A122102 - OEIS

%I #31 Jan 02 2023 12:30:46

%S 16,97,722,3123,17764,46325,129846,260167,540008,1247289,2170810,

%T 4044971,6870732,10289533,15169214,23059695,35177056,49022897,

%U 69174018,94585699,122983940,161934021,209392342,272134583,360663864,464724265,577275146,708354747,849512908

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

%C a(n) is prime for n = {2,32,90,110,134,152,168,180,194,...} = A122127.

%H Vincenzo Librandi, <a href="/A122102/b122102.txt">Table of n, a(n) for n = 1..1000</a>

%H OEIS Wiki, <a href="https://oeis.org/wiki/Sums_of_primes_divisibility_sequences">Sums of powers of primes divisibility sequences</a>

%H V. Shevelev, <a href="http://list.seqfan.eu/oldermail/seqfan/2013-August/011512.html">Asymptotics of sum of the first n primes with a remainder term</a>

%F From _Vladimir Shevelev_, Aug 02 2013: (Start)

%F 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 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 (End)

%p seq(add(ithprime(k)^4, k=1..n), n=1..30); # _G. C. Greubel_, Oct 02 2019

%t Table[Sum[Prime[k]^4,{k,1,n}],{n,1,100}]

%t Accumulate[Prime[Range[30]]^4] (* _Harvey P. Dale_, Aug 07 2021 *)

%o (PARI) a(n)=my(s);forprime(p=2,prime(n),s+=p^4); s \\ _Charles R Greathouse IV_, Aug 02 2013

%o (Magma) [&+[NthPrime(k)^4: k in [1..n]]: n in [1..30]]; // _G. C. Greubel_, Oct 02 2019

%o (Sage) [sum(nth_prime(k)^4 for k in (1..n)) for n in (1..30)] # _G. C. Greubel_, Oct 02 2019

%Y Cf. A007504, A024450, A098999, A122103, A122127.

%Y Partial sums of A030514.

%K nonn

%O 1,1

%A _Alexander Adamchuk_, Aug 20 2006