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A105965 Numbers n such that n = 2^i_1+2^i_2+...2^i_k = b(j_1)*b(j_2)*...b(j_k) for distinct i's and distinct j's, where b is A050376.

%I A105965 #8 Sep 24 2023 04:32:09
%S A105965 2,4,6,10,12,16,18,20,33,34,36,42,48,56,65,68,70,80,84,88,104,120,129,
%T A105965 138,140,144,152,200,210,216,224,256,266,270,272,273,276,290,296,312,
%U A105965 322,328,330,336,352,360,385,390,392,408,416,420,448,456,480,514,518
%N A105965 Numbers n such that n = 2^i_1+2^i_2+...2^i_k = b(j_1)*b(j_2)*...b(j_k) for distinct i's and distinct j's, where b is A050376.
%C A105965 May be called ambipartite additive-multiplicative numbers.
%C A105965 If the exponents in the prime factorization of n are a_1, a_2, ..., a_k, then n is in this sequence iff A000120(n) = sum_{i = 1..k} A000120(a_i).
%C A105965 Numbers n such that A000120(n)=A064547(n).
%C A105965 Numbers n such that n=2^i_1+2^i_2+...2^i_k=b(j_1)*b(j_2)*...b(j_k) for distinct i's and distinct j's, where b is A050376. For all i's = j's, n = A052330(n)= 4, 36, ...? - _Thomas Ordowski_, May 11 2005
%e A105965 16=2^4=2^(2^2), 33=1+32=3*11, 42=2+8+32=2*3*7, 120=8+16+32+64=2*3*4*5.
%e A105965 2 = 2^1 = 2^(2^0)
%e A105965 4 = 2^2 = 2^(2^1)
%e A105965 6 = 2 + 4 = 2 * 3
%e A105965 10 = 2 + 8 = 2 * 5
%e A105965 12 = 4 + 8 = 3 * 4
%e A105965 16 = 2^4 = 2^(2^2)
%e A105965 18 = 2 + 16 = 2 * 9
%e A105965 20 = 4 + 16 = 4 * 5
%e A105965 33 = 1 + 32 = 3 * 11
%e A105965 34 = 2 + 32 = 2 * 17
%e A105965 36 = 4 + 32 = 4 * 9
%e A105965 42 = 2 + 8 + 32 = 2 * 3 * 7
%e A105965 48 = 16 + 32 = 3 * 16
%e A105965 56 = 8 + 16 + 32 = 2 * 4 * 7
%e A105965 65 = 1 + 64 = 5 * 13
%e A105965 68 = 4 + 64 = 4 * 17
%e A105965 70 = 2 + 4 + 64 = 2 * 5 * 7
%e A105965 80 = 16 + 64 = 5 * 16
%e A105965 84 = 4 + 16 + 64 = 3 * 4 * 7
%e A105965 88 = 8 + 16 + 64 = 2 * 4 * 11
%e A105965 104 = 8 + 32 + 64 = 2 * 4 * 13
%e A105965 120 = 8 + 16 + 32 + 64 = 2 * 3 * 4 * 5
%o A105965 (PARI) f(n) =if (n, n%2 + f(n\2), 0);
%o A105965 g(n) = local(a); a = factor(n); f(n) == sum(i = 1, matsize(a)[1], f(a[i, 2]));
%o A105965 for (n = 1, 1000, if (g(n), print1(n, ", "))); \\ _David Wasserman_, Apr 29 2005
%Y A105965 Cf. A000120.
%Y A105965 Cf. A052330, A000120 and A064547.
%K A105965 nonn
%O A105965 1,1
%A A105965 _Thomas Ordowski_, Apr 28 2005
%E A105965 More terms from _David Wasserman_, Apr 29 2005
%E A105965 Examples from _Thomas Ordowski_, May 11 2005