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 1-6 of 6 results.

A073825 Numbers n such that Sum_{k=1..n} k^k is prime.

Original entry on oeis.org

2, 5, 6, 10, 30
Offset: 1

Views

Author

Rick L. Shepherd, Aug 13 2002

Keywords

Comments

Any additional terms are greater than 1320 with the next prime having more than 4120 digits.
No terms out to 3000. The next term would yield a prime with over 10000 digits. - John Sillcox (johnsillcox(AT)hotmail.com), Aug 05 2003
For every n, a(n) must be equal to 1 or 2 (mod 4) because Sum[k^k,{k,a(n)}] must be odd. Any additional terms are greater than 5368 with the next prime having more than 20025 digits. - Farideh Firoozbakht, Aug 09 2003
Soundararajan finds an asymptotic upper bound of log k / log log k prime numbers of the form 1^1 + 2^2 + ... + n^n less than k; that is, n << log a(n) / log log a(n). - Charles R Greathouse IV, Aug 27 2008
According to Andersen, the next term is larger than 28000, see Rivera link. - M. F. Hasler, Mar 01 2009
Conjecture: This sequence is infinite. - Daniel Hoying, Jul 20 2020

Links

Crossrefs

Cf. A073826 (corresponding primes), A001923 (Sum k^k, k=1..n).

Programs

  • Mathematica
    v={}; Do[If[(Mod[n, 4]==1||Mod[n, 4]==2)&&PrimeQ[Sum[k^k, {k, n}]], v=Insert[v, n, -1]; Print[v]], {n, 5368}]
  • PARI
    s=0; for(k=1,1320, s=s+k^k; if(isprime(s), print1(k,", ")))

Formula

log a(n) >> n log^2 n. - Charles R Greathouse IV, May 17 2016

Extensions

Edited by Charles R Greathouse IV, Oct 27 2010

A122166 Numbers n such that 1 + Sum k^k (k=1..n) is prime.

Original entry on oeis.org

1, 52, 124, 431
Offset: 1

Views

Author

Alexander Adamchuk, Aug 23 2006

Keywords

Comments

Primes of the form A001923[n] = Sum k^k (k=1..n) are given in A073826[n] and their indices are given in A073825[n] = {2,5,6,10,30,...}.

Crossrefs

Cf. A073825, A001923.

Programs

  • Mathematica
    s=1; Do[s=s+k^k;If[PrimeQ[s],Print[{k,s}]],{k,1,500}]

A340392 Primes of the form Sum_{k=i..j} k^k.

Original entry on oeis.org

5, 31, 283, 3413, 50069, 17650823, 10405071317, 449317973725128511, 18895749970915969007, 18896062057839748031, 846136323944176515589, 40192544390028896900861, 40192544398944997349117, 40192544399240696440217, 208492413443704093346554910065262730566475781
Offset: 1

Views

Author

J. M. Bergot and Robert Israel, Jan 05 2021

Keywords

Examples

			a(1) = 5 = 1^1 + 2^2 is prime.
a(2) = 31 = 2^2 + 3^3 is prime.
a(3) = 283 = 3^3 + 4^4 is prime.
a(4) = 3413 = 1^1 + 2^2 + 3^3 + 4^4 + 5^5 is prime.
a(5) = 50069 = 1^1 + 2^2 + 3^3 + 4^4 + 5^5 + 6^6 is prime.
a(6) = 17650823 = 3^3 + 4^4 + 5^5 + 6^6 + 7^7 + 8^8 is prime.

Links

Crossrefs

Cf. A073826.

Programs

  • Maple
    B:= [0,seq(i^i,i=1..100)]:
S:= ListTools:-PartialSums(B):
R:=select(t -> t < 101^101 and isprime(t), {seq(seq(S[i]-S[j],j=1..i-1),i=2..101)}):
sort(convert(R,list));

A175232 The smallest prime divisor of 1 + 2^2 + 3^3 + ... + n^n.

Original entry on oeis.org

5, 2, 2, 3413, 50069, 2, 2, 7, 10405071317, 2, 2, 88799, 3, 2, 2, 3, 3, 2, 2, 5, 3, 2, 2, 3, 7, 2, 2, 7, 208492413443704093346554910065262730566475781, 2, 2, 3, 17, 2, 2, 5, 61, 2, 2, 71, 11, 2, 2, 11, 7, 2, 2, 5, 3, 2, 2, 3, 3, 2, 2, 23, 3, 2, 2, 3, 44818693, 2, 2, 5
Offset: 2

Views

Author

Michel Lagneau, Mar 09 2010

Keywords

Examples

			a(2) = 5 divides 1 + 2^2.
a(3) = 2 divides 1 + 2^2 + 3^3 = 32.
a(4) = 2 divides 1 + 2^2 + 3^3 + 4^4 = 288.
a(5) = 3413 divides 1 + 2^2 + 3^3 + 4^4 + 5^5 = 3413.
a(13) = 88799 divides 1 + 2^2 + 3^3 + ... + 13^13 = 88799 * 3514531963.

Crossrefs

Cf. A073826, A122166.

Programs

  • Maple
    with(numtheory): s :=1: for n from 2 to 60 do ;s := s+ n^n: s1 := ifactors(s)[2] : s2 :=s1[i][1], i=1..nops(s1):print(s1[1][1]):od:
  • Mathematica
    a[n_] := FactorInteger[Sum[k^k, {k, 1, n}]][[1, 1]]; Array[a, 20, 2] (* Amiram Eldar, Feb 04 2020 *)

Formula

a(n) = A020639(A001923(n)).

Extensions

Edited by R. J. Mathar, Mar 16 2010
a(61)-a(65) from Amiram Eldar, Feb 04 2020

A214662 Greatest prime divisor of 1 + 2^2 + 3^3 + ... + n^n.

Original entry on oeis.org

5, 2, 3, 3413, 50069, 8089, 487, 2099, 10405071317, 1274641129, 164496735539, 3514531963, 15624709, 23747111, 10343539, 56429700667, 1931869473647715169, 2383792821710269, 144326697012150473, 2053857208873393249, 128801386946535261205906957, 2298815880166789
Offset: 2

Views

Author

Michel Lagneau, Jul 24 2012

Keywords

Examples

			a(2) = 5 divides 1 + 2^2 ;
a(3) = 2 divides 1 + 2^2 + 3^3 = 32 ;
a(4) = 3 divides 1 + 2^2 + 3^3 + 4^4 = 288 = 2^5*3^2 ;
a(5) = 3413 divides 1 + 2^2 + 3^3 + 4^4 + 5^5 = 3413.
a(13) = 3514531963 divides 1 + 2^2 + 3^3 + ... + 13^13 = 88799 * 3514531963.

Links

Crossrefs

Cf. A001923, A006530, A073826, A122166, A175232.

Programs

  • Magma
    [Max(PrimeDivisors(&+[k^k:k in [1..n]])):n in [2..23]]; // Marius A. Burtea, Feb 09 2020
  • Maple
    with (numtheory):
s:= proc(n) option remember; `if`(n=1, 1, s(n-1)+n^n) end:
a:= n-> max(factorset(s(n))[]):
seq (a(n), n=2..23);  # Alois P. Heinz, Jul 24 2012
  • Mathematica
    s = 1; Table[s = s + n^n; FactorInteger[s][[-1, 1]], {n, 2, 24}] (* T. D. Noe, Jul 25 2012 *)
Module[{nn=30,lst},lst=Table[n^n,{n,nn}];Table[FactorInteger[Total[Take[lst,k]]][[-1,1]],{k,2,nn}]] (* Harvey P. Dale, Oct 09 2022 *)
  • PARI
    a(n) = vecmax(factor(sum(k=1, n, k^k))[,1]); \\ Michel Marcus, Feb 09 2020

Formula

a(n) = A006530(A001923(n)).

A308610 Numbers k such that Sum_{j=1..k} composite(j)^composite(j) is prime, where composite(j) is the j-th composite number.

Original entry on oeis.org

7, 194, 219
Offset: 1

Views

Author

Metin Sariyar, Aug 23 2019

Keywords

Comments

If it exists, a(4) > 25800.

Examples

			7 is a term because for the first 7 composites, 4^4 + 6^6 + 8^8 + 9^9 + 10^10 + 12^12 + 14^14 = 11120933330250889 is prime.

Crossrefs

Cf. A000040, A002808, A053782, A073826, A132637.

Programs

  • Mathematica
    f[ n_Integer ] := Block[ {k = n + PrimePi[ n ] + 1}, While[ k - PrimePi[ k ] - 1 != n, k++ ]; k ]; s = 0; Do[ s = s + f[n]^f[n]; If[ PrimeQ[ s ], Print[ n ] ], {n, 1, 1000} ]
Showing 1-6 of 6 results.

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