A122103 -id:A122103 - cp's OEIS frontend

A128169 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^5 = 1 + A122103(k).

1, 2, 4, 6, 10, 12, 22, 58, 155, 363, 464, 665, 1146, 2870, 3048, 4019, 5931, 8724, 21503, 50439, 67560, 476281, 705570, 4050684, 6956459, 7443590, 10449928, 10799546, 15385564, 17735139, 83325458, 245271750, 255583775, 1395860516, 2921734534, 6255577368, 9050771725, 12062893218, 13689205205, 42254229197, 46440930382

Offset: 1

Alexander Adamchuk, Feb 22 2007, Feb 23 2007

a(52) > 3*10^13. - Bruce Garner, Jun 05 2021

a(53) > 1.2*10^14. - Bruce Garner, Mar 28 2022

Bruce Garner, Table of n, a(n) for n = 1..52
OEIS Wiki, Sums of powers of primes divisibility sequences

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

p = 2; k = 0; s = 1; lst = {}; While[k < 521330000, s = s + p^5; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p]

a(31) from Sean A. Irvine, Jan 19 2011
a(32)-a(33) from Robert G. Wilson v, Jan 20 2011
a(34)-a(41) from Robert Price, Dec 18 2013

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

A122125 Indices n such that A122103[n] = Sum[ Prime[k]^5, {k,1,n}] is a prime.

Original entry on oeis.org

66, 148, 150, 164, 174, 214, 238, 264, 312, 328, 354, 440, 516, 536, 616, 624, 724, 744, 774, 836, 940, 1002, 1030, 1034, 1038, 1064, 1086, 1092, 1108, 1162, 1322, 1394, 1424, 1468, 1500, 1630, 1654, 1668, 1684, 1694, 1840, 1886, 1980

Offset: 1

Alexander Adamchuk, Aug 20 2006

nonn

Corresponding primes of form Sum[ Prime[k]^5, {k,1,n} ] are listed in A122126[n] = {32353461605953,9874820441996857,10821208357045699,...}.

Cf. A122103, A122126.

Select[Range[2000],PrimeQ[Sum[Prime[k]^5,{k,1,#1}]]&]

A122126 Primes of form Sum[ Prime[k]^5, {k,1,n} ] or primes in A122103[n].

Original entry on oeis.org

32353461605953, 9874820441996857, 10821208357045699, 20261841065985329, 30672624709674923, 126540093912585209, 267908140423628051, 538379772006780299, 1713808283318707391, 2413049165702037271

Offset: 1

Alexander Adamchuk, Aug 20 2006

nonn

A122103[n] = Sum[ Prime[k]^5, {k,1,n} ] begins {32,275,3400,20207,181258,552551,1972408,4448507,...}. Indices n such that A122103[n] is prime are listed in A122125[n] = {66,148,150,164,174,214,238,264,312,328,354,440,516,536,616,624,724,744,774, 836,940,...}.

Cf. A122103, A122125.

Select[Table[Sum[Prime[k]^5,{k,1,n}],{n,1,1000}],PrimeQ[ #1]&]

a(n) = A122103[ A122125[n] ].

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

Original entry on oeis.org

16, 97, 722, 3123, 17764, 46325, 129846, 260167, 540008, 1247289, 2170810, 4044971, 6870732, 10289533, 15169214, 23059695, 35177056, 49022897, 69174018, 94585699, 122983940, 161934021, 209392342, 272134583, 360663864, 464724265, 577275146, 708354747, 849512908

Offset: 1

Alexander Adamchuk, Aug 20 2006

nonn

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
OEIS Wiki, Sums of powers of primes divisibility sequences
V. Shevelev, Asymptotics of sum of the first n primes with a remainder term

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

Partial sums of A030514.

[&+[NthPrime(k)^4: k in [1..n]]: n in [1..30]]; // G. C. Greubel, Oct 02 2019

seq(add(ithprime(k)^4, k=1..n), n=1..30); # G. C. Greubel, Oct 02 2019

Table[Sum[Prime[k]^4,{k,1,n}],{n,1,100}]
Accumulate[Prime[Range[30]]^4] (* Harvey P. Dale, Aug 07 2021 *)

a(n)=my(s);forprime(p=2,prime(n),s+=p^4); s \\ Charles R Greathouse IV, Aug 02 2013

[sum(nth_prime(k)^4 for k in (1..n)) for n in (1..30)] # G. C. Greubel, Oct 02 2019

From Vladimir Shevelev, Aug 02 2013: (Start)
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).
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))).
(End)

A122138 Indices k such that A122136(k) is a prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 11, 12, 14, 15, 18, 20, 22, 23, 26, 27, 32, 36, 38, 39, 40, 44, 47, 48, 50, 51, 52, 54, 55, 56, 58, 59, 60, 64, 66, 68, 71, 72, 74, 76, 78, 80, 83, 84, 86, 88, 89, 90, 92, 94, 95, 96, 98, 100, 102, 103, 107, 108, 110, 112, 114, 116, 118, 120, 122, 126

Offset: 1

Alexander Adamchuk, Aug 21 2006

nonn

The corresponding primes are listed in A122139.

Cf. A122136, A122137, A122139, A002110, A024450, A007504, A098999, A122102, A122103.

Select[Range[200],PrimeQ[Numerator[Sum[Prime[k]^2,{k,1,#1}]/Product[Prime[k],{k,1,#1}]]]&]

A133550 Sum of fifth powers of n odd primes.

Original entry on oeis.org

243, 3368, 20175, 181226, 552519, 1972376, 4448475, 10884818, 31395967, 60025118, 129369075, 245225276, 392233719, 621578726, 1039774219, 1754698518, 2599294819, 3949419926, 5753649277, 7826720870, 10903777269, 14842817912

Offset: 1

Artur Jasinski, Sep 16 2007

			a(2)=3368 because 3^5+5^5 = 3368.

Cf. A007148, A007504, A024450, A098999, A122102, A122103, A133547, A133548, A133550, A133551.

c = 5; a = {}; b = 0; Do[b = b + Prime[n]^c; AppendTo[a, b], {n, 2, 1000}]; a

a(n) = A122103(n+1)-32.

A118219 Smallest number k>1 such that Sum_{i=1..k} Prime[i]^n divides Product_{i=1..k} Prime[i]^n.

Original entry on oeis.org

3, 30, 17, 248, 515, 49682

Offset: 1

Alexander Adamchuk, Feb 24 2007

a(7)>991430. - Robert G. Wilson v, Mar 02 2007

			a(1) = 3 because 2 + 3 + 5 = 10 divides 2*3*5 = 30 but 2 + 3 = 5 does not divide 2*3 = 6.

Cf. A051838 = Sum of first n primes divides product of first n primes. Cf. A125314 = Smallest number k>1 such that Sum_{i=1..k} i^n divides Product_{i=1..k} i^n. Cf. A007504, A002110, A024450, A098999, A122102, A122103.

f[n_] := Block[{k = 2, p = 2, s = 2^n}, While[p = p*Prime@ k; s = s + Prime@ k^n; PowerMod[p, n, s] != 0, k++ ]; k]; Do[ Print@ f@n, {n, 10}] (* Robert G. Wilson v *)

a(6) from Robert G. Wilson v, Mar 02 2007

A122124 Numbers n such that 25 divides Sum[ Prime[k]^n, {k,1,n}].

Original entry on oeis.org

3, 5, 7, 11, 15, 19, 23, 25, 27, 31, 35, 39, 43, 45, 47, 51, 55, 59, 63, 65, 67, 71, 75, 79, 83, 85, 87, 91, 95, 99, 103, 105, 107, 111, 115, 119, 123, 125, 127, 131, 135, 139, 143, 145, 147, 151, 155, 159, 163, 165, 167, 171, 175, 179, 183, 185, 187, 191, 195, 199

Offset: 1

Alexander Adamchuk, Aug 21 2006, Sep 18 2006, Sep 21 2006

nonn

a(n) up to a(7) = 23 coincides with A007665[n+1] = Tower of Hanoi with 5 pegs. It appears that a(n) includes all A007665[n] = {1, 3, 5, 7, 11, 15, 19, 23, 27, 31, 39, 47, 55, 63, 71, 79, 87, 95, 103, 111, 127, 143, 159, 175, 191, 207, 223, 239, 255, 271, 287, 303, 319, 335, 351, 383, 415, 447, 479, 511, 543, 575, 607, 639, 671, 703, 735, 767, 799, ...} except A007665[1] = 1.

Primes in this sequence include 5 and all primes of the form 4k+3, A002145[n]. Terms include all numbers of the form 10k+5 (with nonnegative k), A017329[n].

			There are 25 primes p < 100, p(n) = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}.
a(1) = because 25 divides Sum[p(n)^3,{n,1,25}] = 2^3 + 3^3 + ... + 89^3 + 97^3 = A098999[25] and does not divide Sum[p(n)^1,{n,1,25}] = A007504[25] and Sum[p(n)^2,{n,1,25}] = A024450[25].
The next a(2) = 5 because 25 divides Sum[p(n)^5,{n,1,25}] = A122103[25] and does not divide Sum[p(n)^4,{n,1,25}] = A122102[25].

Cf. A007504, A024450, A098999, A122102, A122103, A007665.

Cf. A002144, A002145, A017329.

Select[Range[300],IntegerQ[Sum[ Prime[k]^#1, {k,1,25}]/25]&]

for(n=1,100,if(sum(k=1,25,prime(k)^n)%25==0,print1(n,",")));
print;print("Alternative method not using primes:");
for(n=1,100,m=(n-1)%6;print1((n-m)*3+(n-m+if(m>1,(m-1)*12-1,m*6-1))/3,",")) \\ K. Spage, Oct 23 2009

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A128169 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^5 = 1 + A122103(k).

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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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A122125 Indices n such that A122103[n] = Sum[ Prime[k]^5, {k,1,n}] is a prime.

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A122126 Primes of form Sum[ Prime[k]^5, {k,1,n} ] or primes in A122103[n].

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

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A122138 Indices k such that A122136(k) is a prime.

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A133550 Sum of fifth powers of n odd primes.

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A118219 Smallest number k>1 such that Sum_{i=1..k} Prime[i]^n divides Product_{i=1..k} Prime[i]^n.

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