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A381583 Starts of runs of 3 consecutive integers that are all terms in A381581.

%I A381583 #8 Feb 28 2025 12:08:38
%S A381583 1,2,20,55,56,110,304,364,398,846,1024,1084,1744,1854,2044,2104,2105,
%T A381583 2527,2824,2862,3870,4374,5222,5223,5243,5718,5928,6488,6784,6844,
%U A381583 6894,6978,7142,7924,10590,11240,11889,11975,12248,14284,14915,16638,17710,17714,17824
%N A381583 Starts of runs of 3 consecutive integers that are all terms in A381581.
%C A381583 If k is congruent to 1 or 5 mod 12 (A087445), then A001906(k) = Fibonacci(2*k) is a term.
%H A381583 Amiram Eldar, <a href="/A381583/b381583.txt">Table of n, a(n) for n = 1..10000</a>
%e A381583 1 is a term since A291711(1) = 1 divides 1, A291711(2) = 2 divides 2, and A291711(3) = 1 divides 3.
%e A381583 20 is a term since A291711(20) = 4 divides 20, A291711(21) = 1 divides 21, and A291711(22) = 2 divides 22.
%t A381583 f[n_] := f[n] = Fibonacci[2*n]; q[n_] := q[n] = Module[{s = 0, m = n, k}, While[m > 0, k = 1; While[m > f[k], k++]; If[m < f[k], k--]; If[m >= 2*f[k], s += 2; m -= 2*f[k], s++; m -= f[k]]]; Divisible[n, s]]; seq[count_, nConsec_] := Module[{cn = q /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ cn, c++; AppendTo[s, k - nConsec]]; cn = Join[Rest[cn], {q[k]}]; k++]; s]; seq[45, 3]
%o A381583 (PARI) mx = 20; fvec = vector(mx, i, fibonacci(2*i)); f(n) = if(n <= mx, fvec[n], fibonacci(2*n));
%o A381583 is1(n) = {my(s = 0, m = n, k); while(m > 0, k = 1; while(m > f(k), k++); if(m < f(k), k--); if(m >= 2*f(k), s += 2; m -= 2*f(k), s++; m -= f(k))); !(n % s);}
%o A381583 list(lim) = {my(q1 = is1(1), q2 = is1(2), q3); for(k = 3, lim, q3 = is1(k); if(q1 && q2 && q3, print1(k-2, ", ")); q1 = q2; q2 = q3);}
%Y A381583 Cf. A000045, A001906, A087445, A291711.
%Y A381583 Subsequence of A381581 and A381582.
%Y A381583 Subsequences: A381584, A381585.
%Y A381583 Similar sequences: A154701, A328210, A330932, A351721.
%K A381583 nonn,easy,base
%O A381583 1,2
%A A381583 _Amiram Eldar_, Feb 28 2025