cp's OEIS Frontend

A363621 Positive integers whose prime indices have reverse-weighted alternating sum 0.

%I A363621 #10 Jun 15 2023 07:32:55
%S A363621 1,6,21,40,50,54,65,132,133,154,210,224,319,340,351,360,374,392,450,
%T A363621 481,486,507,546,598,624,644,731,825,855,969,1007,1029,1054,1144,1210,
%U A363621 1254,1320,1364,1386,1403,1408,1520,1558,1653,1750,1785,1827,1836,1890,1960
%N A363621 Positive integers whose prime indices have reverse-weighted alternating sum 0.
%C A363621 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A363621 We define the reverse-weighted alternating sum of a sequence (y_1,...,y_k) to be Sum_{i=1..k} (-1)^(k-i) i * y_{k-i+1}.
%e A363621 The prime indices of 360 are {1,1,1,2,2,3}, with reverse-weighted alternating sum 1*3 - 2*2 + 3*2 - 4*1 + 5*1 - 6*1 = 0, so 360 is in the sequence.
%e A363621 The terms together with their prime indices begin:
%e A363621      1: {}
%e A363621      6: {1,2}
%e A363621     21: {2,4}
%e A363621     40: {1,1,1,3}
%e A363621     50: {1,3,3}
%e A363621     54: {1,2,2,2}
%e A363621     65: {3,6}
%e A363621    132: {1,1,2,5}
%e A363621    133: {4,8}
%e A363621    154: {1,4,5}
%e A363621    210: {1,2,3,4}
%e A363621    224: {1,1,1,1,1,4}
%e A363621    319: {5,10}
%e A363621    340: {1,1,3,7}
%e A363621    351: {2,2,2,6}
%e A363621    360: {1,1,1,2,2,3}
%t A363621 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A363621 revaltwtsum[y_]:=Sum[(-1)^(Length[y]-k)*k*y[[-k]],{k,1,Length[y]}];
%t A363621 Select[Range[1000],revaltwtsum[prix[#]]==0&]
%Y A363621 The unweighted version is A000290.
%Y A363621 Partitions of this type are counted by A363532.
%Y A363621 Positions of zeros in A363620 and A363624, reverse A363619 and A363625.
%Y A363621 Compositions of this type are counted by A363626.
%Y A363621 A053632 counts compositions by weighted sum.
%Y A363621 A055396 gives minimum prime index, maximum A061395.
%Y A363621 A112798 lists prime indices, length A001222, sum A056239.
%Y A363621 A264034 counts partitions by weighted sum, reverse A358194.
%Y A363621 A304818 gives weighted sum of prime indices.
%Y A363621 A318283 gives weighted sum of reversed prime indices.
%Y A363621 A320387 counts multisets by weighted sum.
%Y A363621 A344616 gives reverse-alternating sum of prime indices.
%Y A363621 A363623 counts partitions by reverse-weighted alternating sum.
%Y A363621 Cf. A046660, A106529, A124010, A181819, A261079, A316524, A359674, A363622.
%K A363621 nonn
%O A363621 1,2
%A A363621 _Gus Wiseman_, Jun 13 2023