A328210 -id:A328210 - cp's OEIS frontend

A330713 Numbers k such that both k and k+1 are Zeckendorf-Niven numbers (A328208) and lazy-Fibonacci-Niven numbers (A328212).

1, 7475, 10205, 13740, 40754, 52479, 93044, 95984, 141911, 151487, 196416, 198255, 202824, 202895, 213920, 231552, 335535, 339744, 363320, 366876, 404719, 408680, 434259, 446480, 487710, 495159, 504440, 528408, 585599, 607410, 645560, 646575, 665567, 735020, 736280

Offset: 1

Amiram Eldar, Dec 27 2019

nonn

Can 3 consecutive numbers be both Zeckendorf-Niven numbers and lazy-Fibonacci-Niven numbers? Equivalently, are there numbers that are both in A328210 and A328214?

			7475 is a term since A007895(7475) = 5 and A112310(7475) = 13 and both 5 and 13 are divisors of 7475, and A007895(7476) = 6 and A112310(7476) = 12 and both 6 and 12 are divisors of 7476.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Intersection of A328209 and A328213.

Cf. A328208, A328210, A328212, A328214, A330711.

A331090 Positive numbers k such that -k, -(k + 1), and -(k + 2) are 3 consecutive negative negaFibonacci-Niven numbers (A331088).

Original entry on oeis.org

1, 2, 20, 54, 55, 56, 110, 376, 398, 974, 986, 1084, 1744, 2464, 2524, 3304, 3870, 5223, 5718, 6095, 6124, 6184, 6663, 6764, 6844, 7142, 7684, 9035, 9124, 10590, 11598, 11975, 12606, 13444, 13504, 14284, 14915, 17164, 17643, 17710, 17714, 17824, 17884, 18698, 18905, 19494, 23191, 24243, 24785, 25542, 26382, 27390, 29644, 34278, 35464

Offset: 1

Amiram Eldar, Jan 08 2020

Numbers of the form F(6*k + 2) - 1 and F(6*k + 4) - 1, where F(m) is the m-th Fibonacci number, are terms.

If m is of the form F(k) - 1, where k > 2 is congruent to {2, 10} mod 24, then {-m, -(m + 1), -(m + 2), -(m + 3), -(m + 4)} are 5 consecutive negative negaFibonacci-Niven numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000045, A154701, A269819, A328210, A328214, A330932, A331087, A331088.

ind[n_] := Floor[Log[Abs[n]*Sqrt[5] + 1/2]/Log[GoldenRatio]];
f[1] = 1; f[n_] := If[n > 0, i = ind[n - 1]; If[EvenQ[i], i++]; i, i = ind[-n]; If[OddQ[i], i++]; i];
negaFibTermsNum[n_] := Module[{k = n, s = 0}, While[k != 0, i = f[k]; s += 1; k -= Fibonacci[-i]]; s];
negFibQ[n_] := Divisible[n, negaFibTermsNum[-n]];
nConsec = 3; neg = negFibQ /@ Range[nConsec]; seq = {}; c = 0;
k = nConsec+1; While[c < 55, If[And @@ neg, c++; AppendTo[seq, k - nConsec]];neg = Join[Rest[neg], {negFibQ[k]}]; k++]; seq

A377273 Starts of runs of 3 consecutive integers that are terms in A377209.

Original entry on oeis.org

1, 2, 3, 4, 231700599, 1069467839, 1156703470, 1241186868, 2533742848, 2684864798, 3037193808, 5056780650, 7073145000, 7557047134, 9623855878, 12090760318, 12120887700, 13816479742, 14430478270, 15811947072, 16864260048, 20905152190, 22735441078, 23224253128, 23269229774, 23766221400, 25175490262

Offset: 1

Amiram Eldar, Oct 22 2024

			231700599 is a term since 231700599, 231700600 and 231700601 are all terms in A377209: 231700599/A007895(231700599) = 17823123 and 17823123/A007895(17823123) = 1980347 are integers, 231700600/A007895(231700600) = 23170060 and 23170060/A007895(23170060) = 2317006 are integers, and 231700601/A007895(231700601) = 21063691 and 21063691/A007895(21063691) = 1914881 are integers.

Cf. A007895, A376794 (binary analog).

Subsequence of A328208, A328209, A328210, A377209 and A377271.

zeck(n) = if(n<4, n>0, my(k=2, s, t); while(fibonacci(k++)<=n, ); while(k && n, t=fibonacci(k); if(t<=n, n-=t; s++); k--); s); \\ Charles R Greathouse IV at A007895
is1(k) = {my(z = zeck(k)); !(k % z) && !((k/z) % zeck(k/z)); }
lista(kmax) = {my(q1 = is1(1), q2 = is1(2), q3); for(k = 3, kmax, q3 = is1(k); if(q1 && q2 && q3, print1(k-2, ", ")); q1 = q2; q2 = q3);}

A331823 Positive numbers k such that -k, -(k + 1), and -(k + 2) are 3 consecutive negative negabinary-Niven numbers (A331728).

Original entry on oeis.org

2, 8, 32, 54, 114, 128, 174, 234, 294, 370, 413, 414, 474, 512, 534, 580, 654, 774, 894, 954, 1000, 1014, 1134, 1430, 1734, 1794, 1840, 1854, 1914, 1974, 2034, 2048, 2093, 2094, 2154, 2214, 2334, 2574, 2680, 2694, 2814, 2870, 3054, 3100, 3520, 3773, 3774, 3834

Offset: 1

Amiram Eldar, Jan 27 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005352, A027615, A039724, A154701, A328210, A328214, A330932, A331087, A331090, A331728, A331819, A331822.

negaBinWt[n_] := negaBinWt[n] = If[n == 0, 0, negaBinWt[Quotient[n - 1, -2]] + Mod[n, 2]]; negaBinNivenQ[n_] := Divisible[n, negaBinWt[-n]]; nConsec = 3; neg = negaBinNivenQ /@ Range[nConsec]; seq = {}; c = 0; k = nConsec+1; While[c < 50, If[And @@ neg, c++; AppendTo[seq, k - nConsec]]; neg = Join[Rest[neg], {negaBinNivenQ[k]}]; k++]; seq

cp's OEIS Frontend

A330713 Numbers k such that both k and k+1 are Zeckendorf-Niven numbers (A328208) and lazy-Fibonacci-Niven numbers (A328212).

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A331090 Positive numbers k such that -k, -(k + 1), and -(k + 2) are 3 consecutive negative negaFibonacci-Niven numbers (A331088).

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A377273 Starts of runs of 3 consecutive integers that are terms in A377209.

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A331823 Positive numbers k such that -k, -(k + 1), and -(k + 2) are 3 consecutive negative negabinary-Niven numbers (A331728).

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