A109791 -id:A109791 - cp's OEIS frontend

A143682 a(n) = (prime(n)^4 - prime(n^4))/2, where prime(n) is the n-th prime.

7, 14, 103, 391, 5002, 8967, 31065, 45724, 107077, 301276, 382000, 820141, 1246909, 1479730, 2129740, 3534420, 5523879, 6237871, 9209731, 11625564, 12865129, 17844972, 21754756, 28999632, 41437737, 48684207, 52341667, 60941856

Offset: 1

Juri-Stepan Gerasimov, Nov 01 2008

nonn

			a(1) = (prime(1)^2^2 - prime(1^2^2))/2 = (16 - 2)/2 = 14/2 = 7,
a(2) = (prime(2)^2^2 - prime(2^2^2))/2 = (81 - 53)/2 = 28/2 = 14,
a(3) = (prime(3)^2^2 - prime(3^2^2))/2 = (625 - 419)/2 = 206/2 = 103,
a(4) = (prime(4)^2^2 - prime(4^2^2))/2 = (2401 - 1619)/2 = 782/2 = 391 = a(4),
a(5) = (prime(5)^2^2 - prime(5^2^2))/2 = (14641 - 4637)/2 = 10004/2 = 5002,
etc.

G. C. Greubel, Table of n, a(n) for n = 1..750

Cf. A000040.

Cf. A030514, A109791. - R. J. Mathar, Nov 05 2008

[(NthPrime(n)^4 - NthPrime(n^4))/2: n in [1..30]]; // Vincenzo Librandi, Oct 05 2015

A143682 := proc(n) (ithprime(n)^4-ithprime(n^4))/2 ; end: for n from 1 to 50 do printf("%d,",A143682(n)) ; od: # R. J. Mathar, Nov 05 2008

Table[(Prime[n]^4 - Prime[n^4])/2, {n, 40}] (* G. C. Greubel, May 29 2021 *)

a(n) = (prime(n)^4 - prime(n^4))/2; \\ Michel Marcus, Oct 05 2015

[(nth_prime(n)^4 - nth_prime(n^4))/2 for n in (1..40)] # G. C. Greubel, May 29 2021

More terms from R. J. Mathar, Nov 05 2008

A109796 a(n) = prime(1^4) + prime(2^4) + ... + prime(n^4).

Original entry on oeis.org

2, 55, 474, 2093, 6730, 17357, 38748, 77621, 143308, 248037, 407558, 641437, 973380, 1432721, 2052922, 2874563, 3944166, 5314265, 7045924, 9206477, 11874460, 15134597, 19083406, 23826383, 29480190, 36172177, 44039724

Offset: 1

Jonathan Vos Post, Aug 15 2005

nonn

Analog of prime(1^2) + prime(2^2) + ... + prime(n^2) (A109724). For a(n) to be prime for n > 1 it is necessary but not sufficient that n == 0 (mod 4).

			a(1) = 2 because prime(1^4) = prime(1) = 2.
a(2) = 55 because prime(1^4) + prime(2^4) = prime(1) + prime(16) = 2 + 53.
a(3) = 474 because prime(1^4) + prime(2^4) + prime(3^4) = prime(1) + prime(16) + prime(81) = 2 + 53 + 419.
a(4) = 2093 because prime(1^4) + prime(2^4) + prime(3^4) + prime(4^4) = 2 + 53 + 419 + prime(256) = 2 + 53 + 419 + 1619.
a(8) = 2 + 53 + 419 + 1619 + 4637 + 10627 + 21391 + 38873 = 77621 (which is prime).
a(12) = 2 + 53 + 419 + 1619 + 4637 + 10627 + 21391 + 38873 + 65687 + 104729 + 159521 + 233879 = 641437 (which is prime).
a(28) = 2 + 53 + 419 + 1619 + 4637 + 10627 + 21391 + 38873 + 65687 + 104729 + 159521 + 233879 + 331943 + 459341 + 620201 + 821641 + 1069603 + 1370099 + 1731659 + 2160553 + 2667983 + 3260137 + 3948809 + 4742977 + 5653807 + 6691987 + 7867547 + 9195889 = 53235613 (which is prime).
It is a coincidence that a(1), a(2) and a(3) are all palindromes.

Harvey P. Dale, Table of n, a(n) for n = 1..100
Eric Weisstein's World of Mathematics, Biquadratic Number.

First differences are A109791.

Cf. A000040, A000290, A000583, A011757, A109724, A109770.

Accumulate[Table[Prime[n^4],{n,30}]] (* Harvey P. Dale, Feb 02 2019 *)

A109796(n)={
        sum(i=1,n,prime(i^4))
} /* R. J. Mathar, Mar 09 2012 */

a(n) = Sum_{i=1..n} A000040(A000583(i)).

A347000 The (m^n)-th prime, written as square array T(n,m) read by falling antidiagonals.

Original entry on oeis.org

2, 3, 2, 5, 7, 2, 7, 23, 19, 2, 11, 53, 103, 53, 2, 13, 97, 311, 419, 131, 2, 17, 151, 691, 1619, 1543, 311, 2, 19, 227, 1321, 4637, 8161, 5519, 719, 2, 23, 311, 2309, 10627, 28687, 38873, 19289, 1619, 2, 29, 419, 3671, 21391, 79349, 171529, 180503, 65687, 3671, 2

Offset: 1

Hugo Pfoertner, Aug 10 2021

			The array begins
  2   3     5      7     11      13       17 ...
  2   7    23     53     97     151      227 ...
  2  19   103    311    691    1321     2309 ...
  2  53   419   1619   4637   10627    21391 ...
  2 131  1543   8161  28687   79349   185707 ...
  2 311  5519  38873 171529  567871  1549817 ...
  2 719 19289 180503 994837 3950183 12579617 ...

Hugo Pfoertner, Table of k, a(k) for k = 1..351, antidiagonals for m+n<=26, flattened.

Cf. A000040, A033844, A038833, A119772, A011757, A055875, A109791, A062448.

Cf. A003320, A051129.

T[n_,m_]:=Prime[m^n];Flatten[Table[Reverse[Table[T[n-m+1,m],{m,n}]],{n,10}]] (* Stefano Spezia, Aug 10 2021 *)

cp's OEIS Frontend

A143682 a(n) = (prime(n)^4 - prime(n^4))/2, where prime(n) is the n-th prime.

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A109796 a(n) = prime(1^4) + prime(2^4) + ... + prime(n^4).

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A347000 The (m^n)-th prime, written as square array T(n,m) read by falling antidiagonals.

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Mathematica