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A261722 Values of m such that 2^m + 3^m + 5^m + 7^m + 11^m + 13^m is a prime number. 0
1, 7, 25, 91 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
2, 3, 5, 7, 11, 13 are first six consecutive prime numbers.
From Bruno Berselli, Sep 04 2015: (Start)
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
. k == 0, 2, 4 (mod 6), b(k) is divisible by 5;
. k == 1, 5 (mod 6), b(k) is divisible by 377 = 13*29;
. k == 3 (mod 6), b(k) is divisible by 29. (End)
From Jon E. Schoenfield, Mar 02 2018: (Start)
For n odd:
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)):
.
p n mod (p-1) such that p|t(n)
== ============================
2 -
3 -
5 -
7 -
11 9
13 -
17 5
19 9
23 11
29 3
31 15
37 21, 29
41 1, 19
43 11, 33, 37
47 23
53 -
59 29, 55
...
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.
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)
No other terms than the four terms cited less than 25000. - Robert G. Wilson v, Mar 07 2018
LINKS
EXAMPLE
1 is a term because 2^1 + 3^1 + 5^1 + 7^1 + 11^1 + 13^1 = 41 and 41 is a prime number.
MATHEMATICA
Select[Table[{n, Sum[Prime[k]^n, {k, 6}]}, {n, 1000}], PrimeQ[#[[2]]]&] [[All, 1]] (* Michael De Vlieger, Aug 29 2015 *)
PROG
(PARI) for(n=1, 1e3, if(isprime(13^n+11^n+7^n+5^n+3^n+2^n), print1(n", ")))
(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
CROSSREFS
Sequence in context: A245769 A146933 A155258 * A256370 A199116 A304421
KEYWORD
nonn,more
AUTHOR
Altug Alkan, Aug 29 2015
EXTENSIONS
Mathematica scripts updated by Jean-François Alcover, Sep 04 2015
STATUS
approved

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Last modified April 13 02:05 EDT 2024. Contains 371639 sequences. (Running on oeis4.)