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%I #27 Apr 21 2024 22:25:39
%S 1,1,1,2,1,2,4,6,1,2,4,6,8,12,18,24,1,2,4,6,8,12,18,24,16,24,36,48,54,
%T 72,96,120,1,2,4,6,8,12,18,24,16,24,36,48,54,72,96,120,32,48,72,96,
%U 108,144,192,240,162,216,288,360,384,480,600,720,1,2,4,6,8,12,18,24,16,24,36,48,54,72,96,120,32,48,72,96,108,144,192,240,162,216,288,360,384,480
%N a(n) = A005361(A283477(n)).
%C a(n) is the product of elements of the multiset that covers an initial interval of positive integers with multiplicities equal to the parts of the n-th composition in standard order (graded reverse-lexicographic, A066099). This composition is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. For example, the 13th composition is (1,2,1) giving the multiset {1,2,2,3} with product 12, so a(13) = 12. - _Gus Wiseman_, Apr 26 2020
%H Antti Karttunen, <a href="/A284001/b284001.txt">Table of n, a(n) for n = 0..8191</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F a(n) = A005361(A283477(n)).
%F a(n) = A003963(A057335(n)). - _Gus Wiseman_, Apr 26 2020
%F a(n) = A284005(A053645(n)) for n > 0 with a(0) = 1. - _Mikhail Kurkov_, Jun 05 2021 [verification needed]
%t Table[Times @@ FactorInteger[#][[All, -1]] &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]]], {n, 0, 93}] (* _Michael De Vlieger_, Mar 18 2017 *)
%o (PARI)
%o A005361(n) = factorback(factor(n)[, 2]); \\ From A005361
%o A034386(n) = prod(i=1, primepi(n), prime(i));
%o A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) }; \\ From A108951
%o A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
%o A283477(n) = A108951(A019565(n));
%o A284001(n) = A005361(A283477(n));
%o (Scheme) (define (A284001 n) (A005361 (A283477 n)))
%Y Cf. A005361, A283477, A284002, A284005.
%Y Row products of A095684.
%Y All of the following pertain to compositions in standard order (A066099):
%Y - Length is A000120.
%Y - Weighted sum is A029931.
%Y - Necklaces are A065609.
%Y - Sum is A070939.
%Y - Strict compositions are A233564.
%Y - Constant compositions are A272919.
%Y - Lyndon words are A275692.
%Y - Distinct parts are counted by A334028.
%Y Cf. A003963, A034691, A053645, A057335, A305936, A318284, A333764, A333940, A333942, A334030.
%K nonn,look
%O 0,4
%A _Antti Karttunen_, Mar 18 2017
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