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A047432 Numbers that are congruent to {0, 1, 4, 5, 6} mod 8.

%I A047432 #27 Jul 16 2025 17:59:44
%S A047432 0,1,4,5,6,8,9,12,13,14,16,17,20,21,22,24,25,28,29,30,32,33,36,37,38,
%T A047432 40,41,44,45,46,48,49,52,53,54,56,57,60,61,62,64,65,68,69,70,72,73,76,
%U A047432 77,78,80,81,84,85,86,88,89,92,93,94,96,97,100,101,102
%N A047432 Numbers that are congruent to {0, 1, 4, 5, 6} mod 8.
%C A047432 This sequence comprises the only possible sizes of sets of distinct primes for which the sum of their squares is a square. Proof: 2^2 mod 8 = 4, and squares mod 8 of odd numbers are always 1, therefore sums mod 8 of squares of k distinct primes can only be, for k mod 8 = 0..7: {{0, 3}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {0, 5}, {1, 6}, {2, 7}}. But overall, squares mod 8 are always in {0, 1, 4}. These only intersect at k mod 8 in {0, 1, 4, 5, 6}. - _Charles L. Hohn_, Jul 13 2025
%H A047432 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).
%F A047432 G.f.: x^2*(1+x)*(2*x^3-x^2+2*x+1) / ( (x^4+x^3+x^2+x+1)*(x-1)^2 ). - _R. J. Mathar_, Dec 07 2011
%F A047432 From _Wesley Ivan Hurt_, Aug 01 2016: (Start)
%F A047432 a(n) = a(n-1) + a(n-5) - a(n-6) for n > 6, a(n) = a(n-5) + 8 for n > 5.
%F A047432 a(n) = (40*n - 40 + 3*(n mod 5) + 3*((n+1) mod 5) - 7*((n+2) mod 5) + 3*((n+3) mod 5) - 2*((n+4) mod 5))/25.
%F A047432 a(5k) = 8k-2, a(5k-1) = 8k-3, a(5k-2) = 8k-4, a(5k-3) = 8k-7, a(5k-4) = 8k-8. (End)
%p A047432 A047432:=n->8*floor(n/5)+[(0, 1, 4, 5, 6)][(n mod 5)+1]: seq(A047432(n), n=0..100); # _Wesley Ivan Hurt_, Aug 01 2016
%t A047432 Select[Range[0, 100], MemberQ[{0, 1, 4, 5, 6}, Mod[#, 8]] &] (* _Wesley Ivan Hurt_, Aug 01 2016 *)
%o A047432 (Magma) [n : n in [0..150] | n mod 8 in [0, 1, 4, 5, 6]]; // _Wesley Ivan Hurt_, Aug 01 2016
%o A047432 (PARI) a(n)=[-2,0,1,4,5][n%5+1] + n\5*8 \\ _Charles R Greathouse IV_, Aug 01 2016
%K A047432 nonn,easy
%O A047432 1,3
%A A047432 _N. J. A. Sloane_