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A122102 a(n) = Sum_{k=1..n} prime(k)^4.

%I A122102 #32 Dec 23 2024 14:53:42
%S A122102 16,97,722,3123,17764,46325,129846,260167,540008,1247289,2170810,
%T A122102 4044971,6870732,10289533,15169214,23059695,35177056,49022897,
%U A122102 69174018,94585699,122983940,161934021,209392342,272134583,360663864,464724265,577275146,708354747,849512908
%N A122102 a(n) = Sum_{k=1..n} prime(k)^4.
%C A122102 a(n) is prime for n = {2,32,90,110,134,152,168,180,194,...} = A122127.
%H A122102 Vincenzo Librandi, <a href="/A122102/b122102.txt">Table of n, a(n) for n = 1..1000</a>
%H A122102 OEIS Wiki, <a href="https://oeis.org/wiki/Sums_of_primes_divisibility_sequences">Sums of powers of primes divisibility sequences</a>
%H A122102 V. Shevelev, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2013-August/011512.html">Asymptotics of sum of the first n primes with a remainder term</a>
%F A122102 From _Vladimir Shevelev_, Aug 02 2013: (Start)
%F A122102 a(n) = 0.2*n^5*log(n)^4 + O(n^5*log(n)^3*log(log(n))). The proof is similar to proof for A007504(n) (see link of Shevelev).
%F A122102 A generalization: Sum_{i=1..n} prime(i)^k = 1/(k+1)*n^(k+1)*log(n)^k + O(n^(k+1)*log(n)^(k-1)*log(log(n))).
%F A122102 (End)
%p A122102 seq(add(ithprime(k)^4, k=1..n), n=1..30); # _G. C. Greubel_, Oct 02 2019
%t A122102 Table[Sum[Prime[k]^4,{k,1,n}],{n,1,100}]
%t A122102 Accumulate[Prime[Range[30]]^4] (* _Harvey P. Dale_, Aug 07 2021 *)
%o A122102 (PARI) a(n)=my(s);forprime(p=2,prime(n),s+=p^4); s \\ _Charles R Greathouse IV_, Aug 02 2013
%o A122102 (Magma) [&+[NthPrime(k)^4: k in [1..n]]: n in [1..30]]; // _G. C. Greubel_, Oct 02 2019
%o A122102 (Sage) [sum(nth_prime(k)^4 for k in (1..n)) for n in (1..30)] # _G. C. Greubel_, Oct 02 2019
%Y A122102 Cf. A007504, A024450, A098999, A122103, A122127.
%Y A122102 Partial sums of A030514.
%K A122102 nonn
%O A122102 1,1
%A A122102 _Alexander Adamchuk_, Aug 20 2006