A357632 Numbers k such that the skew-alternating sum of the prime indices of k is 0.

1, 4, 9, 16, 25, 36, 49, 64, 81, 90, 100, 121, 144, 169, 196, 210, 225, 256, 289, 324, 360, 361, 400, 441, 462, 484, 525, 529, 550, 576, 625, 676, 729, 784, 840, 841, 858, 900, 910, 961, 1024, 1089, 1155, 1156, 1225, 1296, 1326, 1369, 1440, 1444, 1521, 1600

Offset

1,2

Comments

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....

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.

Links

Table of n, a(n) for n=1..52.

Example

The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     9: {2,2}
    16: {1,1,1,1}
    25: {3,3}
    36: {1,1,2,2}
    49: {4,4}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
    90: {1,2,2,3}
   100: {1,1,3,3}
   121: {5,5}
   144: {1,1,1,1,2,2}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}];
Select[Range[1000], skats[primeMS[#]]==0&]

Crossrefs

The version for original alternating sum is A000290.

The version for standard compositions is A357627, reverse A357628.

Positions of zeros in A357630, reverse A357634.

The half-alternating form is A357631, reverse A357635.

The reverse version is A357636.

These partitions are counted by A357640, half A357639.

A056239 adds up prime indices, row sums of A112798.

A316524 gives alternating sum of prime indices, reverse A344616.

A351005 = alternately equal and unequal partitions, compositions A357643.

A351006 = alternately unequal and equal partitions, compositions A357644.

A357641 counts comps w/ half-alt sum 0, even A357642.

Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357621-A357626, A357629, A357637, A357638.

Sequence in context: A035121 A080151 A292679 * A106545 A169920 A093837

Adjacent sequences: A357629 A357630 A357631 * A357633 A357634 A357635

Keyword

nonn

Author

Gus Wiseman, Oct 09 2022

Status

approved