A377049 - OEIS

A377049

First term of the n-th differences of the nonsquarefree numbers. Inverse zero-based binomial transform of A013929.

4, 4, -3, 5, -6, 4, 3, -15, 25, -10, -84, 369, -1067, 2610, -5824, 12246, -24622, 47577, -88233, 155962, -259086, 393455, -512281, 456609, 191219, -2396571, 8213890, -21761143, 50923029, -110269263, 225991429, -444168664, 844390152, -1561482492, 2817844569

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OFFSET

0,1

LINKS

Harvey P. Dale, Table of n, a(n) for n = 0..400

FORMULA

The inverse zero-based binomial transform of a sequence (q(0), q(1), q(2), ...) is the sequence p given by:

p(j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) q(k)

MATHEMATICA

nn=20;

Table[First[Differences[NestList[NestWhile[#+1&, #+1, SquareFreeQ[#]&]&, 4, 2*nn], k]], {k, 0, nn}]

With[{nsf=Select[Range[1000], !SquareFreeQ[#]&]}, Table[Differences[nsf, n], {n, 0, 40}]][[;; , 1]] (* Harvey P. Dale, Nov 28 2024 *)

CROSSREFS

The version for primes is A007442, noncomposites A030016, composites A377036.

For squarefree instead of nonsquarefree numbers we have A377041.

For antidiagonal-sums we have A377047, absolute A377048.

For first position of 0 in each row we have A377050.

A000040 lists the primes, differences A001223, seconds A036263.

A005117 lists the squarefree numbers.

A013929 lists the nonsquarefree numbers, differences A078147, seconds A376593.

A073576 counts integer partitions into squarefree numbers, factorizations A050320.

Cf. A007674, A053797, A053806, A061398, A072284, A075526, A084758, A112925, A120992, A251092, A376311, A376591.

Sequence in context: A073321 A055620 A072420 * A258075 A286296 A023530

Adjacent sequences: A377046 A377047 A377048 * A377050 A377051 A377052

KEYWORD

sign,changed

AUTHOR

Gus Wiseman, Oct 19 2024

STATUS

approved