A379542 Second term of the n-th differences of the prime numbers.

3, 2, 0, 2, -6, 14, -30, 62, -122, 220, -344, 412, -176, -944, 4112, -11414, 26254, -53724, 100710, -175034, 281660, -410896, 506846, -391550, -401486, 2962260, -9621128, 24977308, -57407998, 120867310, -236098336, 428880422, -719991244, 1096219280

Offset

0,1

Comments

Also the inverse zero-based binomial transform of the odd prime numbers.

Links

Table of n, a(n) for n=0..33.

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * prime(k+2).

Mathematica

nn=40; Table[Differences[Prime[Range[nn+2]], n][[2]], {n, 0, nn}]

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n, k) * prime(k+2)); \\ Michel Marcus, Jan 12 2025

Crossrefs

For all primes (not just odd) we have A007442.

Including 1 in the primes gives A030016.

Column n=2 of A095195.

The version for partitions is A320590 (first column A281425), see A175804, A053445.

For nonprime instead of prime we have A377036, see A377034-A377037.

Arrays of differences: A095195, A376682, A377033, A377038, A377046, A377051.

A000040 lists the primes, differences A001223, A036263.

A002808 lists the composite numbers, differences A073783, A073445.

A008578 lists the noncomposite numbers, differences A075526.

Cf. A064113, A065890, A084758, A140119, A173390, A258025, A258026, A293467, A333214, A333254, A377041.

Sequence in context: A393706 A077814 A131728 * A373088 A075115 A273528

Adjacent sequences: A379539 A379540 A379541 * A379543 A379544 A379545

Keyword

sign

Author

Gus Wiseman, Jan 12 2025

Status

approved