This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A096278 #32 Feb 23 2025 14:43:29
%S A096278 33,50,72,96,120,144,172,206,240,274,308,336,364,402,444,480,514,548,
%T A096278 578,610,648,692,742,786,816,840,864,900,960,1024,1070,1108,1152,1196,
%U A096278 1236,1278,1320,1362,1404,1444,1488,1530,1560,1592,1650,1728,1790,1824
%N A096278 Sums of successive sums of successive sums of successive primes.
%C A096278 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]
%H A096278 Seiichi Manyama, <a href="/A096278/b096278.txt">Table of n, a(n) for n = 1..10000</a>
%F A096278 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.
%F A096278 a(n) = A096277(n) + A096277(n+1). - _M. F. Hasler_, Jun 02 2017
%F A096278 a(n) = prime(n)+3*prime(n+1)+3*prime(n+2)+prime(n+3). - _Robert Israel_, Dec 28 2022
%e A096278 The first two terms of SS order 1 is 13 and 20. 13+20 = 33 the first term of the sequence.
%p A096278 Ss:= L -> L[1..-2]+L[2..-1]:
%p A096278 (Ss@@3)([seq(ithprime(i),i=1..100)]); # _Robert Israel_, Dec 28 2022
%t A096278 Nest[ListConvolve[{1,1},#]&,Prime[Range[100]],3] (* _Paolo Xausa_, Oct 31 2023 *)
%o A096278 (PARI)
%o A096278 f(n) = return(prime(n)+prime(n+1))
%o A096278 f1(n) = return(f(n)+f(n+1))
%o A096278 f2(n) = return(f1(n)+f1(n+1))
%o A096278 g(n) = for(x=1,n,print1(f2(x)","))
%o A096278 (PARI) A096278(n,m=3)=for(k=0,m,prime(n+k)*binomial(m,k)) \\ or, to get a list:
%o A096278 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
%Y A096278 Cf. A000040, A096277, A001043.
%K A096278 easy,nonn
%O A096278 1,1
%A A096278 _Cino Hilliard_, Jun 22 2004