A241541 Exponent of 11 in prime factorization of (2^n + 3^n + 5^n + 7^n).

0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 0

Author

Zak Seidov, Apr 25 2014

Keywords

• nonn

Comments

11^a(n) is the largest power of 11 dividing (2^n + 3^n + 5^n + 7^n);

(2^n + 3^n + 5^n + 7^n) is divisible by 11^2 = 121 for n == 5 mod 10.

Among first 10000 nonzero terms there are {8182, 1652, 150, 14, 1, 1} terms with values {2, 3, 4, 5, 6, 7}, respectively.

Record values are a(5) = 2, a(45) = 3, a(595) = 5, a(40525) = 7, a(6482565) = 8, a(97435855) = 9, a(927694285) = 10, a(11789738455) = 11, a(129687123005) = 12, a(508958242255) = 13, a(11921425066695) = 14, a(74689992601115) = 15, a(1110371356919045) = 16, a(20886240847078255) = 17, a(229748649317860805) = 18, etc. - Charles R Greathouse IV, Apr 25 2014

Examples

			at n = 5, 2^n + 3^n + 5^n + 7^n = 20207 = 11^2*167,
at n = 15, 2^n + 3^n + 5^n + 7^n = 4778093469743 = 11^2*587*67271509.
		

Links

• Zak Seidov, Table n, a(-5 + 10*n) for n = 1..10^4.

Crossrefs

Cf. A006519, A007814, A023512.

Programs

Mathematica

Table[IntegerExponent[2^n + 3^n + 5^n + 7^n, 11], {n, 0, 100}]

PARI

a(n)=valuation(2^n+3^n+5^n+7^n,11) \\ Charles R Greathouse IV, Apr 25 2014

PARI

a(n,e=8)=my(m=11^e, o=valuation(Mod(2,m)^n +Mod(3,m)^n +Mod(5,m)^n +Mod(7,m)^n, 11)); if(oCharles R Greathouse IV, Apr 25 2014