A109791 -id:A109791

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A143682

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

EXAMPLE

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.

MAPLE

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

MATHEMATICA

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

PROG

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

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

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

CROSSREFS

Cf. A000040.

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

KEYWORD

nonn

AUTHOR

Juri-Stepan Gerasimov, Nov 01 2008

EXTENSIONS

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

STATUS

approved

A109796

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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).

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..100

Eric Weisstein's World of Mathematics, Biquadratic Number.

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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

PROG

(PARI) A109796(n)={

sum(i=1, n, prime(i^4))

} /* R. J. Mathar, Mar 09 2012 */

CROSSREFS

First differences are A109791.

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

KEYWORD

nonn

AUTHOR

Jonathan Vos Post, Aug 15 2005

STATUS

approved

A347000

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

+10
1

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

EXAMPLE

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 ...

MATHEMATICA

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

CROSSREFS

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

Cf. A003320, A051129.

KEYWORD

nonn,tabl

AUTHOR

Hugo Pfoertner, Aug 10 2021

STATUS

approved

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