 Table[numSetPtnsOfType[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,100}] (* _Gus Wiseman_, Sep 12 2018 *) %o A124794 (PARI) a(n) = my(f=factor(n)); sum(k=1, #f~, primepi(f[k,1])*f[k,2])!/(prod(k=1, #f~, f[k,2]!)*prod(k=1, #f~, primepi(f[k,1])!^f[k,2])); \\ _Michel Marcus_, Oct 11 2023 %Y A124794 Cf. A000110, A000258, A000670, A005651, A008277, A008480, A056239, A094416, A124794, A215366, A318762, A319182, A319225. %Y A124794 Cf. A065475, A100484. %K A124794 nonn %O A124794 1,6 %A A124794 _Max Alekseyev_, Nov 07 2006 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE