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}]
A164280
a(n) is the sum of the first A129749(n) nonprimes.
Original entry on oeis.org
1, 64, 800, 1452, 207143, 14934568, 18656679, 42721785, 889239450, 1991045973, 6684665064, 11847007536, 127880982508860, 262907426075600, 435646635220423, 58536266213035868, 1777764795352216572, 5690843954660983949, 190101452851104748107, 71537286176000166753366
Offset: 1
A129749(4) = 44; sum of first 44 nonprimes is 1452, so a(4) = 1452.
A179861
a(n) is the sum of the first A179859(n) noncomposites.
Original entry on oeis.org
1, 6, 42, 143100, 775304, 1891890, 14646554832, 130985694070, 188757015148, 419047914740, 1055777525624570390, 1492138298614167680, 70288308055831268412, 91779857115464381780, 571686203669195590338, 56978071532766214007450, 83023388015844408083484
Offset: 1
A179859(3) = 7; sum of first 7 noncomposites is 42, so a(3) = 42.
A223936
Numbers prime(m), such that (Sum_{i=1..m} prime(i)^3) / m is an integer.
Original entry on oeis.org
2, 97, 3877, 4943, 50741, 1487159, 3356117, 131047091863, 449627893189, 906460844407, 61168531626487, 141835115384731, 749668095960389, 1259394274876189, 3849791511371129, 6669425423437787, 11674340378841221, 75041264698436783
Offset: 1
a(2) = 97, because 97 is the 25th prime and the sum of the first 25 primes^3 = 4696450 when divided by 25 equals 187858 which is an integer.
Cf.
A085450 (smallest m > 1 that divides Sum_{k=1..m} prime(k)^n).
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k = 1; p = 2; s = 0; lst = {}; While[p < 1000000000, s = s + p^3; If[ Mod[s, k++] == 0, AppendTo[lst, p]]; p = NextPrime@ p]; lst
A224083
Prime(m), where m is such that (Sum_{i=1..m} prime(i)^5) / m is an integer.
Original entry on oeis.org
2, 97, 6449, 49943, 1220347, 3821963, 60252541, 61785991, 10678796441, 47363940857, 830546726491, 2639027583253, 4087115060797, 4645513891321, 711935349228079, 3393070609976863
Offset: 1
a(2) = 97, because 97 is the 25th prime and the sum of the first 25 primes^5 = 29014217650 when divided by 25 equals 1160568706 which is an integer.
Cf.
A085450 = smallest m > 1 such that m divides Sum_{k=1..m} prime(k)^n.
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t = {}; sm = 0; Do[sm = sm + Prime[n]^5; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
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