cp's OEIS Frontend

A349159 Numbers whose sum of prime indices is twice their alternating sum.

%I A349159 #8 Dec 10 2021 11:14:20
%S A349159 1,12,63,66,112,190,255,325,408,434,468,609,805,832,931,946,1160,1242,
%T A349159 1353,1380,1534,1539,1900,2035,2067,2208,2296,2387,2414,2736,3055,
%U A349159 3108,3154,3330,3417,3509,3913,4185,4340,4503,4646,4650,4664,4864,5185,5684,5863
%N A349159 Numbers whose sum of prime indices is twice their alternating sum.
%C A349159 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 A349159 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%C A349159 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are also Heinz numbers of partitions whose sum is twice their alternating sum.
%F A349159 A056239(a(n)) = 2*A316524(a(n)).
%F A349159 A346697(a(n)) = 3*A346698(a(n)).
%e A349159 The terms and their prime indices begin:
%e A349159      1: ()
%e A349159     12: (2,1,1)
%e A349159     63: (4,2,2)
%e A349159     66: (5,2,1)
%e A349159    112: (4,1,1,1,1)
%e A349159    190: (8,3,1)
%e A349159    255: (7,3,2)
%e A349159    325: (6,3,3)
%e A349159    408: (7,2,1,1,1)
%e A349159    434: (11,4,1)
%e A349159    468: (6,2,2,1,1)
%e A349159    609: (10,4,2)
%e A349159    805: (9,4,3)
%e A349159    832: (6,1,1,1,1,1,1)
%e A349159    931: (8,4,4)
%e A349159    946: (14,5,1)
%e A349159   1160: (10,3,1,1,1)
%t A349159 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A349159 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A349159 Select[Range[1000],Total[primeMS[#]]==2*ats[primeMS[#]]&]
%Y A349159 These partitions are counted by A000712 up to 0's.
%Y A349159 An ordered version is A348614, negative A349154.
%Y A349159 The negative version is A348617.
%Y A349159 The reverse version is A349160, counted by A006330 up to 0's.
%Y A349159 A025047 counts alternating or wiggly compositions, complement A345192.
%Y A349159 A027193 counts partitions with rev-alt sum > 0, ranked by A026424.
%Y A349159 A034871, A097805, and A345197 count compositions by alternating sum.
%Y A349159 A035363 = partitions with alt sum 0, ranked by A066207, complement A086543.
%Y A349159 A056239 adds up prime indices, row sums of A112798, row lengths A001222.
%Y A349159 A103919 counts partitions by alternating sum, reverse A344612.
%Y A349159 A116406 counts compositions with alternating sum >= 0, ranked by A345913.
%Y A349159 A138364 counts compositions with alternating sum 0, ranked by A344619.
%Y A349159 A325534 counts separable partitions, ranked by A335433.
%Y A349159 A325535 counts inseparable partitions, ranked by A335448.
%Y A349159 A344607 counts partitions with rev-alt sum >= 0, ranked by A344609.
%Y A349159 A346697 adds up odd-indexed prime indices.
%Y A349159 A346698 adds up even-indexed prime indices.
%Y A349159 Cf. A000070, A000290, A001700, A028260, A045931, A120452, A195017, A241638, A257991, A257992, A325698, A345958, A349155.
%K A349159 nonn
%O A349159 1,2
%A A349159 _Gus Wiseman_, Nov 23 2021