A122140 -id:A122140 - cp's OEIS frontend

A223937 a(n) is the sum of the cubes of the first A122140(n) primes.

8, 4696450, 7024453131396, 17761740387522, 155912686127038650, 87598780898450312031408, 2147216863131055036604400, 2908950240914054780101441371333254159676520, 384422969812280951687876430655304031054262132, 6187047308209705064673104196645071104957480508

Offset: 1

Robert Price, Mar 29 2013

nonn

Paul W. Dyson, Table of n, a(n) for n = 1..18 (terms 1..10 from Robert Price, 11 from Paul W. Dyson, 12..15 from Bruce Garner, 16 from Paul W. Dyson, 17 from Bruce Garner)
OEIS Wiki, Sums of powers of primes divisibility sequences

Cf. A085450 (smallest m > 1 that divides Sum_{k=1..m} prime(k)^n), A122140.
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Title corrected by Hugo Pfoertner, Feb 09 2021

A223939 Integer averages of first k primes cubed for some k (a(n) = A223937(n)/A122140(n)).

Original entry on oeis.org

8, 187858, 13080918308, 26871014202, 29988975981350, 773478679579793136, 8923646993118036400, 545048444084018901462938808502760, 22049455928935679528789623492181708, 180819643079146957138056211903672348

Offset: 1

Robert Price, Mar 29 2013

nonn

			The integer, 187858 is the average of the first 25 primes^3 (4696450/25=187858).

Paul W. Dyson, Table of n, a(n) for n = 1..18 (terms 1..11, 16, 18 from Paul W. Dyson, terms 11..15, 17 from Bruce Garner)
OEIS Wiki, Sums of powers of primes divisibility sequences

Cf. A122140, A223937.
Cf. A085450 (smallest m > 1 that divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

A085450 a(n) is the smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.

Original entry on oeis.org

23, 19, 25, 2951, 25, 4357, 25, 43, 281525, 269, 25, 37, 23, 295, 17, 3131, 395191, 37, 25, 19, 139, 1981, 23, 37, 25, 455, 25, 41, 124403, 61, 17, 511, 193, 535, 23, 5209, 1951, 19, 25, 301, 891, 9805, 25, 527, 23, 83, 17, 37, 131, 43, 25, 193, 53, 37, 25, 19

Offset: 1

Farideh Firoozbakht, Aug 14 2003

nonn

By definition a(1) is A045345(2).
This sequence has a very interesting behavior. If Mod(n, 2)(Mod(n, 20)-1)(Mod(n, 20)-9)(Mod(n, 20)-13)(Mod(n, 20)-17)!=0, a(n)=17, 23 or 25; in other cases a(n) may be too large. If Mod[n, 16] = 15, a(n) = 17. For example, a(n) = 17 for n = 15, 31, 47, 63, 79, 95, 111, 127, 143, 159, 175, 191, ...; also, a(n) = 23 for n = 1, 13, 23, 35, 45, 57, 67, 89, 101, 123, 133, 145, 155, 167, 177, 189, 199, ...; a(n) = 25 for n = 3, 5, 7, 11, 19, 25, 27, 39, 43, 51, 55, 59, 65, 71, 75, ..., . For a(n) = 19 for n = 2, 20, 38, 56, 74, 92, 110, 128, 146, 164, 182, 200, 218, ..., == 2 (mod 18).
From Alexander Adamchuk, Jul 20 2008: (Start)
Conjectures:
a(n) exists for all n; a(n) >= 17.
a(325)-a(575) = {25,19,25,5851,1843,61,23,821,89,301,17,37,131,455,25,1607,297,37,23,19,25,
325,25,37,353,47,17,1663,23,691,25,691,509269,155,25,269,105893,19,25,3971,
23,213215,17,26021,327,79,25,37,151,83,23,161,101,37,25,19,327,265,17,37,25,
43,23,41,169,61,25,113,21761,6289,25,47,23,19,17,4073,1137,565,25,527,25,
325,25,37,23,455,25,431,13195,37,17,19,53,155,23,37,89,455,25,18839,25,6221,
25,41,18597,229,17,811,623173,19,25,193,2079,673,25,881,23,47,25,37,25,97,
17,79,131,37,25,19,23,56501,25,37,299,455,25,167,2707,446963,17,157,25,325,
25,41,53,19,25,5917,103,1051,23,607,101,155,17,37,6233,455,25,9049,23,37,25,
19,327,5359,25,37,43,455,17,9187,23,193,25,1861,7923,301,25,113,25,19,23,41,
89,61,17,43,1785,131,25,37,1417,455,23,151,53,37,25,19,25,79,17,37,23,455,
25,289,59,47,25,511,47,83,25,739,23,19,17,301,25,269,25,41,707,2735,23,37,
299,43,25,283,69723,37,17,19,1785,479,23,37,25,455,25,1867,131,61,25,31799,
23,161,17}.
a(n) is currently unknown and a(n)>10^7 for n = {324, 576, ...}. (End)
All but one of the terms up to n=1000 are known and they are less than 10^8. Currently the only unknown term for n<=1000 is a(656)>10^8. - Alexander Adamchuk, May 24 2009
More terms: a(324) = 18642551, a(576) = 12824827. - Alexander Adamchuk, May 24 2009
a(656) > 23,491,000,000. - Robert Price, Apr 22 2014
a(656) > 10^12. - Paul W. Dyson, Nov 23 2024
From Paul W. Dyson, Jan 18 2025: (Start)
If n == 15 (mod 16), a(n) = 17; otherwise if n == 2 (mod 18), a(n) = 19; otherwise if n mod 22 = 1 or 13, a(n) = 23; otherwise if n mod 20 = 3, 5, 7, 11, 15 or 19, a(n) = 25; otherwise if n mod 36 = 12, 18 or 24, a(n) = 37; etc. These follow from the fact that a(n) will also be a divisor for a prime sum with power j when j == n (mod psi(a(n))) and both n and j are greater than or equal to the maximum exponent in the prime factorization of a(n), where psi is the reduced totient function (A002322). E.g. for n=15, a(n)=17 and psi(a(n)) = 16. So j = 31, 47, 63, ..., and a(31) = a(47) = a(63) = a(15) = 17. For proof, see the comment dated Dec 09 2022 in A111441.
If a(n) exists, a(n) >= 17. For k < 17, psi(k) <= 12 and the maximum exponent in a prime factorization is 4 (as 16=2^4). So any a(n) < 17 would appear with periodicity <= 12, and would be seen in the first 15 (=12+4-1) terms of the sequence. (End)

			a(3) = 25 because 2^3+3^3+5^3...+prime(25)^3 == 0 (mod 25) and for 1 < n < 25 2^3+3^3+...+prime(n)^3 is not congruent to zero (mod n).

Alexander Adamchuk and Robert Price, Table of n, a(n) for n = 1..655 (first 323 terms from Alexander Adamchuk)

Cf. A045345, A111441, A122140, A125907, A122142, A125825, A125826, A125828, A131263, A131264, A125827, A131272, A131273, A131274, A131275, A131276, A131277, A131278, A131279.

a[n_] := Block[{m = 2, s = 2^n}, While[s = s + Prime[m]^n; Mod[s, m] != 0, m++ ]; m]; Table[ a[n], {n, 1, 56}]
a[n_] := Block[{m = 2, s = 2^n}, While[s = s + Prime[m]^n; Mod[s, m] != 0&& m<10000000, m++ ]; m]; Table[ a[n], {n, 1, 1000}] (* Alexander Adamchuk, Jul 20 2008 *)

a(n)=my(s=2^n,m=1); forprime(p=3,, if((s+=p^n)%m++==0, return(m))) \\ Charles R Greathouse IV, Feb 06 2015

For[m=2, Mod[Sum[Prime[k]^n, {k, m}], m]!=0, m++ ]; m

Edited and extended by Robert G. Wilson v, Aug 14 2003

A122142 Numbers m such that m divides sum of 5th powers of the first m primes A122103(m).

Original entry on oeis.org

1, 25, 837, 5129, 94375, 271465, 3576217, 3661659, 484486719, 2012535795, 31455148645, 95748332903, 145967218799, 165153427677, 21465291596581, 97698929023845

Offset: 1

Alexander Adamchuk, Aug 21 2006

No other terms up to 10^8. - Stefan Steinerberger, Jun 06 2007
a(11) > 6*10^9. - Donovan Johnson, Oct 15 2012
a(13) > 10^11. - Robert Price, Mar 30 2013
a(15) > 10^12. - Paul W. Dyson, Jan 04 2021
a(16) > 2.2*10^13. - Bruce Garner, May 09 2021
a(17) > 10^14. - Paul W. Dyson, Feb 04 2022
a(17) > 10^15. - Paul W. Dyson, Nov 19 2024

			a(2) = 25 because 25 is the first number n>1 that divides A122103[n] = Sum[ Prime[k]^5, {k,1,n} ].
Mod[ A122103[25], 25] = Mod[ 2^5 + 3^5 + 5^5 + ... + 89^5 + 97^5, 25 ] = 0.

OEIS Wiki, Sums of powers of primes divisibility sequences

Cf. A122103, A098999, A007504, A045345, A024450, A111441, A122102, A122140.

s = 0; t = {}; Do[s = s + Prime[n]^5; If[ Mod[s, n] == 0, AppendTo[t, n]], {n, 1000000}]; t
Module[{nn = 4*10^6},Select[Thread[{Range[nn], Accumulate[ Prime[ Range[ nn]]^5]}], Divisible[#[[2]], #[[1]]] &]][[All, 1]] (* Generates the first 8 terms; to generate more, increase the value of nn, but the program may take a long time to run. *) (* Harvey P. Dale, Aug 26 2019 *)

2 more terms from Stefan Steinerberger, Jun 06 2007
a(9)-a(10) from Donovan Johnson, Oct 15 2012
a(11)-a(12) from Robert Price, Mar 30 2013
a(13)-a(14) from Paul W. Dyson, Jan 04 2021
a(15) from Bruce Garner, May 09 2021
a(16) from Paul W. Dyson, Feb 04 2022

cp's OEIS Frontend

A223937 a(n) is the sum of the cubes of the first A122140(n) primes.

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A223939 Integer averages of first k primes cubed for some k (a(n) = A223937(n)/A122140(n)).

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A085450 a(n) is the smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.

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A122142 Numbers m such that m divides sum of 5th powers of the first m primes A122103(m).

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A221867 Let m = A153022(n); a(n) = (1 + sum_{i=1..m} prime(i)^2)/(1+m).

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