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

Author

Gus Wiseman, Jan 12 2025

Keywords

• sign

Comments

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

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.

Programs

Mathematica

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

PARI

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

Formula

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