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A381584 Starts of runs of 4 consecutive integers that are all terms in A381581.

%I A381584 #8 Feb 28 2025 12:08:47
%S A381584 1,55,2104,5222,24784,63510,64264,69487,95463,121393,184327,327303,
%T A381584 374589,463110,468168,561069,572550,596868,671407,740310,759030,
%U A381584 819948,902670,956680,1023009,1036230,1065030,1259817,1274910,1359552,1683154,1714470,1731750,2182023
%N A381584 Starts of runs of 4 consecutive integers that are all terms in A381581.
%C A381584 If k is congruent to 1 or 5 mod 12 (A087445), then A001906(k) = Fibonacci(2*k) is a term.
%H A381584 Amiram Eldar, <a href="/A381584/b381584.txt">Table of n, a(n) for n = 1..1000</a>
%e A381584 1 is a term since A291711(1) = 1 divides 1, A291711(2) = 2 divides 2, A291711(3) = 1 divides 3, and A291711(4) = 2 divides 4.
%e A381584 55 is a term since A291711(55) = 1 divides 55, A291711(56) = 2 divides 56, A291711(57) = 3 divides 57, and A291711(58) = 2 divides 58.
%t A381584 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[12, 4]
%o A381584 (PARI) mx = 20; fvec = vector(mx, i, fibonacci(2*i)); f(n) = if(n <= mx, fvec[n], fibonacci(2*n));
%o A381584 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 A381584 list(lim) = {my(q1 = is1(1), q2 = is1(2), q3 = is1(3), q4); for(k = 4, lim, q4 = is1(k); if(q1 && q2 && q3 && q4, print1(k-3, ", ")); q1 = q2; q2 = q3; q3 = q4);}
%Y A381584 Cf. A000045, A001906, A087445, A291711.
%Y A381584 Subsequence of A381581, A381582 and A381583.
%Y A381584 A381585 is a subsequence.
%Y A381584 Similar sequences: A141769, A328211, A328215, A330933.
%K A381584 nonn,base
%O A381584 1,2
%A A381584 _Amiram Eldar_, Feb 28 2025