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.

Showing 1-10 of 15 results. Next

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

Views

Author

Alexander Adamchuk, Feb 22 2007

Keywords

Comments

a(280) > 5*10^13. - Bruce Garner, Jun 05 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]^4; If[ Mod[s, n] == 0, Print[n]], {n, 17500}]

A122127 Indices n such that A122102[n] = Sum[ Prime[k]^4, {k,1,n}] is a prime.

Original entry on oeis.org

2, 32, 90, 110, 134, 152, 168, 180, 194, 212, 222, 234, 270, 290, 300, 302, 368, 530, 540, 548, 570, 582, 630, 650, 698, 810, 848, 888, 914, 920, 930, 968, 990, 1100, 1122, 1148, 1218, 1232, 1242, 1320, 1340, 1350, 1364, 1388, 1418, 1448, 1452, 1514
Offset: 1

Views

Author

Alexander Adamchuk, Aug 20 2006

Keywords

Comments

Corresponding primes of form Sum[ Prime[k], {k,1,n} ] are listed in A122128[n] = {97,1567204831,771745495049,...}.

Crossrefs

Cf. A122102, A122128.

Programs

  • Mathematica
    Select[Range[1514],PrimeQ[Sum[Prime[k]^4,{k,1,#1}]]&]

A122128 Primes of form Sum[ Prime[k]^4, {k,1,n} ] or primes in A122102[n].

Original entry on oeis.org

97, 1567204831, 771745495049, 2523423764749, 7766703722053, 16167518745751, 28962816187367, 42932141486099, 65562552714433, 109392979058371, 142311158323421, 194472382292633, 434809071709709, 654363068549089
Offset: 1

Views

Author

Alexander Adamchuk, Aug 20 2006

Keywords

Comments

A122102[n] = Sum[ Prime[k]^4, {k,1,n} ] begins {16,97,722,3123,17764,46325,...}. Indices n such that A122102[n] is prime are listed in A122127[n] = {2,32,90,110,134,152,168,180,194,212,222,234,270,290,300,...}.

Crossrefs

Cf. A122102, A122127.

Programs

  • Mathematica
    Select[Table[Sum[Prime[k]^4,{k,1,n}],{n,1,700}],PrimeQ[ #1]&]

Formula

a(n) = A122102[ A122127[n] ].

A264900 Integers k such that A122102(k) + 1 is prime.

Original entry on oeis.org

1, 13, 43, 71, 101, 149, 163, 191, 233, 257, 259, 277, 307, 311, 373, 389, 421, 439, 463, 563, 571, 617, 647, 743, 751, 763, 871, 899, 907, 971, 1099, 1171, 1223, 1429, 1517, 1577, 1621, 1631, 1687, 1691, 1709, 1741, 1757, 1759, 1777, 1841, 1871, 1963
Offset: 1

Views

Author

Altug Alkan, Nov 27 2015

Keywords

Comments

Only a(11) = 259 is a composite number for n <= 25.
Initial corresponding primes are 17, 6870733, 9723349723 and 190977764951.

Examples

			a(1) = 1 because 2^4 + 1 = 17 is prime.

Crossrefs

Cf. A122102.

Programs

  • Mathematica
    Select[Range@ 2000, PrimeQ[Sum[Prime[k]^4, {k, #}] + 1] &] (* Michael De Vlieger, Nov 28 2015 *)
  • PARI
    a(n) = sum(k=1, n, prime(k)^4)+1;
for(n=0, 3000, if(ispseudoprime(a(n)) , print1(n, ", ")))

A122103 Sum of the fifth powers of the first n primes.

Original entry on oeis.org

32, 275, 3400, 20207, 181258, 552551, 1972408, 4448507, 10884850, 31395999, 60025150, 129369107, 245225308, 392233751, 621578758, 1039774251, 1754698550, 2599294851, 3949419958, 5753649309, 7826720902, 10903777301, 14842817944
Offset: 1

Views

Author

Alexander Adamchuk, Aug 20 2006

Keywords

Comments

a(n) is prime for n = {66, 148, 150, 164, 174, 214, 238, 264, 312, 328, 354, 440, 516, 536, 616, 624, 724, 744, 774, 836, 940, ...} = A122125. Primes of this form are listed in A122126 = {32353461605953, 9874820441996857, 10821208357045699, ...}.

Examples

			a(2) = 275 because the first two primes are 2 and 3, the fifth powers of which are 32 and 243, and 32 + 243 = 275.
a(3) = 3400, because the third prime is 5, its fifth power if 3125 and 275 + 3125 = 3400.

Links

Crossrefs

Cf. A050997, A007504, A024450, A098999, A122102, A122125, A122126.

Programs

  • Mathematica
    Table[Sum[Prime[k]^5, {k, n}], {n, 100}]
  • PARI
    a(n)=sum(i=1,n,prime(i)) \\ Charles R Greathouse IV, Nov 30 2013

Formula

a(n) = sum(k = 1 .. n, prime(k)^5).
a(n) = 1/6*n^6*log(n)^5 + O(n^6*log(n)^4*log(log(n))). The proof is similar to proof for A007504(n) (see link of Shevelev). For a generalization, see comment in A122102. - Vladimir Shevelev, Aug 14 2013

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

Views

Author

Alexander Adamchuk, Feb 04 2007

Keywords

Comments

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

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.
Cf. A122102, A232869, A128168, A233893.

Programs

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

Extensions

a(4) from Bruce Garner, Feb 26 2021
a(5) from Paul W. Dyson, May 09 2024

A122138 Indices k such that A122136(k) is a prime.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 11, 12, 14, 15, 18, 20, 22, 23, 26, 27, 32, 36, 38, 39, 40, 44, 47, 48, 50, 51, 52, 54, 55, 56, 58, 59, 60, 64, 66, 68, 71, 72, 74, 76, 78, 80, 83, 84, 86, 88, 89, 90, 92, 94, 95, 96, 98, 100, 102, 103, 107, 108, 110, 112, 114, 116, 118, 120, 122, 126
Offset: 1

Views

Author

Alexander Adamchuk, Aug 21 2006

Keywords

Comments

The corresponding primes are listed in A122139.

Crossrefs

Cf. A122136, A122137, A122139, A002110, A024450, A007504, A098999, A122102, A122103.

Programs

  • Mathematica
    Select[Range[200],PrimeQ[Numerator[Sum[Prime[k]^2,{k,1,#1}]/Product[Prime[k],{k,1,#1}]]]&]

A122142 Numbers m such that m divides sum of 5th powers of the first m primes A122103(m).

Original entry on oeis.org

1, 25, 837, 5129, 94375, 271465, 3576217, 3661659, 484486719, 2012535795, 31455148645, 95748332903, 145967218799, 165153427677, 21465291596581, 97698929023845
Offset: 1

Views

Author

Alexander Adamchuk, Aug 21 2006

Keywords

Comments

No other terms up to 10^8. - Stefan Steinerberger, Jun 06 2007
a(11) > 6*10^9. - Donovan Johnson, Oct 15 2012
a(13) > 10^11. - Robert Price, Mar 30 2013
a(15) > 10^12. - Paul W. Dyson, Jan 04 2021
a(16) > 2.2*10^13. - Bruce Garner, May 09 2021
a(17) > 10^14. - Paul W. Dyson, Feb 04 2022
a(17) > 10^15. - Paul W. Dyson, Nov 19 2024

Examples

			a(2) = 25 because 25 is the first number n>1 that divides A122103[n] = Sum[ Prime[k]^5, {k,1,n} ].
Mod[ A122103[25], 25] = Mod[ 2^5 + 3^5 + 5^5 + ... + 89^5 + 97^5, 25 ] = 0.

Links

Crossrefs

Cf. A122103, A098999, A007504, A045345, A024450, A111441, A122102, A122140.

Programs

  • Mathematica
    s = 0; t = {}; Do[s = s + Prime[n]^5; If[ Mod[s, n] == 0, AppendTo[t, n]], {n, 1000000}]; t
Module[{nn = 4*10^6},Select[Thread[{Range[nn], Accumulate[ Prime[ Range[ nn]]^5]}], Divisible[#[[2]], #[[1]]] &]][[All, 1]] (* Generates the first 8 terms; to generate more, increase the value of nn, but the program may take a long time to run. *) (* Harvey P. Dale, Aug 26 2019 *)

Extensions

2 more terms from Stefan Steinerberger, Jun 06 2007
a(9)-a(10) from Donovan Johnson, Oct 15 2012
a(11)-a(12) from Robert Price, Mar 30 2013
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

A133550 Sum of fifth powers of n odd primes.

Original entry on oeis.org

243, 3368, 20175, 181226, 552519, 1972376, 4448475, 10884818, 31395967, 60025118, 129369075, 245225276, 392233719, 621578726, 1039774219, 1754698518, 2599294819, 3949419926, 5753649277, 7826720870, 10903777269, 14842817912
Offset: 1

Views

Author

Artur Jasinski, Sep 16 2007

Keywords

Examples

			a(2)=3368 because 3^5+5^5 = 3368.

Crossrefs

Cf. A007148, A007504, A024450, A098999, A122102, A122103, A133547, A133548, A133550, A133551.

Programs

  • Mathematica
    c = 5; a = {}; b = 0; Do[b = b + Prime[n]^c; AppendTo[a, b], {n, 2, 1000}]; a

Formula

a(n) = A122103(n+1)-32.

A133549 Sum of the fourth powers of the first n odd primes.

Original entry on oeis.org

81, 706, 3107, 17748, 46309, 129830, 260151, 539992, 1247273, 2170794, 4044955, 6870716, 10289517, 15169198, 23059679, 35177040, 49022881, 69174002, 94585683, 122983924, 161934005, 209392326, 272134567, 360663848, 464724249, 577275130
Offset: 1

Views

Author

Artur Jasinski, Sep 16 2007

Keywords

Examples

			a(2)=706 because 3^4 + 5^4 = 706.

Crossrefs

Cf. A007148, A007504, A024450, A098999, A122102, A133547, A133548, A133550, A133551.

Programs

  • Maple
    a:=proc (n) options operator, arrow: add(ithprime(j)^4, j=2..n+1) end proc: seq(a(n),n=1..26); # Emeric Deutsch, Oct 02 2007
  • Mathematica
    c = 4; a = {}; b = 0; Do[b = b + Prime[n]^c; AppendTo[a, b], {n, 2, 1000}]; a

Formula

a(n) = A122102(n+1) - 16. - Michel Marcus, Nov 05 2013

Extensions

Comment corrected by Michel Marcus, Nov 05 2013
Showing 1-10 of 15 results. Next

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