A261722 Values of m such that 2^m + 3^m + 5^m + 7^m + 11^m + 13^m is a prime number.

1, 7, 25, 91

Offset: 1

Author

Altug Alkan, Aug 29 2015

Keywords

• nonn
• more

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

No other terms than the four terms cited less than 100000. - Michael S. Branicky, Sep 28 2024

Examples

			1 is a term because 2^1 + 3^1 + 5^1 + 7^1 + 11^1 + 13^1 = 41 and 41 is a prime number.
		

Crossrefs

Cf. A000040, A076515, A268064, A268067.

Programs

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

Mathematica

Select[Table[{n, Sum[Prime[k]^n, {k, 6}]}, {n, 1000}], PrimeQ[#[[2]]]&] [[All, 1]] (* Michael De Vlieger, Aug 29 2015 *)

PARI

for(n=1, 1e3, if(isprime(13^n+11^n+7^n+5^n+3^n+2^n), print1(n", ")))

Extensions

Mathematica scripts updated by Jean-François Alcover, Sep 04 2015