A357851 - OEIS

A357851

Numbers k such that the half-alternating sum of the prime indices of k is 1.

2, 8, 18, 32, 45, 50, 72, 98, 105, 128, 162, 180, 200, 231, 242, 275, 288, 338, 392, 420, 429, 450, 455, 512, 578, 648, 663, 720, 722, 800, 833, 882, 924, 935, 968, 969, 1050, 1058, 1100, 1125, 1152, 1235, 1250, 1311, 1352, 1458, 1463, 1568, 1680, 1682, 1716

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OFFSET

1,1

COMMENTS

We define the half-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..51.

EXAMPLE

The terms together with their prime indices begin:

2: {1}

8: {1,1,1}

18: {1,2,2}

32: {1,1,1,1,1}

45: {2,2,3}

50: {1,3,3}

72: {1,1,1,2,2}

98: {1,4,4}

105: {2,3,4}

128: {1,1,1,1,1,1,1}

162: {1,2,2,2,2}

180: {1,1,2,2,3}

200: {1,1,1,3,3}

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];

Select[Range[1000], halfats[primeMS[#]]==1&]

CROSSREFS

The version for k = 0 is A357631, standard compositions A357625-A357626.

The version for original alternating sum is A001105.

Positions of ones in A357629, reverse A357633.

The skew version for k = 0 is A357632, reverse A357636.

Partitions with these Heinz numbers are counted by A035444, skew A035544.

The reverse version is A357635, k = 0 version A000583.

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-length A357642.

Cf. A003963, A053251, A055932, A345958, A357621-A357624, A357630, A357634, A357637, A357639, A357640.

Sequence in context: A293296 A055044 A356209 * A067051 A074629 A209303

Adjacent sequences: A357848 A357849 A357850 * A357852 A357853 A357854

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 28 2022

STATUS

approved