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

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, 37, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 89, 92, 97, 98, 99, 100, 101, 102, 103, 105, 107

Offset: 1

Author

Gus Wiseman, May 30 2021

Keywords

• nonn

Comments

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.

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.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

Examples

			The sequence of terms together with their prime indices begins:
      1: {}            20: {1,1,3}         45: {2,2,3}
      2: {1}           23: {9}             47: {15}
      3: {2}           25: {3,3}           48: {1,1,1,1,2}
      4: {1,1}         27: {2,2,2}         49: {4,4}
      5: {3}           28: {1,1,4}         50: {1,3,3}
      7: {4}           29: {10}            52: {1,1,6}
      8: {1,1,1}       30: {1,2,3}         53: {16}
      9: {2,2}         31: {11}            59: {17}
     11: {5}           32: {1,1,1,1,1}     61: {18}
     12: {1,1,2}       36: {1,1,2,2}       63: {2,2,4}
     13: {6}           37: {12}            64: {1,1,1,1,1,1}
     16: {1,1,1,1}     41: {13}            66: {1,2,5}
     17: {7}           42: {1,2,4}         67: {19}
     18: {1,2,2}       43: {14}            68: {1,1,7}
     19: {8}           44: {1,1,5}         70: {1,3,4}
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.
		

Crossrefs

The opposite (nonpositive) version is A028260, counted by A027187.

The strict case (n > 0) is counted by A067659, odd bisection A344650.

Permutations of prime indices of these terms are counted by A116406.

Complement of A119899, Heinz numbers of the partitions counted by A344608.

Positions of nonnegative terms in A316524 or A344617.

Heinz numbers of the partitions counted by A344607.

A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.

A000070 counts partitions with alternating sum 1.

A000097 counts partitions with alternating sum 2.

A056239 adds up prime indices, row sums of A112798.

A103919 counts partitions by sum and alternating sum.

A120452 counts partitions with reverse-alternating sum 2.

A316524 is the alternating sum of the prime indices of n (reverse: A344616).

A335433/A335448 rank separable/inseparable partitions.

A344604 counts wiggly compositions with twins.

A344610 counts partitions by sum and positive reverse-alternating sum.

A344612 counts partitions by sum and reverse-alternating sum.

A344618 gives reverse-alternating sums of standard compositions.

Cf. A001222, A001250, A003242, A005649, A026424, A071321/A071322, A124754, A239829, A343938, A344611, A344651, A344653/A344742, A344739.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[100],ats[primeMS[#]]>=0&]