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A324383 a(n) is the minimal number of primorials that add to A322827(n).

%I A324383 #24 Feb 28 2019 18:54:05
%S A324383 1,1,1,2,2,1,2,2,2,6,1,6,4,2,4,4,8,6,6,10,8,1,10,22,4,6,2,12,8,4,4,2,
%T A324383 8,16,6,4,24,6,8,14,26,18,1,26,20,6,18,30,6,12,2,14,16,2,10,16,8,6,4,
%U A324383 8,6,2,4,4,12,14,14,18,18,12,16,32,42,28,6,22,32,24,24,42,46,32,18,20,30,1,24,54,38,26,14,44,34,8
%N A324383 a(n) is the minimal number of primorials that add to A322827(n).
%C A324383 a(n) is odd if and only if n is one of the terms of A000975: 1, 2, 5, 10, 21, 42, 85, ..., in which case A322827(n) will be one of primorials (A002110), and a(n) = 1. This happens because A276150 is even for all multiples of four, and a product of two or more primorials > 1 is always a multiple of 4. Note that the same property does not hold in factorial system: 36 = 3!*3!, but A034968(36) = 3 as 36 = 4!+3!+3!.
%H A324383 Antti Karttunen, <a href="/A324383/b324383.txt">Table of n, a(n) for n = 0..10922</a>
%H A324383 Antti Karttunen, <a href="/A324383/a324383.txt">Data supplement: n, a(n) computed for n = 0..65537</a>
%H A324383 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H A324383 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%H A324383 <a href="/index/Pri#primorial_numbers">Index entries for sequences related to primorial numbers</a>
%F A324383 a(n) = A276150(A322827(n)).
%F A324383 a(n) = A324386(A003188(n)).
%o A324383 (PARI)
%o A324383 A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
%o A324383 A322827(n) = if(!n,1,my(bits = Vecrev(binary(n)), rl=1, o = List([])); for(i=2,#bits,if(bits[i]==bits[i-1], rl++, listput(o,rl))); listput(o,rl); my(es=Vecrev(Vec(o)), m=1); for(i=1,#es,m *= prime(i)^es[i]); (m));
%o A324383 A324383(n) = A276150(A322827(n));
%Y A324383 Cf. A000975 (positions of ones), A002110, A003188, A025487, A276150, A322827, A324342, A324382.
%Y A324383 Cf. also A324386, A324387 (permutations of this sequence).
%K A324383 nonn
%O A324383 0,4
%A A324383 _Antti Karttunen_, Feb 27 2019