A377033 - OEIS

A377033

Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the composite numbers (A002808).

4, 6, 2, 8, 2, 0, 9, 1, -1, -1, 10, 1, 0, 1, 2, 12, 2, 1, 1, 0, -2, 14, 2, 0, -1, -2, -2, 0, 15, 1, -1, -1, 0, 2, 4, 4, 16, 1, 0, 1, 2, 2, 0, -4, -8, 18, 2, 1, 1, 0, -2, -4, -4, 0, 8, 20, 2, 0, -1, -2, -2, 0, 4, 8, 8, 0, 21, 1, -1, -1, 0, 2, 4, 4, 0, -8, -16, -16

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OFFSET

0,1

COMMENTS

Row n is the k-th differences of A002808 = the composite numbers.

LINKS

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

FORMULA

A(i,j) = sum_{k=0..j} (-1)^(j-k) binomial(j,k) A002808(i+k).

EXAMPLE

Array begins:

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

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

k=0: 4 6 8 9 10 12 14 15 16

k=1: 2 2 1 1 2 2 1 1 2

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

k=3: -1 1 1 -1 -1 1 1 -1 -1

k=4: 2 0 -2 0 2 0 -2 0 2

k=5: -2 -2 2 2 -2 -2 2 2 -2

k=6: 0 4 0 -4 0 4 0 -4 -1

k=7: 4 -4 -4 4 4 -4 -4 3 10

k=8: -8 0 8 0 -8 0 7 7 -29

k=9: 8 8 -8 -8 8 7 0 -36 63

Triangle begins:

6 2

8 2 0

9 1 -1 -1

10 1 0 1 2

12 2 1 1 0 -2

14 2 0 -1 -2 -2 0

15 1 -1 -1 0 2 4 4

16 1 0 1 2 2 0 -4 -8

18 2 1 1 0 -2 -4 -4 0 8

20 2 0 -1 -2 -2 0 4 8 8 0

21 1 -1 -1 0 2 4 4 0 -8 -16 -16

MATHEMATICA

nn=9;

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

CROSSREFS

Initial rows: A002808, A073783, A073445.

The version for primes is A095195 or A376682.

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

Triangle row-sums are A377034, absolute version A377035.

Column n = 1 is A377036, for primes A007442 or A030016.

First position of 0 in each row is A377037.

Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).

A000040 lists the primes, differences A001223, seconds A036263.

A008578 lists the noncomposites, differences A075526.

Cf. A065310, A065890, A084758, A173390, A350004, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680.

Sequence in context: A153124 A066252 A065165 * A088516 A114538 A181096

Adjacent sequences: A377030 A377031 A377032 * A377034 A377035 A377036

KEYWORD

sign,tabl

AUTHOR

Gus Wiseman, Oct 17 2024

STATUS

approved