cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 21-30 of 32 results. Next

A381582 Numbers k such that k and k+1 are both terms in A381581.

Original entry on oeis.org

1, 2, 3, 20, 21, 27, 44, 55, 56, 57, 75, 95, 110, 111, 115, 152, 175, 207, 264, 287, 291, 304, 305, 344, 364, 365, 377, 380, 395, 398, 399, 404, 425, 435, 455, 534, 584, 605, 815, 846, 847, 864, 888, 930, 987, 992, 1004, 1011, 1024, 1025, 1064, 1084, 1085, 1145, 1182
Offset: 1

Views

Author

Amiram Eldar, Feb 28 2025

Keywords

Comments

If k is not divisible by 3 (A001651), then A001906(k) = Fibonacci(2*k) is a term.

Examples

			1 is a term since A291711(1) = 1 divides 1 and A291711(2) = 2 divides 2.
20 is a term since A291711(20) = 4 divides 20 and A291711(21) = 1 divides 21.

Links

Crossrefs

Cf. A000045, A001906, A001651, A291711.
Subsequence of A381581.
Subsequences: A381583, A381584, A381585.
Similar sequences: A328209, A330927, A330931, A351720.

Programs

  • Mathematica
    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]]; Select[Range[1200], q[#] && q[#+1] &]
  • PARI
    mx = 20; fvec = vector(mx, i, fibonacci(2*i)); f(n) = if(n <= mx, fvec[n], fibonacci(2*n));
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);}
list(lim) = {my(q1 = is1(1), q2); for(k = 2, lim, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A331089 Positive numbers k such that -k and -(k + 1) are both negative negaFibonacci-Niven numbers (A331088).

Original entry on oeis.org

1, 2, 3, 15, 20, 21, 44, 50, 54, 55, 56, 57, 75, 104, 110, 111, 115, 128, 141, 152, 175, 207, 264, 291, 304, 308, 335, 351, 363, 376, 377, 380, 392, 398, 399, 435, 452, 455, 534, 584, 594, 605, 623, 654, 735, 740, 744, 753, 795, 804, 875, 884, 897, 924, 964, 968
Offset: 1

Views

Author

Amiram Eldar, Jan 08 2020

Keywords

Comments

The Fibonacci numbers F(6*k + 2) and F(6*k + 4) are terms.

Links

Crossrefs

Cf. A328209, A328213, A330927, A330931, A331086, A331088.

Programs

  • Mathematica
    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 = 2; 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

A364007 Numbers k such that k and k+1 are both Wythoff-Niven numbers (A364006).

Original entry on oeis.org

3, 6, 7, 20, 39, 51, 54, 55, 90, 135, 143, 294, 305, 321, 356, 365, 369, 374, 375, 376, 784, 800, 924, 978, 979, 980, 986, 1904, 1945, 1970, 2043, 2199, 2232, 2289, 2394, 2424, 2439, 2499, 2525, 2562, 2580, 2583, 4185, 4598, 4707, 4774, 4790, 4796, 4879, 5004
Offset: 1

Views

Author

Amiram Eldar, Jul 01 2023

Keywords

Comments

A035508(n) = Fibonacci(2*n+2) - 1 is a term for n >= 2 since A135818(Fibonacci(2*n+2) - 1) = A135818(Fibonacci(2*n+2)) = 1.

Links

Crossrefs

Cf. A035508, A135818.
Subsequence of A364006.
Subsequences: A364008, A364009.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090, A352108, A352321, A352509.

Programs

  • Mathematica
    seq[count_, nConsec_] := Module[{cn = wnQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ cn, c++; AppendTo[s, k - nConsec]]; cn = Join[Rest[cn], {wnQ[k]}]; k++]; s]; seq[50, 2] (* using the function wnQ[n] from A364006 *)

A364124 Numbers k such that k and k+1 are both Stolarsky-Niven numbers (A364123).

Original entry on oeis.org

8, 56, 84, 159, 195, 224, 384, 399, 405, 995, 1140, 1224, 1245, 1295, 1309, 1419, 1420, 1455, 1474, 1507, 2585, 2597, 2600, 2680, 2681, 2727, 2744, 2750, 2799, 2855, 3122, 3311, 3339, 3345, 3618, 3707, 3795, 4004, 6770, 6774, 6984, 6985, 7014, 7074, 7154, 7405
Offset: 1

Views

Author

Amiram Eldar, Jul 07 2023

Keywords

Links

Crossrefs

Subsequence of A364123.
Subsequences: A364125, A364126.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090, A352108, A352321, A352343, A352509.

Programs

  • Mathematica
    seq[count_, nConsec_] := Module[{cn = stolNivQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ cn, c++; AppendTo[s, k - nConsec]]; cn = Join[Rest[cn], {stolNivQ[k]}]; k++]; s]; seq[50, 2] (* using the function stolNivQ[n] from A364123 *)
  • PARI
    lista(count, nConsec) = {my(cn = vector(nConsec, i, isStolNivQ(i)), c = 0, k = nConsec + 1); while(c < count, if(vecsum(cn) == nConsec, c++; print1(k-nConsec, ", ")); cn = concat(vecextract(cn, "^1"), isStolNivQ(k)); k++);} \\ using the function isA364123(n) from A364123
lista(50, 2)

A376795 Numbers k such that k and k+1 are both in A376617.

Original entry on oeis.org

1, 10624, 13824, 1114112, 2625664, 4563999, 6554624, 16843904, 17266688, 17368064, 20003840, 27137024, 32375160, 32679360, 42993664, 44643599, 63732096, 69222464, 69424640, 70083584, 80778752, 84783104, 85458944, 90256383, 92478000, 116469899, 118063231, 121900544
Offset: 1

Views

Author

Amiram Eldar, Oct 04 2024

Keywords

Examples

			10624 is a term since both 10624 and 10625 are in A376617: 10624/A000120(10624) = 2656, 2656/A000120(2656) = 664, and 664/A000120(664) = 166 are integers, and 10625/A000120(10625) = 2125, 2125/A000120(2125) = 425, and 425/A000120(425) = 85 are integers.

Links

Crossrefs

Subsequence of A330931, A376617 and A376793.
Cf. A000120.

Programs

  • Mathematica
    q[k_] := q[k] = Module[{w = DigitCount[k, 2, 1], w2, m, n}, IntegerQ[m = k/w] && Divisible[m, w2 = DigitCount[m, 2, 1]] && Divisible[n = m/w2, DigitCount[n, 2, 1]]]; Select[Range[1.2*10^6], q[#] && q[#+1] &]
  • PARI
    s(n) = {my(w = hammingweight(n)); if(w == 1, 0, if(n % w, 1, 1 + s(n/w)));}
is1(k) = {my(sk = s(k)); sk == 0 || sk >= 4;}
lista(kmax) = {my(q1 = is1(1), q2); for(k = 2, kmax, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A334345 Numbers k such that k and k+1 are both binary Moran numbers (A334344).

Original entry on oeis.org

115, 355, 1266, 1555, 1686, 1795, 4195, 4206, 4962, 5155, 5298, 6978, 9235, 10002, 11230, 13315, 18822, 21752, 22602, 23106, 26072, 29816, 40616, 42258, 60056, 60730, 64690, 68802, 83586, 87272, 91736, 94616, 100990, 107526, 108910, 109448, 113192, 121112, 125436
Offset: 1

Views

Author

Amiram Eldar, Apr 23 2020

Keywords

Examples

			115 is a term since 115/A000120(115) = 23 and 116/A000120(116) = 29 are both prime numbers.

Links

Crossrefs

Subsequence of A330931 and A334344.
Cf. A000120, A085775.

Programs

  • Maple
    q:= n-> (p-> is(p, integer) and isprime(p))(n/add(i, i=Bits[Split](n))):
select(k-> q(k) and q(k+1), [$1..126000])[];  # Alois P. Heinz, Apr 23 2020
  • Mathematica
    binMoranQ[n_] := PrimeQ[n / DigitCount[n, 2, 1]]; Select[Range[10^5], binMoranQ[#] && binMoranQ[# + 1] &]

A338514 Numbers k such that k and k+1 are both divisible by the total binary weight of their divisors (A093653).

Original entry on oeis.org

1, 2, 54, 2119, 11100, 13727, 14382, 15799, 16399, 20159, 20950, 33421, 34617, 36328, 36396, 39400, 42198, 42438, 42650, 46253, 46873, 50370, 55368, 56600, 58793, 67013, 67320, 69023, 72325, 76057, 86393, 90781, 92906, 93216, 105909, 132088, 134028, 134823, 140466
Offset: 1

Views

Author

Amiram Eldar, Oct 31 2020

Keywords

Comments

Numbers k such that k and k+1 are both in A093705, or, equivalently, k is divisible by A093653(k) and k+1 is divisible by A093653(k+1).

Examples

			1 is a term since 1 and 2 are both terms of A093705.

Links

Crossrefs

Cf. A000120, A093653, A093705.
Similar sequences: A330927, A330931, A334345, A338452.

Programs

  • Mathematica
    divQ[n_] := Divisible[n, DivisorSum[n, DigitCount[#, 2, 1] &]]; q1 = divQ[1]; Reap[Do[q2 = divQ[n]; If[q1 && q2, Sow[n - 1]]; q1 = q2, {n, 2, 10^5}]][[2, 1]]
SequencePosition[Table[If[Divisible[n,Total[DigitCount[Divisors[n],2,1]]],1,0],{n,150000}],{1,1}][[All,1]] (* Harvey P. Dale, Jun 14 2022 *)

A363790 Numbers k such that k and k+1 are both primitive binary Niven numbers (A363787).

Original entry on oeis.org

115, 155, 204, 284, 355, 395, 404, 555, 564, 595, 675, 804, 835, 846, 1075, 1124, 1164, 1182, 1266, 1315, 1434, 1555, 1604, 1686, 1795, 1938, 2075, 2124, 2195, 2244, 2315, 2324, 2358, 2435, 2595, 3084, 3204, 3282, 3366, 4124, 4195, 4206, 4235, 4244, 4364, 4458
Offset: 1

Views

Author

Amiram Eldar, Jun 22 2023

Keywords

Examples

			115 is a term since 115 and 116 are both primitive binary Niven numbers.

Links

Crossrefs

Subsequence of A049445, A330931 and A363787.
Subsequences: A363791, A363792.

Programs

  • Mathematica
    binNivQ[n_] := Divisible[n, DigitCount[n, 2, 1]]; q[n_] := binNivQ[n] && ! (EvenQ[n] && binNivQ[n/2]); Select[Range[5000], q[#] && q[# + 1] &]
  • PARI
    isbinniv(n) = !(n % hammingweight(n));
isprim(n) = isbinniv(n) && !(!(n%2) && isbinniv(n/2));
is(n) = isprim(n) && isprim(n+1);

A363791 Starts of runs of 3 consecutive integers that are primitive binary Niven numbers (A363787).

Original entry on oeis.org

4184046, 5234670, 6285294, 7861230, 8123886, 8255214, 8255215, 8320878, 8353710, 8370126, 8379247, 12238830, 12451631, 12572622, 13623246, 13629935, 14515182, 14646510, 14673870, 14673871, 14679342, 15040494, 15335375, 15449071, 15531759, 15708078, 15986543, 16178670
Offset: 1

Views

Author

Amiram Eldar, Jun 22 2023

Keywords

Examples

			4184046 is a term since 4184046, 4184047 and 4184048 are all primitive binary Niven numbers.

Links

Crossrefs

Subsequence of A049445, A330931, A330932, A363787 and A363790.
A363792 is a subsequence.

Programs

  • Mathematica
    binNivQ[n_] := Divisible[n, DigitCount[n, 2, 1]]; primBinNivQ[n_] := binNivQ[n] && ! (EvenQ[n] && binNivQ[n/2]);
seq[kmax_] := Module[{tri = primBinNivQ /@ Range[3], s = {}, k = 4}, While[k < kmax, If[And @@ tri, AppendTo[s, k - 3]]; tri = Join[Rest[tri], {primBinNivQ[k]}]; k++]; s]; seq[10^7]
  • PARI
    isbinniv(n) = !(n % hammingweight(n));
isprim(n) = isbinniv(n) && !(!(n%2) && isbinniv(n/2));
lista(kmax) = {my(tri = vector(3, i, isprim(i)), k = 4); while(k < kmax, if(vecsum(tri) == 3, print1(k-3, ", ")); tri = concat(vecextract(tri, "^1"), isprim(k)); k++); }

A363792 Starts of runs of 4 consecutive integers that are primitive binary Niven numbers (A363787).

Original entry on oeis.org

8255214, 14673870, 29092590, 33185646, 41743854, 47697390, 48069486, 56348622, 56999790, 58116078, 59604462, 60534702, 60813774, 61837038, 62581230, 64069614, 64999854, 65371950, 66581262, 66674286, 75232494, 83418606, 86767470, 88069806, 92255886, 95418702, 96441966, 99511758, 99604782
Offset: 1

Views

Author

Amiram Eldar, Jun 22 2023

Keywords

Comments

There are no runs of 5 or more consecutive integers that are primitive binary Niven numbers (see the second comment in A330933).

Examples

			8255214 is a term since 8255214, 8255215, 8255216 and 8255217 are all primitive binary Niven numbers.

Links

Crossrefs

Subsequence of A049445, A330931, A330932, A330933, A363787, A363790 and A363791.

Programs

  • Mathematica
    binNivQ[n_] := Divisible[n, DigitCount[n, 2, 1]]; primBinNivQ[n_] := binNivQ[n] && ! (EvenQ[n] && binNivQ[n/2]);
seq[kmax_] := Module[{quad = primBinNivQ /@ Range[4], s = {}, k = 5}, While[k < kmax, If[And @@ quad, AppendTo[s, k - 4]]; quad = Join[Rest[quad], {primBinNivQ[k]}]; k++]; s]; seq[3*10^7]
  • PARI
    isbinniv(n) = !(n % hammingweight(n));
isprim(n) = isbinniv(n) && !(!(n%2) && isbinniv(n/2));
lista(kmax) = {my(quad = vector(4, i, isprim(i)), k = 5); while(k < kmax, if(vecsum(quad) == 4, print1(k-4, ", ")); quad = concat(vecextract(quad, "^1"), isprim(k)); k++); }
Previous Showing 21-30 of 32 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.