A157932 Numbers k such that (3^(35*k) + 5^(21*k) + 7^(15*k)) mod 105 is prime.

0, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 48, 52, 54, 56, 60, 64, 66, 68, 72, 76, 78, 80, 84, 88, 90, 92, 96, 100, 102, 104, 108, 112, 114, 116, 120, 124, 126, 128, 132, 136, 138, 140, 144, 148, 150, 152, 156, 160, 162, 164, 168, 172, 174

Offset

1,2

Comments

Let b(k) = (3^(35*k) + 5^(21*k) + 7^(15*k)) mod 105, then sequence {b(k)} is 3, repeat (60, 68, 75, 17, 30, 23, 60, 47, 75, 38, 30, 2), with primes 3, 17, 23, 47, 2. First differences of {a(n)} are 4, 2, 2, 4, 4, 2, 2, 4, .... - Michel Marcus, Aug 15 2013

3^(35*k) + 5^(21*k) + 7^(15*k) = (4^k)*(3^k + 5^k + 7^k) mod 105, then by the division algorithm a simple proof yields that only numbers k of the form 24*m, 24*m+4, 24*m+6, 24*m+8, 24*m+12, 24*m+16, 24*m+18, 24*m+20 will be congruent to a prime modulo 105. Thus the pattern 4, 2, 2, 4, 4, 2, 2, ... will repeat infinitely. - Kyle D. Balliet, Jan 01 2014

Even numbers that can be written as the sum of 3 of their divisors, not necessarily distinct (see A355200). Also, numbers k of the form 12*m, 12*m+4, 12*m+6, 12*m+8. - Bernard Schott, Sep 08 2023

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Formula

3n - 4 <= a(n) <= 3n - 2. - Charles R Greathouse IV, Dec 27 2013

From Colin Barker, Oct 19 2015: (Start)

a(n) = (-6 - (-i)^n - i^n + 6*n)/2, where i = sqrt(-1).

G.f.: 2*x^2*(2*x^2-x+2) / ((x-1)^2*(x^2+1)). (End)

Example

a(4) = 3^(35*4) + 5^(21*4) + 7^(15*4) mod 105 = 17 (prime).

Mathematica

Select[Range[0, 180], PrimeQ[Mod[3^(35#)+5^(21#)+7^(15#), 105]]&] (* Harvey P. Dale, Oct 10 2017 *)

Prog

(PARI) isok(n) = isprime((3^(35*n)+5^(21*n)+7^(15*n)) % 105); \\ Michel Marcus, Aug 15 2013
(PARI) a(n)=n\4*12+[-4, 0, 4, 6][n%4+1] \\ Charles R Greathouse IV, Dec 27 2013
(PARI) is(n)=n%=12; n==0||n==4||n==6||n==8 \\ Charles R Greathouse IV, Dec 27 2013
(PARI) a(n) = (-6-(-I)^n-I^n+6*n)/2 \\ Colin Barker, Oct 19 2015
(PARI) concat(0, Vec(2*x^2*(2*x^2-x+2)/((x-1)^2*(x^2+1)) + O(x^100))) \\ Colin Barker, Oct 19 2015

Crossrefs

Equals {0} Union (A355200 \ A016945) <=> subsequence of even terms in A355200.

Sequence in context: A337812 A100390 A199768 * A097619 A113709 A076082

Adjacent sequences: A157929 A157930 A157931 * A157933 A157934 A157935

Keyword

nonn,easy

Author

Kyle D. Balliet, Mar 09 2009

Extensions

More terms from Michel Marcus, Aug 15 2013

Status

approved