cp's OEIS Frontend

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

%I A157932 #40 Oct 06 2023 15:57:44
%S A157932 0,4,6,8,12,16,18,20,24,28,30,32,36,40,42,44,48,52,54,56,60,64,66,68,
%T A157932 72,76,78,80,84,88,90,92,96,100,102,104,108,112,114,116,120,124,126,
%U A157932 128,132,136,138,140,144,148,150,152,156,160,162,164,168,172,174
%N A157932 Numbers k such that (3^(35*k) + 5^(21*k) + 7^(15*k)) mod 105 is prime.
%C A157932 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
%C A157932 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
%C A157932 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
%H A157932 Colin Barker, <a href="/A157932/b157932.txt">Table of n, a(n) for n = 1..1000</a>
%H A157932 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).
%F A157932 3n - 4 <= a(n) <= 3n - 2. - _Charles R Greathouse IV_, Dec 27 2013
%F A157932 From _Colin Barker_, Oct 19 2015: (Start)
%F A157932 a(n) = (-6 - (-i)^n - i^n + 6*n)/2, where i = sqrt(-1).
%F A157932 G.f.: 2*x^2*(2*x^2-x+2) / ((x-1)^2*(x^2+1)). (End)
%e A157932 a(4) = 3^(35*4) + 5^(21*4) + 7^(15*4) mod 105 = 17 (prime).
%t A157932 Select[Range[0,180],PrimeQ[Mod[3^(35#)+5^(21#)+7^(15#),105]]&] (* _Harvey P. Dale_, Oct 10 2017 *)
%o A157932 (PARI) isok(n) = isprime((3^(35*n)+5^(21*n)+7^(15*n)) % 105); \\ _Michel Marcus_, Aug 15 2013
%o A157932 (PARI) a(n)=n\4*12+[-4,0,4,6][n%4+1] \\ _Charles R Greathouse IV_, Dec 27 2013
%o A157932 (PARI) is(n)=n%=12;n==0||n==4||n==6||n==8 \\ _Charles R Greathouse IV_, Dec 27 2013
%o A157932 (PARI) a(n) = (-6-(-I)^n-I^n+6*n)/2 \\ _Colin Barker_, Oct 19 2015
%o A157932 (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
%Y A157932 Equals {0} Union (A355200 \ A016945) <=> subsequence of even terms in A355200.
%K A157932 nonn,easy
%O A157932 1,2
%A A157932 _Kyle D. Balliet_, Mar 09 2009
%E A157932 More terms from _Michel Marcus_, Aug 15 2013