A377046 - OEIS

A377046

Array read by downward antidiagonals where A(n,k) is the n-th term of the k-th differences of nonsquarefree numbers.

4, 8, 4, 9, 1, -3, 12, 3, 2, 5, 16, 4, 1, -1, -6, 18, 2, -2, -3, -2, 4, 20, 2, 0, 2, 5, 7, 3, 24, 4, 2, 2, 0, -5, -12, -15, 25, 1, -3, -5, -7, -7, -2, 10, 25, 27, 2, 1, 4, 9, 16, 23, 25, 15, -10, 28, 1, -1, -2, -6, -15, -31, -54, -79, -94, -84, 32, 4, 3, 4, 6, 12, 27, 58, 112, 191, 285, 369

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OFFSET

0,1

COMMENTS

Row k is the k-th differences of A013929.

LINKS

Table of n, a(n) for n=0..77.

FORMULA

A(i,j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) A013929(i+k).

EXAMPLE

Array form:

n=1: n=2: n=3: n=4: n=5: n=6: n=7: n=8: n=9:

---------------------------------------------------------

k=0: 4 8 9 12 16 18 20 24 25

k=1: 4 1 3 4 2 2 4 1 2

k=2: -3 2 1 -2 0 2 -3 1 -1

k=3: 5 -1 -3 2 2 -5 4 -2 4

k=4: -6 -2 5 0 -7 9 -6 6 -7

k=5: 4 7 -5 -7 16 -15 12 -13 10

k=6: 3 -12 -2 23 -31 27 -25 23 -13

k=7: -15 10 25 -54 58 -52 48 -36 13

k=8: 25 15 -79 112 -110 100 -84 49 1

k=9: -10 -94 191 -222 210 -184 133 -48 -57

Triangle form:

8 4

9 1 -3

12 3 2 5

16 4 1 -1 -6

18 2 -2 -3 -2 4

20 2 0 2 5 7 3

24 4 2 2 0 -5 -12 -15

25 1 -3 -5 -7 -7 -2 10 25

27 2 1 4 9 16 23 25 15 -10

28 1 -1 -2 -6 -15 -31 -54 -79 -94 -84

32 4 3 4 6 12 27 58 112 191 285 369

MATHEMATICA

nn=9;

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

Table[t[[j, i-j+1]], {i, nn}, {j, i}]

CROSSREFS

Initial rows: A013929, A078147, A376593.

The version for primes is A095195, noncomposites A376682, composites A377033.

A version for partitions is A175804, cf. A053445, A281425, A320590.

For squarefree numbers we have A377038, sums A377039, absolute A377040.

Triangle row-sums are A377047, absolute version A377048.

Column n = 1 is A377049, for squarefree A377041, for prime A007442 or A030016.

First position of 0 in each row is A377050.

For prime-power instead of nonsquarefree we have A377051.

A000040 lists the primes, differences A001223, seconds A036263.

A005117 lists the squarefree numbers.

A073576 counts integer partitions into squarefree numbers, factorizations A050320.

Cf. A000961, A007674, A053797, A053806, A061398, A072284, A112925, A120992, A376311, A376591, A376592.

Sequence in context: A059163 A091198 A200641 * A245780 A165267 A092159

Adjacent sequences: A377043 A377044 A377045 * A377047 A377048 A377049

KEYWORD

sign,tabl

AUTHOR

Gus Wiseman, Oct 19 2024

STATUS

approved