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A335404 Numbers k such that the k-th composition in standard order (A066099) has the same product as sum.

%I A335404 #6 Jun 09 2020 07:23:13
%S A335404 1,2,4,8,10,16,32,37,38,41,44,50,52,64,128,139,141,142,163,171,173,
%T A335404 174,177,181,182,184,186,197,198,209,213,214,216,218,226,232,234,256,
%U A335404 295,307,313,316,403,409,412,457,460,484,512,535,539,541,542,647,707,737
%N A335404 Numbers k such that the k-th composition in standard order (A066099) has the same product as sum.
%C A335404 The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
%H A335404 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%F A335404 A124758(a(n)) = A070939(a(n)).
%e A335404 The sequence together with the corresponding compositions begins:
%e A335404     1: (1)
%e A335404     2: (2)
%e A335404     4: (3)
%e A335404     8: (4)
%e A335404    10: (2,2)
%e A335404    16: (5)
%e A335404    32: (6)
%e A335404    37: (3,2,1)
%e A335404    38: (3,1,2)
%e A335404    41: (2,3,1)
%e A335404    44: (2,1,3)
%e A335404    50: (1,3,2)
%e A335404    52: (1,2,3)
%e A335404    64: (7)
%e A335404   128: (8)
%e A335404   139: (4,2,1,1)
%e A335404   141: (4,1,2,1)
%e A335404   142: (4,1,1,2)
%e A335404   163: (2,4,1,1)
%e A335404   171: (2,2,2,1,1)
%t A335404 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A335404 Select[Range[0,100],Times@@stc[#]==Plus@@stc[#]&]
%Y A335404 The lengths of standard compositions are given by A000120.
%Y A335404 Sum of binary indices is A029931.
%Y A335404 Sum of prime indices is A056239.
%Y A335404 Sum of standard compositions is A070939.
%Y A335404 Product of standard compositions is A124758.
%Y A335404 Taking GCD instead of product gives A131577.
%Y A335404 The version for prime indices is A301987.
%Y A335404 The version for prime indices of nonprime numbers is A301988.
%Y A335404 These compositions are counted by A335405.
%Y A335404 Cf. A001055, A003963, A066099, A096111, A124767, A228351, A233249, A272919, A333219, A333220, A331579.
%K A335404 nonn
%O A335404 1,2
%A A335404 _Gus Wiseman_, Jun 06 2020