A096278 Sums of successive sums of successive sums of successive primes.

33, 50, 72, 96, 120, 144, 172, 206, 240, 274, 308, 336, 364, 402, 444, 480, 514, 548, 578, 610, 648, 692, 742, 786, 816, 840, 864, 900, 960, 1024, 1070, 1108, 1152, 1196, 1236, 1278, 1320, 1362, 1404, 1444, 1488, 1530, 1560, 1592, 1650, 1728, 1790, 1824

Offset: 1

Author

Cino Hilliard, Jun 22 2004

Keywords

• easy
• nonn

Comments

If we consider the m-fold iterated "take sums of successive terms" operation acting on the primes, then for all m >= 1, the first term is always odd (and the only odd term); it is prime for m=1, 2, 4, 8, 21, 24, 27, 31, 40, 98,..., but not for m=3 (the present sequence). [Edited by M. F. Hasler, Jun 02 2017]

Examples

			The first two terms of SS order 1 is 13 and 20. 13+20 = 33 the first term of the sequence.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A096277, A001043.

Programs

Maple

Ss:= L -> L[1..-2]+L[2..-1]:
(Ss@@3)([seq(ithprime(i),i=1..100)]); # Robert Israel, Dec 28 2022

Mathematica

Nest[ListConvolve[{1,1},#]&,Prime[Range[100]],3] (* Paolo Xausa, Oct 31 2023 *)

PARI

f(n) = return(prime(n)+prime(n+1))
f1(n) = return(f(n)+f(n+1))
f2(n) = return(f1(n)+f1(n+1))
g(n) = for(x=1,n,print1(f2(x)","))

PARI

A096278(n,m=3)=for(k=0,m,prime(n+k)*binomial(m,k)) \\ or, to get a list:
A096278_vec(Nmax,m=3,v=primes(Nmax+m))=sum(k=0,m,binomial(m,k)*v[1+k,k-1-m]) \\ Alternatively, do m times v=v[^1]+v[^-1]. - M. F. Hasler, Jun 02 2017

Formula

Let f(n) = prime(n) + prime(n+1) f1(n) = f(n)+f(n+1) : SS of order 1 Then f2(n) = f1(n)+f1(n) : SS of order 2 is the general term of this sequence.

a(n) = A096277(n) + A096277(n+1). - M. F. Hasler, Jun 02 2017

a(n) = prime(n)+3*prime(n+1)+3*prime(n+2)+prime(n+3). - Robert Israel, Dec 28 2022