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 A123855 #20 Sep 08 2022 08:45:28
%S A123855 2,18,208,3730,201092,7335762,526460272,26465563878,2363769149128,
%T A123855 487833920370774,40049421223880084,7972075784185713954,
%U A123855 1235006486302921316794,124887894202756460238954
%N A123855 a(n) = Sum_{j=1..n} Sum_{i=1..n} prime(i)^j.
%C A123855 Primes p that divide a(p-1) are listed in A123856.
%C A123855 Nonprime numbers n that divide a(n-1) are listed in A123857.
%C A123855 It appears that 2^k divides a(2^k-1) for all k > 0 (confirmed for 0 < k < 10).
%C A123855 The summation over j can be carried out first and expressed analytically, leading to the given formula and Maple program. - _M. F. Hasler_, Nov 09 2006
%H A123855 M. F. Hasler, Nov 09 2006, <a href="/A123855/b123855.txt">Table of n, a(n) for n = 1..25</a>
%F A123855 a(n) = Sum_{j=1..n} Sum_{i=1..n} prime(i)^j.
%F A123855 a(p) = Sum_{i=1..p} (prime(i)^p - 1)/(prime(i) - 1)*prime(i). - _M. F. Hasler_, Nov 09 2006
%e A123855 a(1) = prime(1)^1 = 2.
%e A123855 a(2) = prime(1)^1 + prime(1)^2 + prime(2)^1 + prime(2)^2 = 2^1 + 2^2 + 3^1 + 3^2 = 18.
%p A123855 A123855 := p-> sum((ithprime(i)^p-1)/(ithprime(i)-1)*ithprime(i),i = 1 .. p); map(%,[$1..20]); # _M. F. Hasler_, Nov 09 2006
%t A123855 Table[Sum[Sum[Prime[i]^j,{i,1,n}],{j,1,n}],{n,1,20}]
%o A123855 (PARI) vector(20, n, sum(i=1,n, sum(j=1,n, prime(i)^j )) ) \\ _G. C. Greubel_, Aug 08 2019
%o A123855 (Magma) [(&+[ (&+[ NthPrime(i)^j: j in [1..n]]): i in [1..n]]): n in [1..20]]; // _G. C. Greubel_, Aug 08 2019
%o A123855 (Sage) [sum(sum( nth_prime(i)^j for j in (1..n)) for i in (1..n)) for n in (1..20)] # _G. C. Greubel_, Aug 08 2019
%Y A123855 Cf. A123856, A123857.
%Y A123855 Cf. A086787 (Sum_{i=1..n} Sum_{j=1..n} i^j).
%K A123855 nonn
%O A123855 1,1
%A A123855 _Alexander Adamchuk_, Oct 13 2006