cp's OEIS Frontend

A358170 Heinz number of the partial sums of the n-th composition in standard order (A066099).

%I A358170 #6 Dec 21 2022 13:18:17
%S A358170 1,2,3,6,5,15,10,30,7,35,21,105,14,70,42,210,11,77,55,385,33,231,165,
%T A358170 1155,22,154,110,770,66,462,330,2310,13,143,91,1001,65,715,455,5005,
%U A358170 39,429,273,3003,195,2145,1365,15015,26,286,182,2002,130,1430,910,10010
%N A358170 Heinz number of the partial sums of the n-th composition in standard order (A066099).
%C A358170 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.
%C A358170 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%H A358170 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%e A358170 The terms together with their prime indices begin:
%e A358170            1: {}
%e A358170            2: {1}
%e A358170            3: {2}
%e A358170            6: {1,2}
%e A358170            5: {3}
%e A358170           15: {2,3}
%e A358170           10: {1,3}
%e A358170           30: {1,2,3}
%e A358170            7: {4}
%e A358170           35: {3,4}
%e A358170           21: {2,4}
%e A358170          105: {2,3,4}
%e A358170           14: {1,4}
%e A358170           70: {1,3,4}
%e A358170           42: {1,2,4}
%e A358170          210: {1,2,3,4}
%t A358170 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A358170 Times@@Prime/@#&/@Table[Accumulate[stc[n]],{n,0,100}]
%Y A358170 See link for sequences related to standard compositions.
%Y A358170 Applying A001221 or A001222 gives A000120.
%Y A358170 The image is A005117 (squarefree numbers).
%Y A358170 The reverse version is A019565, triangular version A048793.
%Y A358170 Greatest prime index of a(n) is A029837 or A070939.
%Y A358170 Least prime index of a(n) is A065120.
%Y A358170 The adjusted version is A253565, inverse A253566, reverse A005940.
%Y A358170 These are the Heinz numbers of the rows of A358134.
%Y A358170 Sum of prime indices of a(n) is A359042.
%Y A358170 A066099 lists standard compositions.
%Y A358170 A112798 list prime indices, sum A056239.
%Y A358170 Cf. A000720, A001511, A029931, A059893, A061395, A241916, A242628, A355536, A358133, A358135, A358137.
%K A358170 nonn
%O A358170 0,2
%A A358170 _Gus Wiseman_, Dec 20 2022