A050248 -id:A050248 - cp's OEIS frontend

A236226 Sum of the eighteenth powers of the first n primes.

262144, 387682633, 3815084948258, 1632228682858707, 5561549542175090188, 118016956494132483317, 14181101408561857474326, 118308451706473099007167, 3362459361601721384307536, 213819743726773841714612697, 912873363725818880253782938

Offset: 1

Robert Price, Jan 20 2014

Robert Price, 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. 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.
Partial sums of A030637.

Table[Sum[Prime[k]^18, {k, n}], {n, 1000}]
Accumulate[Prime[Range[20]]^18] (* Harvey P. Dale, Jul 08 2024 *)

a(n) = sum(k = 1 .. n, prime(k)^18).

A236227 Sum of the nineteenth powers of the first n primes.

Original entry on oeis.org

524288, 1162785755, 19074649113880, 11417969834487023, 61170508418249033314, 1523090798793695143991, 240595526483945019991144, 2219015182144258609115123, 76834486109734969171023610, 6180095732699726458749873279, 27850757952670122653464150750

Offset: 1

Robert Price, Jan 20 2014

Robert Price, 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. 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.

Table[Sum[Prime[k]^19, {k, n}], {n, 1000}]

a(n) = sum(k = 1 .. n, prime(k)^19).

A160759 Integer averages of n values of pi(n^2) for some n, where pi(n) is the number of primes <= n.

Original entry on oeis.org

0, 1, 2, 3, 201, 235, 265, 431, 1705, 11744, 33946, 622755, 3446493, 8134880, 14287916, 208340425, 223689468

Offset: 1

Daniel Tisdale, May 25 2009

For values of n: 1, 2, 3, 4, 62, 68, 73, 97, 213, 624, 1116, 5364, 13350, 21048, 28351, 116151, 120562, ..., . [Robert G. Wilson v, Jun 05 2009]

Cf. A050248, Integer averages of n primes for some n.

lst = {}; s = 0; k = 1; While[k < 239600, s = s + PrimePi[k^2]; If[ Mod[s, k] == 0, AppendTo[lst, s/c]]; k++ ]; lst (* Robert G. Wilson v, Jun 05 2009 *)

1/k*Sum {j=1..k} Pi(j^2) is an integer. [Robert G. Wilson v, Jun 05 2009]

More terms from Robert G. Wilson v, Jun 05 2009

A164619 Integers of the form A164577(k)/3.

Original entry on oeis.org

4, 15, 54, 75, 132, 169, 320, 459, 735, 847, 1104, 1250, 1764, 2175, 2904, 3179, 3780, 4107, 5200, 6027, 7425, 7935, 9024, 9604, 11492, 12879, 15162, 15979, 17700, 18605, 21504, 23595, 26979, 28175, 30672, 31974, 36100, 39039, 43740, 45387, 48804

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

The sequence members are the third of the average of a set of smallest cubes, if integer.

			A third of the average of the first cube, A164577(1)/3=1/3, is not an integer and does not contribute to the sequence.
A third of the average of the first two cubes, A164577(2)/3=4, is an integer and defines a(1)=4 of the sequence.

Index entries for linear recurrences with constant coefficients, signature (2,-1,-1,2,-1,2,-4,2,2,-4,2,-1,2,-1,-1,2,-1).

Cf. A050248, A051456, A078617, A078618, A136116, A154293, A164286, A164576, A164577, A164578, A164579.

s=0;lst={};Do[a=(s+=(n^3)/3)/n;If[Mod[a,1]==0,AppendTo[lst,a]],{n,2*5!}]; lst
LinearRecurrence[{2,-1,-1,2,-1,2,-4,2,2,-4,2,-1,2,-1,-1,2,-1},{4,15,54,75,132,169,320,459,735,847,1104,1250,1764,2175,2904,3179,3780},50] (* Harvey P. Dale, Apr 06 2016 *)

Vec(x*(x^14 +x^13 +16*x^12 +10*x^11 +47*x^10 -22*x^9 +61*x^8 +10*x^7 +88*x^6 +8*x^5 +43*x^4 -14*x^3 +28*x^2 +7*x +4) / ((x -1)^4*(x +1)^3*(x^2 -x +1)^3*(x^2 +x +1)^2) + O(x^100)) \\ Colin Barker, Oct 27 2014

a(n) = +2*a(n-1) -a(n-2) -a(n-3) +2*a(n-4) -a(n-5) +2*a(n-6) -4*a(n-7) +2*a(n-8) +2*a(n-9) -4*a(n-10) +2*a(n-11) -a(n-12) +2*a(n-13) -a(n-14) -a(n-15) +2*a(n-16) -a(n-17). - R. J. Mathar, Jan 25 2011

G.f.: x*(x^14 +x^13 +16*x^12 +10*x^11 +47*x^10 -22*x^9 +61*x^8 +10*x^7 +88*x^6 +8*x^5 +43*x^4 -14*x^3 +28*x^2 +7*x +4) / ((x -1)^4*(x +1)^3*(x^2 -x +1)^3*(x^2 +x +1)^2). - Colin Barker, Oct 27 2014

Edited by R. J. Mathar, Aug 20 2009

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

Original entry on oeis.org

54, 218, 222054, 669806, 155593313228, 31860927184920, 37843679840313254, 5349233671440437948, 65075392901385088766, 102744428793110424984, 251471854505406311064463, 1074272348712875302655077, 1114427338015137279788981

Offset: 1

Robert Price, Apr 10 2013

nonn

			For n=2, m=10, a(2) = 2398/11=218.

Cf. A007504, A045345, A171399, A050247, A050248, A024450, A111441, A217599, A217600, A122140, A223936, A223937, A024525, A153022.

Definition corrected by N. J. A. Sloane, Apr 20 2013

A232770 Prime(k), where k is such that (Sum_{i=1..k} prime(i)^13) / k is an integer.

Original entry on oeis.org

2, 83, 1979, 2081, 2326469, 6356923, 7170679, 63812027, 4652001719, 241949473277, 163220642765623, 1260677492111911, 8150959175977039

Offset: 1

Robert Price, Nov 29 2013

a(13) > 1352363608564489. - Bruce Garner, Aug 30 2021

a(14) > 18205684894350047. - Paul W. Dyson, Dec 06 2024

			a(2) = 83, because 83 is the 23rd prime and the sum of the first 23 primes^13 = 17226586990098074754709144 when divided by 23 equals 748982043047742380639528 which is an integer.

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.

t = {}; sm = 0; Do[sm = sm + Prime[n]^13; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)

is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^13); s==0 \\ Charles R Greathouse IV, Nov 30 2013

a(11) from Bruce Garner, Mar 23 2021
a(12) from Bruce Garner, Aug 30 2021
a(13) from Paul W. Dyson, Apr 20 2023

A232848 Prime(k), where k divides Sum_{i=1..k} prime(i)^15.

Original entry on oeis.org

2, 59, 97, 127, 12517, 54581, 83921, 89273, 1396411, 2562719, 4952183, 29201281, 35562101, 47567557, 111213143, 184201627, 1172476337, 7309217299, 287609314877, 5173838081669, 408907258717171, 1357729730868191, 66413899001789557

Offset: 1

Robert Price, Dec 09 2013

			a(2) = 59, because 59 is the 17th prime and the sum of the first 17 primes^15 = 455708280934100194626604550 when divided by 17 equals 26806369466711776154506150 which is an integer.

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), A131275.
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

t = {}; sm = 0; Do[sm = sm + Prime[n]^15; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)

is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^15); s==0 \\ Charles R Greathouse IV, Nov 30 2013

S=n=0;forprime(p=1,,(S+=p^15)%n++||print1(p",")) \\ M. F. Hasler, Dec 01 2013

a(n) = prime(A131275(n)).

a(20) from Karl-Heinz Hofmann, Feb 17 2021
a(21) from Bruce Garner, Apr 30 2021
a(22) from Bruce Garner, Jan 07 2022
a(23) from Paul W. Dyson, Apr 18 2024

A232963 Prime(m), where m is such that (sum_{i=1..m} prime(i)^14) / m is an integer.

Original entry on oeis.org

2, 1933, 3217, 41681, 114311, 2743691233, 7252463461, 28682755720447, 2839633449523319

Offset: 1

Robert Price, Dec 02 2013

The primes correspond to indices n = 1, 295, 455, 4361, 10817, 132680789, 334931875, 957643538339 = A131274.

a(10) > 7.6*10^16. - Paul W. Dyson, Jan 17 2025

			a(2) = 1933, because 1193391 is the 295th prime and the sum of the first 295 primes^14 = 172657243368537051859007103457435197295421033550 when divided by 295 equals 585278791079786616471210520194695584052274690 which is an integer.

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.

t = {}; sm = 0; Do[sm = sm + Prime[n]^14; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)

is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^14); s==0 \\ Charles R Greathouse IV, Nov 30 2013

S=n=0;forprime(p=1,,(S+=p^14)%n++||print1(p",")) \\ M. F. Hasler, Dec 01 2013

a(n) = prime(A131274(n)).

a(8) from Paul W. Dyson, Jan 03 2021

a(9) from Bruce Garner, Mar 28 2022

A233192 Prime(k), where k is such that (Sum_{j=1..k} prime(j)^11) / k is an integer.

Original entry on oeis.org

2, 97, 277, 23311, 61583, 6133811, 210952097, 359643241, 5451597181, 42641466149, 51575229001, 199655689679, 248181386429, 61646670874849, 82153230089767, 212374157550341, 11432141933990629, 15031011453909223

Offset: 1

Robert Price, Dec 05 2013

a(17) > 257180056649941. - Bruce Garner, Mar 29 2021

a(19) > 18205684894350047. - Paul W. Dyson, Jan 16 2025

			a(2) = 97, because 97 is the 25th prime and the sum of the first 25 primes^11 = 12718098700540100969050 when divided by 25 equals 508723948021604038762 which is an integer.

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.

t = {}; sm = 0; Do[sm = sm + Prime[n]^11; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)

is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^11); s==0 \\ Charles R Greathouse IV, Nov 30 2013

S=n=0;forprime(p=1,,(S+=p^11)%n++||print1(p",")) \\ M. F. Hasler, Dec 01 2013

a(n) = prime(A125827(n)).

a(14) from Paul W. Dyson, Jan 08 2021
a(15) from Bruce Garner, Mar 08 2021
a(16) from Bruce Garner, Mar 29 2021
a(17) from Paul W. Dyson, Jan 03 2023
a(18) from Paul W. Dyson, Dec 20 2024

A233460 Prime(n), where n is such that (sum_{i=1..n} prime(i)^16) / n is an integer.

Original entry on oeis.org

2, 28751, 62639, 4620757, 6478193, 2298168044423, 128195718927553

Offset: 1

Robert Price, Dec 10 2013

a(8) > 128971810356197. - Bruce Garner, Mar 24 2021

a(8) > 7.6*10^16. - Paul W. Dyson, Nov 27 2024

			a(1) = 2, because 2 is the 1st prime and the sum of the first 1 primes^16 = 65536 when divided by 1 equals 65536 which is an integer.

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.

t = {}; sm = 0; Do[sm = sm + Prime[n]^16; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)

is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^16); s==0 \\ Charles R Greathouse IV, Nov 30 2013

S=n=0;forprime(p=1,,(S+=p^16)%n++||print1(p",")) \\ M. F. Hasler, Dec 01 2013

a(n) = prime(A131276(n)).

a(6)-a(7) from Bruce Garner, Mar 24 2021

cp's OEIS Frontend

A236226 Sum of the eighteenth powers of the first n primes.

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A236227 Sum of the nineteenth powers of the first n primes.

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A160759 Integer averages of n values of pi(n^2) for some n, where pi(n) is the number of primes <= n.

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A164619 Integers of the form A164577(k)/3.

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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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A232770 Prime(k), where k is such that (Sum_{i=1..k} prime(i)^13) / k is an integer.

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A232848 Prime(k), where k divides Sum_{i=1..k} prime(i)^15.

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A232963 Prime(m), where m is such that (sum_{i=1..m} prime(i)^14) / m is an integer.

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