A050248 -id:A050248 - cp's OEIS frontend

A233862 Prime(k), where k is such that (1 + Sum_{i=1..k} prime(i)^2) / k is an integer.

2, 3, 5, 7, 13, 23, 37, 41, 101, 107, 197, 317, 1033, 2029, 2357, 2473, 2879, 5987, 6173, 35437, 56369, 81769, 195691, 199457, 793187, 850027, 1062931, 1840453, 2998421, 4217771, 6200923, 9914351, 10153807, 13563889, 18878099, 60767923, 118825361, 170244929

Offset: 1

Robert Price, Dec 16 2013

nonn

a(51) > 1428199016921.

a(67) > 2407033812270611. - Bruce Garner, May 05 2021

			a(5) = 13, because 13 is the 6th prime and the sum of the first 6 primes^2+1 = 378 when divided by 6 equals 63 which is an integer.

Bruce Garner, Table of n, a(n) for n = 1..66 (first 50 terms from Robert Price)
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 = 1; Do[sm = sm + Prime[n]^2; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
Module[{nn=9600000},Prime[#]&/@Transpose[Select[Thread[{Range[nn], 1+ Accumulate[ Prime[Range[nn]]^2]}],IntegerQ[Last[#]/First[#]]&]][[1]]] (* Harvey P. Dale, Sep 09 2014 *)

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

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

Original entry on oeis.org

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

A164576 Integer averages of the set of the first positive squares up to some n^2.

Original entry on oeis.org

1, 11, 20, 46, 63, 105, 130, 188, 221, 295, 336, 426, 475, 581, 638, 760, 825, 963, 1036, 1190, 1271, 1441, 1530, 1716, 1813, 2015, 2120, 2338, 2451, 2685, 2806, 3056, 3185, 3451, 3588, 3870, 4015, 4313, 4466, 4780, 4941, 5271, 5440, 5786, 5963, 6325, 6510

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2009

Integers of the form A000330(k)/k, k listed in A007310. - R. J. Mathar, Aug 20 2009

			a(1) = 1^2/1 is an integer. The average of the first two squares is (1^2+2^2)/2=5/2, not integer.
The average of the first three squares is (1^2+2^2+3^2)/3=14/3, not integer.
The average of the first five squares is (1^2+2^2+3^2+4^2+5^2)/ 5=11, integer, and constitutes a(2).

Zak Seidov, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A050248, A078617, A154293.

s=0;lst={};Do[a=(s+=n^2)/n;If[Mod[a,1]==0,AppendTo[lst,a]],{n,6!}];lst
Flatten[Table[{(1 + 3 k) (1 + 4 k), (1 + k) (11 + 12 k)}, {k, 0, 499}]] (* Zak Seidov, Aug 15 2012 *)
Module[{nn=150,sq},sq=Range[nn]^2;Select[Table[Mean[Take[sq,n]],{n,nn}],IntegerQ]] (* or *) LinearRecurrence[{1,2,-2,-1,1},{1,11,20,46,63},50] (* Harvey P. Dale, Oct 31 2013 *)

a(n) = 1/4*(12*n^2 - 6*n + (-1)^n*(4*n-1) + 1) \\ Colin Barker, Dec 26 2015

a(n) = A000330(A007310(n)) / A007310(n) = A175485(A007310(n)). - Jaroslav Krizek, May 28 2010
G.f. ( -x*(1+10*x+7*x^2+6*x^3) ) / ( (1+x)^2*(x-1)^3 ). - R. J. Mathar, Jan 25 2011
a(n) = 1/4*(12*n^2 - 6*n + (-1)^n*(4*n-1) + 1). - Colin Barker, Dec 26 2015

Edited by R. J. Mathar, Aug 20 2009

A134125 Integral quotients of partial sums of primes divided by the number of summations.

Original entry on oeis.org

5, 5, 7, 11, 16, 107, 338, 1011, 2249, 22582, 35989, 39167, 61019, 186504, 248776, 367842, 977511, 1790714, 7104697, 15450640, 42428590, 81262621, 232483021, 319278215, 364554172, 419271517, 4432367717, 14591939203, 46911464601, 78572862347, 277369665793, 281386467553

Offset: 1

Enoch Haga, Oct 09 2007

nonn

With 1 summation, the partial sum is 2+3 = 5 and 5/1 = 5 is an integer, added to sequence. With 2 summations, the partial sum is 2+3+5 = 10 and 10/2 = 5 is an integer, added to the sequence. After 3 summations, 2+3+5+7 = 17 and 17/3 = 5.6... is not an integer, no contribution to the sequence.

These are all integers of the form A007504(k+1)/k, occurring at k in A134126. Similar to A050248, which looks at A007504(k)/k. - R. J. Mathar, Oct 23 2007

			a(1) = 5 because 2+3 = 5 and 5/1 = 5, an integral quotient.
a(3) = A007504(5)/4 = 28/4 = 7.
a(4) = A007504(8)/7 = 77/7 = 11.

Cf. A007504, A050248, A134126, A134127, A134128, A134129.

With[{nn=50000000},Select[Rest[Accumulate[Prime[Range[nn]]]]/Range[nn-1],IntegerQ]] (* Harvey P. Dale, Jul 25 2013 *)

lista(pmax) = {my(k = 0, s = 2); forprime(p = 3, pmax, k++; s += p; if(!(s % k), print1(s/k, ", ")));} \\ Amiram Eldar, Apr 30 2024

10 'primes using counters 20 N=3:C=1:R=5:print 2;3,5 30 A=3:S=sqrt(N) 40 B=N\A 50 if B*A=N then N=N+2:goto 30 60 A=A+2:O=A 70 if A<=sqrt(N) then 40 80 C=C+1 90 R=R+N:T=R/C:U=R-N 100 if T=int(T) then print C;U;N;R;T:stop 110 N=N+2:goto 30

a(n) = A007504(k+1)/k where k = A134126(n).

a(21) from R. J. Mathar, Oct 23 2007
Edited by R. J. Mathar, Apr 17 2009
a(22)-a(29) from Max Alekseyev, Jan 28 2012
a(30)-a(32) from Amiram Eldar, Apr 30 2024

A125907 Numbers k such that k divides 2^4 + 3^4 + 5^4 + ... + prime(k)^4.

Original entry on oeis.org

1, 2951, 38266951, 3053263643573, 3798632877308897

Offset: 1

Alexander Adamchuk, Feb 04 2007

No more terms to 10^13. - Charles R Greathouse IV, Mar 21 2011
a(4) is less than 10^13 contradicting the previous comment. It was found using the primesieve library by Kim Walisch and gmplib. - Bruce Garner, Feb 26 2021
a(6) > 4*10^15. - Paul W. Dyson, Nov 19 2024

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.
Cf. A122102, A232869, A128168, A233893.

a(1) = 1; s = 2^4; Do[s = s + Prime[2n]^4+Prime[2n+1]^4; If[ Mod[s, 2n+1] == 0, Print[2n+1]], {n,1, 20000000}]

s=0; n=0; forprime(p=2, 4e9, s+=p^4; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Mar 21 2011

a(4) from Bruce Garner, Feb 26 2021

a(5) from Paul W. Dyson, May 09 2024

A128168 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^4 = 1 + A122102(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 20, 24, 27, 30, 32, 39, 40, 45, 48, 58, 60, 80, 88, 90, 96, 100, 120, 138, 168, 180, 207, 216, 240, 328, 342, 353, 360, 456, 470, 480, 496, 564, 591, 768, 840, 1040, 1215, 1276, 1355, 1360, 1395, 1440, 1600, 2208, 2576, 2904

Offset: 1

Alexander Adamchuk, Feb 22 2007

nonn

a(280) > 5*10^13. - Bruce Garner, Jun 05 2021

Bruce Garner, Table of n, a(n) for n = 1..279 (terms 1..215 from Robert Price)
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.

s = 1; Do[s = s + Prime[n]^4; If[ Mod[s, n] == 0, Print[n]], {n, 17500}]

A164577 Integer averages of the first perfect cubes up to some n^3.

Original entry on oeis.org

1, 12, 25, 45, 112, 162, 225, 396, 507, 637, 960, 1156, 1377, 1900, 2205, 2541, 3312, 3750, 4225, 5292, 5887, 6525, 7936, 8712, 9537, 11340, 12321, 13357, 15600, 16810, 18081, 20812, 22275, 23805, 27072, 28812, 30625, 34476, 36517, 38637, 43120

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 16 2009

Integers of the form A000537(k)/k, created by the k>0 listed in A042965. - R. J. Mathar, Aug 20 2009

Integers of the form (1/4)*n*(n+1)^2 for some n. - Zak Seidov, Aug 17 2009

			The average of the first cube is 1^3/1=1=a(1).
The average of the first two cubes is (1^3+2^3)/2=9/2, not integer, and does not contribute to the sequence.
The average of the first three cubes is (1^3+2^3+3^3)/3=12, integer, and defines a(2).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,3,-3,0,-3,3,0,1,-1).

Cf. A000537, A050248, A051456, A078617, A078618, A154293, A164576

Timing[s=0;lst={};Do[a=(s+=n^3)/n;If[Mod[a,1]==0,AppendTo[lst,a]],{n, 5!}];lst]
With[{nn=80},Select[#[[1]]/#[[2]]&/@Thread[{Accumulate[Range[ nn]^3],Range[ nn]}],IntegerQ]] (* or *) LinearRecurrence[{1,0,3,-3,0,-3,3,0,1,-1},{1,12,25,45,112,162,225,396,507,637},50] (* Harvey P. Dale, Mar 14 2020 *)

G.f.: ( x*(1+11*x+13*x^2+17*x^3+34*x^4+11*x^5+6*x^6+3*x^7) ) / ( (1+x+x^2)^3*(x-1)^4 ). - R. J. Mathar, Jan 25 2011

Changed comments to examples - R. J. Mathar, Aug 20 2009

A233893 Prime(n), where n is such that (1+sum_{i=1..n} prime(i)^4) / n is an integer.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 23, 29, 37, 47, 53, 71, 89, 103, 113, 131, 167, 173, 197, 223, 271, 281, 409, 457, 463, 503, 541, 659, 787, 997, 1069, 1279, 1321, 1511, 2203, 2297, 2381, 2423, 3221, 3331, 3413, 3541, 4093, 4327, 5849, 6473, 8291, 9851, 10429, 11177

Offset: 1

Robert Price, Dec 17 2013

nonn

a(280) > 1701962315686097. - Bruce Garner, Jun 05 2021

			a(6) = 13, because 13 is the 6th prime and the sum of the first 6 primes^4+1 = 46326 when divided by 6 equals 7721 which is an integer.

Bruce Garner, Table of n, a(n) for n = 1..279 (terms 1..215 from Robert Price)
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 = 1; Do[sm = sm + Prime[n]^4; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
Module[{nn=1400,t},t=Accumulate[Prime[Range[nn]]^4]+1;Prime[#]&/@ Transpose[Select[Thread[{Range[nn],t}],IntegerQ[#[[2]]/#[[1]]]&]][[1]]](* Harvey P. Dale, Sep 06 2015 *)

is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^4); s==0 \\ Charles R Greathouse IV, Nov 30 2013

A160758 Integer averages of first n nonprime numbers for some n.

Original entry on oeis.org

1, 8, 25, 33, 359, 2948, 3291, 4959, 22350, 33357, 60907, 80962, 8276347, 11856980, 15254419, 176009996, 967538242, 1729774831, 9977169279, 193005936726

Offset: 1

Daniel Tisdale, May 25 2009

A variant of A050248 for nonprimes.

Numbers n such that (1/n)*sum(j=1..n, A018252(j)) is an integer. - Robert G. Wilson v, Jun 05 2009

			The sum of the first 44 nonprimes is 1452. 1452 / 44 = 33, hence 33 is in the sequence.

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

Cf. A018252, A051349, A164280, A129749.

S:=[]; a:=0; c:=0; for n in [1..40000000] do if not IsPrime(n) then a+:=n; c+:=1; if a mod c eq 0 then Append(~S, a div c); end if; end if; end for; S; // Klaus Brockhaus, Aug 11 2009

lst = {}; s = 0; c = 0; k = 1; While[k < 2700000000, If[ !PrimeQ@k, c++; s = s + k; If[Mod[s, c] == 0, AppendTo[lst, s/c]]]; k++ ]; lst (* Robert G. Wilson v, Jun 05 2009 *)
a=0;lst={};Do[If[ !PrimeQ[n],m=n;a+=m;If[a/n==IntegerPart[a/n],AppendTo[lst,a/n]]],{n,9!}];lst

a(n) = A164280(n) / A129749(n).

a(6) - a(16) from Robert G. Wilson v, Jun 05 2009
a(17) - a(19) from Donovan Johnson, Sep 16 2009
Edited by N. J. A. Sloane, May 11 2010
a(20) from Donovan Johnson, May 20 2010

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

Original entry on oeis.org

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

cp's OEIS Frontend

A233862 Prime(k), where k is such that (1 + Sum_{i=1..k} prime(i)^2) / k is an integer.

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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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A164576 Integer averages of the set of the first positive squares up to some n^2.

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A134125 Integral quotients of partial sums of primes divided by the number of summations.

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A125907 Numbers k such that k divides 2^4 + 3^4 + 5^4 + ... + prime(k)^4.

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A128168 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^4 = 1 + A122102(k).

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A164577 Integer averages of the first perfect cubes up to some n^3.

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