A098999 -id:A098999 - cp's OEIS frontend

A122140 Numbers m that divide the sum of cubes of the first m primes A098999(m).

1, 25, 537, 661, 5199, 113253, 240621, 5337048977, 17434578479, 34216676921, 1991831965911, 4495321247369, 22567781434431, 37328883555791, 110447613624133, 188368390470877, 324587968952249, 1983705516917661

Offset: 1

Alexander Adamchuk, Aug 21 2006

a(19) > 2*10^15. - Paul W. Dyson, Feb 18 2024

			a(2) = 25 because 25 is the first number n>1 that divides sum of cubes of the first n primes. A098999(25) mod 25 = (2^3 + 3^3 + 5^3 + ... + 89^3 + 97^3) mod 25 = 0.

OEIS Wiki, Sums of powers of primes divisibility sequences.

Cf. A098999.

s = 0; t = {}; Do[s = s + Prime[n]^3; If[ Mod[s, n] == 0, AppendTo[t, n]], {n, 1000000}]; t

a(8) from Donovan Johnson, Oct 15 2012
a(9)-a(10) from Robert Price, Mar 29 2013
a(11) from Paul W. Dyson, Jan 05 2021
a(12) from Bruce Garner, Feb 28 2021
a(13) from Bruce Garner, Apr 06 2021
a(14) from Bruce Garner, May 13 2021
a(15) from Bruce Garner, Jan 08 2022
a(16) from Paul W. Dyson, Jan 17 2022
a(17) from Bruce Garner, Jul 31 2022
a(18) from Paul W. Dyson, Feb 18 2024

A128167 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^3 = 1 + A098999(k).

Original entry on oeis.org

1, 2, 4, 5, 6, 10, 12, 14, 55, 82, 87, 935, 973, 1168, 1181, 1457, 5457, 7372, 11250, 17978, 25664, 182717, 472931, 2385026, 3002594, 9249715, 21843515, 37468158, 64403264, 87803374, 140933482, 281907048, 342460116, 1543515106, 1995156064

Offset: 1

Alexander Adamchuk, Feb 22 2007, Feb 23 2007

a(55) > 2.2*10^14. - Bruce Garner, Mar 28 2022

Bruce Garner, Table of n, a(n) for n = 1..54
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.

k = 0; s = 1; p = 2; lst = {}; While[k < 516862000, s = s + p^3; If[Mod[s, ++k] == 0, AppendTo[lst, k]; Print[{k, p}]]; p = NextPrime@ p]; lst

Four more terms from Sean A. Irvine, Jan 19 2011
a(32) & a(33) from Robert G. Wilson v
a(34)-a(35) from Robert Price, Dec 16 2013

A264897 Integers n such that A002110(n) is divisible by A098999(n).

Original entry on oeis.org

138, 163, 873, 1054, 1079, 1604, 1825, 1990, 2079, 2493, 2509, 2810, 2950, 3494, 3800, 3910, 4300, 4462, 4470, 4564, 4593, 4957, 5140, 5450, 5558, 5572, 5581, 5834, 6391, 6792, 6969, 7444, 7892, 8321, 8530, 8581, 9254, 9299, 9522, 9832, 9847, 10082, 10850

Offset: 1

Altug Alkan, Nov 27 2015

nonn

A002110(138) has 327 digits.

What is the minimum value of a(n) - a(n-1)?

Cf. A002110, A051838, A098999, A122137.

Select[Range@ 10000, Divisible[Product[Prime@ k, {k, #}], Sum[Prime[k]^3, {k, #}]] &] (* Michael De Vlieger, Nov 28 2015 *)

for(n=1, 11000, if(prod(k=1, n, prime(k)) % sum(k=1, n, prime(k)^3) == 0, print1(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)

A122103 Sum of the fifth powers of the first n primes.

Original entry on oeis.org

32, 275, 3400, 20207, 181258, 552551, 1972408, 4448507, 10884850, 31395999, 60025150, 129369107, 245225308, 392233751, 621578758, 1039774251, 1754698550, 2599294851, 3949419958, 5753649309, 7826720902, 10903777301, 14842817944

Offset: 1

Alexander Adamchuk, Aug 20 2006

a(n) is prime for n = {66, 148, 150, 164, 174, 214, 238, 264, 312, 328, 354, 440, 516, 536, 616, 624, 724, 744, 774, 836, 940, ...} = A122125. Primes of this form are listed in A122126 = {32353461605953, 9874820441996857, 10821208357045699, ...}.

			a(2) = 275 because the first two primes are 2 and 3, the fifth powers of which are 32 and 243, and 32 + 243 = 275.
a(3) = 3400, because the third prime is 5, its fifth power if 3125 and 275 + 3125 = 3400.

OEIS Wiki, Sums of powers of primes divisibility sequences
V. Shevelev, Asymptotics of sum of the first n primes with a remainder term

Cf. A050997, A007504, A024450, A098999, A122102, A122125, A122126.

Table[Sum[Prime[k]^5, {k, n}], {n, 100}]

a(n)=sum(i=1,n,prime(i)) \\ Charles R Greathouse IV, Nov 30 2013

a(n) = sum(k = 1 .. n, prime(k)^5).

a(n) = 1/6*n^6*log(n)^5 + O(n^6*log(n)^4*log(log(n))). The proof is similar to proof for A007504(n) (see link of Shevelev). For a generalization, see comment in A122102. - Vladimir Shevelev, Aug 14 2013

A098563 Numbers n such that the sum of the cubes of the first n primes is prime.

Original entry on oeis.org

4, 8, 38, 48, 98, 102, 118, 128, 130, 132, 156, 168, 172, 178, 180, 190, 202, 208, 308, 346, 358, 364, 424, 482, 540, 600, 602, 614, 646, 676, 722, 748, 768, 776, 782, 792, 838, 902, 1016, 1028, 1036, 1058, 1062, 1082, 1086, 1100, 1102, 1132, 1144, 1176

Offset: 1

Rick L. Shepherd, Sep 14 2004

nonn

n must clearly be even.

			4 is a term as the sum of the cubes of the first four primes is 2^3 + 3^3 + 5^3 + 7^3 = 503, which is prime.

Nathaniel Johnston, Table of n, a(n) for n = 1..1000

Cf. A066525 (corresponding primes), A098561 (sums of squares of primes), A013916 (sums of primes), A098999 (sums of cubes of primes).

with(numtheory): P:=proc(n) add(ithprime(k)^3, k=1..n): end:
A098563 := proc(n)local m: option remember: if(n=0)then return 0: fi: m:=procname(n-1)+2: while true do if(isprime(P(m)))then return m:fi: m:=m+2:od: end:
seq(A098563(n), n=1..50); # Nathaniel Johnston, Apr 21 2011

Select[Range[1000], PrimeQ[Sum[Prime[i]^3, {i, #}]] &] (* Carl Najafi, Aug 22 2011 *)

lista(nn) = {s = 0; ip = 0; forprime (p=1, nn, ip++; if (isprime(s+=p^3), print1(ip, ", ")););} \\ Michel Marcus, Aug 22 2015

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}]]]&]

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

A133548 a(n) = sum of cubes of first n odd primes.

Original entry on oeis.org

27, 152, 495, 1826, 4023, 8936, 15795, 27962, 52351, 82142, 132795, 201716, 281223, 385046, 533923, 739302, 966283, 1267046, 1624957, 2013974, 2507013, 3078800, 3783769, 4696442, 5726743, 6819470, 8044513, 9339542, 10782439, 12830822, 15078913, 17650266

Offset: 1

Artur Jasinski, Sep 16 2007, corrected Jun 08 2008

			a(3)=495 because 3^3+5^3+7^3=495.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007148, A007504, A024450, A098999, A133547, A133549, A133550, A133551.

c = 3; a = {}; b = 0; Do[b = b + Prime[n]^c; AppendTo[a, b], {n, 2, 1000}]; a
Accumulate[Prime[Range[2,40]]^3] (* Harvey P. Dale, Oct 31 2024 *)

a(n) = sum(i=2, n+1, prime(i)^3); \\ Michel Marcus, Nov 05 2013

a(n) = A098999(n+1) - 8.

More terms from Michel Marcus, Nov 05 2013

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.

cp's OEIS Frontend

A122140 Numbers m that divide the sum of cubes of the first m primes A098999(m).

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A128167 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^3 = 1 + A098999(k).

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A264897 Integers n such that A002110(n) is divisible by A098999(n).

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

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A122103 Sum of the fifth powers of the first n primes.

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A098563 Numbers n such that the sum of the cubes of the first n primes is prime.

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

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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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