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

0,6

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

Links

Table of n, a(n) for n=0..85.

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

Example

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.

Mathematica

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

Prog

(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(o<e, o, a(n, 2*e)) \\ Charles R Greathouse IV, Apr 25 2014

Crossrefs

Cf. A006519, A007814, A023512.

Sequence in context: A352727 A332015 A173667 * A086008 A083910 A069843

Adjacent sequences: A241538 A241539 A241540 * A241542 A241543 A241544

Keyword

nonn

Author

Zak Seidov, Apr 25 2014

Status

approved