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.
%I A261722 #44 Sep 28 2024 15:52:21
%S A261722 1,7,25,91
%N A261722 Values of m such that 2^m + 3^m + 5^m + 7^m + 11^m + 13^m is a prime number.
%C A261722 2, 3, 5, 7, 11, 13 are first six consecutive prime numbers.
%C A261722 From _Bruno Berselli_, Sep 04 2015: (Start)
%C A261722 All terms are odd. In fact, assuming m even and b(k) = 4^k + 9^k + 25^k + 49^k + 121^k + 169^k, for
%C A261722 . k == 0, 2, 4 (mod 6), b(k) is divisible by 5;
%C A261722 . k == 1, 5 (mod 6), b(k) is divisible by 377 = 13*29;
%C A261722 . k == 3 (mod 6), b(k) is divisible by 29. (End)
%C A261722 From _Jon E. Schoenfield_, Mar 02 2018: (Start)
%C A261722 For n odd:
%C A261722 Let t(n) = 2^n + 3^n + 5^n + 7^n + 11^n + 13^n; then t(n) is divisible by prime p for certain pairs (p, n mod (p-1)):
%C A261722 .
%C A261722 p n mod (p-1) such that p|t(n)
%C A261722 == ============================
%C A261722 2 -
%C A261722 3 -
%C A261722 5 -
%C A261722 7 -
%C A261722 11 9
%C A261722 13 -
%C A261722 17 5
%C A261722 19 9
%C A261722 23 11
%C A261722 29 3
%C A261722 31 15
%C A261722 37 21, 29
%C A261722 41 1, 19
%C A261722 43 11, 33, 37
%C A261722 47 23
%C A261722 53 -
%C A261722 59 29, 55
%C A261722 ...
%C A261722 The smallest prime p that divides t(n) at more than three values of n mod (p-1) is 313: 313|t(n) when n mod 312 is any of the four values {39, 117, 195, 273}, i.e., when n mod (312/4 = 78) = 39.
%C A261722 The smallest prime p that divides t(n) at more than four values of n mod (p-1) is 3041: 3041|t(n) when n mod 3040 is any of the 16 values {95, 285, 475, 665, 855, 1045, 1235, 1425, 1615, 1805, 1995, 2185, 2375, 2565, 2755, 2945}, i.e., when n mod (3040/16 = 190) = 95. (End)
%C A261722 No other terms than the four terms cited less than 25000. - _Robert G. Wilson v_, Mar 07 2018
%C A261722 No other terms than the four terms cited less than 100000. - _Michael S. Branicky_, Sep 28 2024
%e A261722 1 is a term because 2^1 + 3^1 + 5^1 + 7^1 + 11^1 + 13^1 = 41 and 41 is a prime number.
%t A261722 Select[Table[{n, Sum[Prime[k]^n, {k, 6}]}, {n, 1000}], PrimeQ[#[[2]]]&] [[All, 1]] (* _Michael De Vlieger_, Aug 29 2015 *)
%o A261722 (PARI) for(n=1, 1e3, if(isprime(13^n+11^n+7^n+5^n+3^n+2^n), print1(n", ")))
%o A261722 (Magma) [n: n in [0..1000] | IsPrime(a) where a is 2^n+3^n+5^n+ 7^n+11^n+13^n]; // _Vincenzo Librandi_, Aug 30 2015
%Y A261722 Cf. A000040, A076515, A268064, A268067.
%K A261722 nonn,more
%O A261722 1,2
%A A261722 _Altug Alkan_, Aug 29 2015
%E A261722 Mathematica scripts updated by _Jean-François Alcover_, Sep 04 2015