A131263
Numbers k such that k divides 2^9 + 3^9 + 5^9 + ... + prime(k)^9.
Original entry on oeis.org
1, 281525, 1011881, 13721649, 309777093, 417800903, 12252701193, 27377813605, 37762351523, 245773819141, 51230573255953, 82578361848569, 277900491430385
Offset: 1
Cf.
A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
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s = 0; Do[s = s + Prime[n]^9; If[ Mod[s, n] == 0, Print[n]], {n, 1100000}]
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s=0; n=0; forprime(p=2, 1e9, s+=p^9; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Apr 14 2011
A131264
Numbers k such that k divides 2^10 + 3^10 + 5^10 + ... + prime(k)^10.
Original entry on oeis.org
1, 269, 41837, 36626159, 154578947, 2155054465, 19410890423, 30691222355, 247555091527, 2201220228533, 227735225320519, 478444326378215
Offset: 1
Cf.
A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
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s = 0; Do[s = s + Prime[n]^10; If[ Mod[s, n] == 0, Print[n]], {n, 1000000}]
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s=0;n=0;forprime(p=2, 1e9, s+=p^10; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Apr 14 2011
A131272
Numbers k such that k divides Sum_{j=1..k} prime(j)^12.
Original entry on oeis.org
1, 37, 7187, 3140407, 4986959, 5139161, 751213639, 163007938237, 5134788477263, 36197588005399, 940901369608517
Offset: 1
Cf.
A085450 = smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.
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s = 0; Do[s = s + Prime[n]^12; If[ Mod[s, n] == 0, Print[n]], {n, 1000000}]
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s=0; n=0; forprime(p=2,1e9,s+=p^12; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Apr 14 2011
A131274
Numbers m such that m divides Sum_{k=1..m} prime(k)^14.
Original entry on oeis.org
1, 295, 455, 4361, 10817, 132680789, 334931875, 957643538339, 82185210732157
Offset: 1
Cf.
A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
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s = 0; Do[s = s + Prime[n]^14; If[ Mod[s, n] == 0, Print[n]], {n, 660000000}] (* Robert G. Wilson v, Jul 01 2007 *)
With[{nn=11000},Select[Thread[{Accumulate[Prime[Range[nn]]^14],Range[ nn]}],Divisible[ #[[1]],#[[2]]]&]][[All,2]] (* The program generates the first 5 terms of the sequence. To generate more, increase the value of nn. *) (* Harvey P. Dale, Jun 25 2021 *)
A131275
Numbers k such that k divides Sum_{j=1..k} prime(j)^15.
Original entry on oeis.org
1, 17, 25, 31, 1495, 5555, 8185, 8647, 106841, 187329, 345377, 1811351, 2179119, 2863775, 6368703, 10250821, 59137893, 337430815, 11349203711, 183233304195, 12538656829431, 40154010310477, 1761333303516473
Offset: 1
Cf.
A085450 = smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.
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s = 0; Do[s = s + Prime[n]^15; If[ Mod[s, n] == 0, Print[n]], {n, 400000}]
With[{nn = 3*10^6},Select[Thread[{Accumulate[Prime[ Range[nn]]^15],Range[ nn]}],Divisible[#[[1]], #[[2]]] &]][[All, 2]] (* This will generate the first 14 terms of the sequence; to generate more, increase the value of nn, but it may take a long time to run. *) (* Harvey P. Dale, Oct 03 2016 *)
A131276
Numbers m such that m divides Sum_{k=1..m} prime(k)^16.
Original entry on oeis.org
1, 3131, 6289, 323807, 443371, 83802527023, 4076111200313
Offset: 1
Cf.
A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
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s = 0; Do[s = s + Prime[n]^16; If[ Mod[s, n] == 0, Print[n]], {n, 500000}]
Transpose[Select[With[{nn=500000},Thread[{Range[nn],Accumulate[ Prime[ Range[nn]]^16]}]], Divisible[ #[[2]],#[[1]]]&]][[1]]
A131277
Numbers m that divide Sum_{k=1..m} prime(k)^17.
Original entry on oeis.org
1, 395191, 697717, 1078323, 2050797, 10543929, 386099691, 2467825171, 4488040933, 17387575533, 39641205433, 825688143387, 2800262033655, 3214748608393, 5174884331693, 16485974355373, 20683624349423, 34390023299149, 629341300687639
Offset: 1
Cf.
A085450 (smallest m > 1 that divide Sum_{k=1..m} prime(k)^n).
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s = 0; Do[s = s + Prime[n]^17; If[ Mod[s, n] == 0, Print[n]], {n, 1100000}]
A131278
Numbers m such that m divides the sum of the 18th powers of the first m primes.
Original entry on oeis.org
1, 37, 265, 17207, 9382589, 970248431, 2427811793, 156281194823, 2955922292131, 372012276565795
Offset: 1
Cf.
A085450 = smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.
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s = 0; Do[s = s + Prime[n]^18; If[ Mod[s, n] == 0, Print[n]], {n, 10^6}]
With[{nn = 18000}, Transpose[With[{c = Thread[{Range[nn], Accumulate[Prime[ Range[nn]]^18]}]}, Select[c, Divisible[Last[#], First[#]] &]]][[1]]] (* Harvey P. Dale, Dec 19 2011 *)
A131279
Numbers k such that k divides Sum_{j=1..k} prime(j)^19.
Original entry on oeis.org
1, 25, 453, 677, 839, 1015, 3735, 4175, 4413, 10369, 14239, 43311, 452567, 1274185, 14102849, 37801813, 71271705, 93524231, 386557609, 2151748733, 261349938459, 761474469415, 1284262332971, 5115376212971, 17863411895047, 122189141425495
Offset: 1
Cf.
A085450 (smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n).
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s = 0; Do[s = s + Prime[n]^19; If[ Mod[s, n] == 0, Print[n]], {n, 50000}]
A131590
Sum of the squares of the first 2^n primes.
Original entry on oeis.org
4, 13, 87, 1027, 13275, 155995, 1789395, 19523155, 204330315, 2081006083, 20605602003, 199966727443, 1908356153955, 17942429101363, 166591116531123, 1529578004981731, 13917470067182067, 125565110929591171, 1124685106917162579, 10009134886727192611
Offset: 0
The sum of the squares of the first 2^2 primes is a(2) = 4 + 9 + 25 + 49 = 87.
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Array[Total[Prime[Range[2^#]]^2]&,20,0] (* James C. McMahon, Feb 25 2025 *)
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sumprimesq(n,b) = { local(x,y,s,a); for(y=0,n, s=0; for(x=1,b^y, s+=prime(x)^2; ); print1(s","); ) }
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lista(pmax) = {my(s = 0, k = 0, pow = 1); forprime(p = 1, pmax, k++; s += p^2; if(k == pow, print1(s, ", "); pow *= 2));} \\ Amiram Eldar, Jul 06 2024
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from sympy import sieve, prime
def a(n): return sum(p*p for p in sieve.primerange(1, prime(2**n)+1))
print([a(n) for n in range(20)]) # Michael S. Branicky, Apr 13 2021
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