cp's OEIS Frontend

A344609 Numbers whose alternating sum of prime indices is >= 0.

%I A344609 #15 Jun 09 2021 06:24:06
%S A344609 1,2,3,4,5,7,8,9,11,12,13,16,17,18,19,20,23,25,27,28,29,30,31,32,36,
%T A344609 37,41,42,43,44,45,47,48,49,50,52,53,59,61,63,64,66,67,68,70,71,72,73,
%U A344609 75,76,78,79,80,81,83,89,92,97,98,99,100,101,102,103,105,107
%N A344609 Numbers whose alternating sum of prime indices is >= 0.
%C A344609 Also Heinz numbers of partitions whose reverse-alternating sum is >= 0. These are partitions whose conjugate parts are all even or whose length is odd.
%C A344609 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 A344609 The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.
%e A344609 The sequence of terms together with their prime indices begins:
%e A344609       1: {}            20: {1,1,3}         45: {2,2,3}
%e A344609       2: {1}           23: {9}             47: {15}
%e A344609       3: {2}           25: {3,3}           48: {1,1,1,1,2}
%e A344609       4: {1,1}         27: {2,2,2}         49: {4,4}
%e A344609       5: {3}           28: {1,1,4}         50: {1,3,3}
%e A344609       7: {4}           29: {10}            52: {1,1,6}
%e A344609       8: {1,1,1}       30: {1,2,3}         53: {16}
%e A344609       9: {2,2}         31: {11}            59: {17}
%e A344609      11: {5}           32: {1,1,1,1,1}     61: {18}
%e A344609      12: {1,1,2}       36: {1,1,2,2}       63: {2,2,4}
%e A344609      13: {6}           37: {12}            64: {1,1,1,1,1,1}
%e A344609      16: {1,1,1,1}     41: {13}            66: {1,2,5}
%e A344609      17: {7}           42: {1,2,4}         67: {19}
%e A344609      18: {1,2,2}       43: {14}            68: {1,1,7}
%e A344609      19: {8}           44: {1,1,5}         70: {1,3,4}
%e A344609 For example, the prime indices of 70 are {1,3,4} with alternating sum 1 - 3 + 4 = 2, so 70 is in the sequence. On the other hand, the prime indices of 24 are {1,1,1,2} with alternating sum 1 - 1 + 1 - 2 = -1, so 24 is not in the sequence.
%t A344609 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A344609 ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
%t A344609 Select[Range[100],ats[primeMS[#]]>=0&]
%Y A344609 The opposite (nonpositive) version is A028260, counted by A027187.
%Y A344609 The strict case (n > 0) is counted by A067659, odd bisection A344650.
%Y A344609 Permutations of prime indices of these terms are counted by A116406.
%Y A344609 Complement of A119899, Heinz numbers of the partitions counted by A344608.
%Y A344609 Positions of nonnegative terms in A316524 or A344617.
%Y A344609 Heinz numbers of the partitions counted by A344607.
%Y A344609 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A344609 A000070 counts partitions with alternating sum 1.
%Y A344609 A000097 counts partitions with alternating sum 2.
%Y A344609 A056239 adds up prime indices, row sums of A112798.
%Y A344609 A103919 counts partitions by sum and alternating sum.
%Y A344609 A120452 counts partitions with reverse-alternating sum 2.
%Y A344609 A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A344609 A335433/A335448 rank separable/inseparable partitions.
%Y A344609 A344604 counts wiggly compositions with twins.
%Y A344609 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A344609 A344612 counts partitions by sum and reverse-alternating sum.
%Y A344609 A344618 gives reverse-alternating sums of standard compositions.
%Y A344609 Cf. A001222, A001250, A003242, A005649, A026424, A071321/A071322, A124754, A239829, A343938, A344611, A344651, A344653/A344742, A344739.
%K A344609 nonn
%O A344609 1,2
%A A344609 _Gus Wiseman_, May 30 2021