cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

Views

Author

Alexander Adamchuk, Jun 25 2007

Keywords

Comments

a(26) > 2*10^13. - Bruce Garner, Jun 30 2021
a(27) > 5*10^14. - Paul W. Dyson, Dec 31 2024

Links

Crossrefs

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.

Programs

  • Mathematica
    s = 0; Do[s = s + Prime[n]^19; If[ Mod[s, n] == 0, Print[n]], {n, 50000}]

Extensions

More terms from Ryan Propper, Oct 09 2007
a(19)-a(20) from Robert Price, Dec 15 2013
a(21) from Karl-Heinz Hofmann, Feb 24 2021
a(22) from Bruce Garner, Mar 01 2021
a(23) from Bruce Garner, Mar 08 2021
a(24) from Bruce Garner, Apr 14 2021
a(25) from Bruce Garner, Jun 30 2021
a(26) from Paul W. Dyson, Jun 27 2023

A179860 Integer averages of first n noncomposites for some n.

Original entry on oeis.org

1, 2, 6, 636, 1592, 2574, 292656, 917042, 1108972, 1678508, 3334890730, 3981285760, 28567166356, 32739591796, 83332116034, 871263881618, 1055495274756
Offset: 1

Views

Author

Ray Chandler, Jul 29 2010

Keywords

Comments

A variant of A050248 (primes), A073263 (composites) and A160758 (nonprimes).

Examples

			Sum of first 7 noncomposites is 42; 42 / 7 = 6 is in the sequence.

Crossrefs

Cf. A008578 (noncomposites), A014284 (sum of first n noncomposites).
Cf. A179859, A179861, A158682.

Formula

a(n) = A179861(n) / A179859(n) = A014284(A179859(n)) / A179859(n).

Extensions

a(16)-a(17) from Robert Price, Apr 21 2013

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

Views

Author

Robert Price, Mar 29 2013

Keywords

Examples

			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.

Links

Crossrefs

Cf. A085450 (smallest m > 1 that divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

  • Mathematica
    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

Extensions

a(11) from Paul W. Dyson, Jan 05 2021
a(12) from Bruce Garner, Mar 01 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

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

Views

Author

Robert Price, Mar 30 2013

Keywords

Comments

a(17) > 3.7*10^16. - Paul W. Dyson, Jan 17 2025

Examples

			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.

Links

Crossrefs

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.

Programs

  • Mathematica
    t = {}; sm = 0; Do[sm = sm + Prime[n]^5; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)

Extensions

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

A232822 Prime(m), where m is such that (Sum_{k=1..m} prime(k)^8) / m is an integer.

Original entry on oeis.org

2, 191, 12599173, 53029063, 22806625723729, 27568116247823, 41455846079203, 289700908580893, 1194728983756489, 6275148480751847
Offset: 1

Views

Author

Robert Price, Nov 30 2013

Keywords

Comments

The primes correspond to indices m = 1, 43, 824747, 3171671, ... = A125828. - M. F. Hasler, Dec 01 2013
a(10) > 1352363608564489. - Bruce Garner, Jul 07 2021
a(11) > 18205684894350047. - Paul W. Dyson, Dec 03 2024

Examples

			a(2) = 191, because 191 is the 43rd prime and the sum of the first 43 primes^8 = 7287989395992721002 = 43 * 169488125488202814.

Links

Crossrefs

Cf. A125828.
Cf. A085450 (smallest m > 1 that divides Sum_{k=1..m} prime(k)^n).
Cf. A007504, A045345, A171399, A128165, A233523, A050247, A050248.
Cf. A024450, A111441, A217599, A128166, A233862, A217600, A217601.

Programs

  • Mathematica
    t = {}; sm = 0; Do[sm = sm + Prime[n]^8; If[Mod[sm, n] == 0, AppendTo[t, Prime[n]]], {n, 100000}]; t (* Derived from A217599 *)
  • PARI
    is(n)=if(!isprime(n),return(0)); my(t=primepi(n),s); forprime(p=2,n,s+=Mod(p,t)^8); s==0 \\ Charles R Greathouse IV, Nov 30 2013
  • PARI
    S=n=0;forprime(p=1,,(S+=p^8)%n++||print1(p",")) \\ M. F. Hasler, Dec 01 2013

Formula

a(n) = prime(A125828(n)). - M. F. Hasler, Dec 01 2013

Extensions

a(5)-a(6) from Paul W. Dyson, Jan 01 2021
a(7) from Bruce Garner, Mar 02 2021
a(8) from Bruce Garner, Mar 30 2021
a(9) from Bruce Garner, Jul 07 2021
a(10) from Paul W. Dyson, Jul 07 2023

A232865 Primes p such that the average of the seventh power of primes up to p is an integer.

Original entry on oeis.org

2, 97, 14293, 247997, 7979737, 15749303, 344468591, 25934255929, 40224745543, 16495569405383, 53941465463489, 84897825837611, 244949151647509, 757938163218799, 1594375071689591, 1674528348898463, 5347819657753523, 6152744788157173, 47008163075851819
Offset: 1

Views

Author

M. F. Hasler, Dec 01 2013

Keywords

Comments

Otherwise said, prime(m) such that m divides prime(1)^7 + ... + prime(m)^7.
a(17) > 1877564517734839. - Bruce Garner, Aug 30 2021
a(18) > 5493145969370039. - Paul W. Dyson, Mar 02 2022
a(19) > 6728882502496787. - Bruce Garner, Sep 18 2022

Links

Crossrefs

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.

Programs

  • PARI
    S=n=0;forprime(p=1,,(S+=p^7)%n++||print1(p","))

Formula

a(n) = prime(A125826(n)).

Extensions

a(10) from Paul W. Dyson, Jan 05 2021
a(11)-a(12) from Bruce Garner, Feb 26 2021
a(13) from Bruce Garner, Mar 23 2021
a(14) from Bruce Garner, May 19 2021
a(15)-a(16) from Bruce Garner, Aug 30 2021
a(17) from Paul W. Dyson, Mar 02 2022
a(18) from Bruce Garner, Sep 18 2022
a(19) from Paul W. Dyson, Jan 17 2024

A236182 Sum of the sixth powers of the first n primes.

Original entry on oeis.org

64, 793, 16418, 134067, 1905628, 6732437, 30870006, 77915887, 225951776, 820775097, 1708278778, 4274005187, 9024109428, 15345472477, 26124687806, 48289048935, 90469582576, 141989956937, 232448339106, 360548623027, 511882849316, 754970304837, 1081910678206
Offset: 1

Views

Author

Robert Price, Jan 19 2014

Keywords

Links

Crossrefs

Cf. A030516 (6th powers of primes).
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.

Programs

  • Mathematica
    Table[Sum[Prime[k]^6, {k, n}], {n, 100}]
Accumulate[Prime[Range[30]]^6] (* Harvey P. Dale, Oct 29 2023 *)
  • PARI
    a(n) = sum(k=1, n, prime(k)^6); \\ Michel Marcus, Mar 01 2022

Formula

a(n) = Sum_{k = 1..n} prime(k)^6.

A363477 Numbers that are integer averages of first k odd primes for some k.

Original entry on oeis.org

3, 4, 5, 133, 169, 1117, 2406, 3564, 6141, 7429, 8220, 8475, 14193, 33543, 121049, 211785, 877650, 5948070, 8494543, 27820975, 41428418, 130490020, 139053727, 200325407, 291720414, 893706168, 977748014, 2103851425, 2173904606, 5996888467, 15790305181
Offset: 1

Views

Author

Ya-Ping Lu, Jun 07 2023

Keywords

Examples

			5 is a term because 5 is the average of the first 3 odd primes, 3, 5 and 7.
133 is a term because 133 is the average of the first 60 odd primes, 3, 5, 7, 11, ..., 281 and 283.

Crossrefs

Cf. A050248, A071148, A097961.

Programs

  • Python
    from sympy import sieve
L = sieve.primerange(3, 4*10**10); s, k = 0, 0
for p in L:
    s += p;  k += 1
    if s%k == 0: print(s//k, end = ", ")

Formula

a(n) = Sum_{i=1..A097961(n)} prime(i)/n.

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

Views

Author

Alexander Adamchuk, Feb 22 2007, Feb 23 2007

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Mathematica
    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

Extensions

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

A128170 Numbers k such that k divides 1 + Sum_{j=1..k} prime(j)^6.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 18, 21, 24, 27, 28, 29, 34, 36, 42, 45, 48, 54, 56, 63, 72, 84, 112, 126, 159, 168, 174, 198, 204, 216, 252, 310, 312, 336, 360, 400, 408, 441, 504, 505, 540, 588, 591, 657, 672, 716, 864, 900, 1080, 1152, 1316, 1350, 1380, 1680, 1722
Offset: 1

Views

Author

Alexander Adamchuk, Feb 22 2007

Keywords

Comments

a(301) > 1.4*10^13. - Bruce Garner, Apr 07 2021

Links

Crossrefs

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.

Programs

  • Mathematica
    s = 1; Do[s = s + Prime[n]^6; If[ Mod[s, n] == 0, Print[n]], {n, 10000}]
Module[{nn=1750,pr},pr=Accumulate[Prime[Range[nn]]^6];Select[Thread[ {Range[ nn],pr}],Divisible[#[[2]]+1,#[[1]]]&]][[All,1]] (* Harvey P. Dale, Feb 10 2019 *)
  • PARI
    n=0; s=1; forprime(p=2,,s+=p^6; if(s%n++==0, print1(n", "))) \\ Charles R Greathouse IV, Dec 03 2013
Previous Showing 31-40 of 105 results. Next

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