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A130335 Smallest k > 0 such that gcd(n*(n+1)/2, (n+k)*(n+k+1)/2) = 1.

%I A130335 #22 Jan 27 2025 06:19:41
%S A130335 1,2,7,2,2,4,2,2,4,2,2,10,2,2,7,2,2,4,2,2,4,2,2,13,2,2,10,2,2,7,2,2,4,
%T A130335 2,2,10,2,2,7,2,2,4,2,2,7,2,2,10,2,2,7,2,2,4,2,2,4,2,2,13,2,2,10,2,2,
%U A130335 4,2,2,4,2,2,10,2,2,7,2,2,4,2,2,4,2,2,22,2,2,7,2,2,16,2,2,4,2,2,10,2,2,7,2
%N A130335 Smallest k > 0 such that gcd(n*(n+1)/2, (n+k)*(n+k+1)/2) = 1.
%C A130335 First occurrence of 3k+1, k=0.. or 0 if unknown, limit = 2^31: 1, 6, 3, 12, 24, 90, 231, 84, 792, 0, 195, 3432, 780, 0, 3255, 6075, 73644, 51482970, 0, 924, 183540, 0, 45219, 0, 509124, 3842375445, 29259, 71484, 0, 0, 0, 2311539, 238547880, 0, 55380135, 893907420, 23303784, 0, 0, 208260975, 0, 0, 1744264599, 0, 0, 0, 1487657079, 665710275, 0, 0, 1963994955, 0, 319589424, 0, 0, 0, 4181294964, 0, 0, 383229924, ..., . - _Robert G. Wilson v_, Jun 03 2007
%H A130335 Reinhard Zumkeller, <a href="/A130335/b130335.txt">Table of n, a(n) for n = 1..10000</a>
%F A130335 a(n) = Min{k>0: A050873(A000217(n+k),A000217(n))=1};
%F A130335 a(n) = A130334(n) - n;
%F A130335 a(n) > 1 for n>1; a(n) > 2 iff n mod 3 = 0: a(A001651(n))=2, a(A008585(n)) > 2 for n > 1.
%F A130335 a(n) == 1 (mod 3) if a(n) != 2. - _Robert G. Wilson v_, Jun 03 2007
%t A130335 f[n_] := Block[{k = If[ n == 1 || Mod[n, 3] == 0, 1, 2]}, While[ GCD[n(n + 1)/2, (n + k)(n + k + 1)/2] != 1, k += 3 ]; k]; Array[f, 100] (* _Robert G. Wilson v_, Jun 03 2007 *)
%o A130335 (Python)
%o A130335 from math import gcd
%o A130335 def A130335(n):
%o A130335     k, Tn, Tm = 1, n*(n+1)//2, (n+1)*(n+2)//2
%o A130335     while gcd(Tn,Tm) != 1:
%o A130335         k += 1
%o A130335         Tm += k+n
%o A130335     return k # _Chai Wah Wu_, Sep 16 2021
%o A130335 (PARI) a(n) = my(k=1); while (gcd(n*(n+1)/2, (n+k)*(n+k+1)/2) != 1, k++); k;
%Y A130335 Cf. A000217, A050873.
%Y A130335 Cf. A001651, A008585.
%Y A130335 Cf. A130334.
%Y A130335 See A130336 and A130337 for record values and where they occur.
%K A130335 nonn
%O A130335 1,2
%A A130335 _Reinhard Zumkeller_, May 28 2007