A328213 -id:A328213 - cp's OEIS frontend

A352090 Numbers k such that k and k+1 are both tribonacci-Niven numbers (A352089).

1, 6, 7, 12, 13, 20, 26, 27, 39, 68, 75, 80, 81, 87, 115, 128, 135, 149, 176, 184, 185, 195, 204, 215, 224, 230, 236, 243, 264, 278, 284, 291, 344, 364, 399, 447, 506, 507, 519, 548, 555, 560, 575, 595, 615, 635, 656, 664, 665, 684, 704, 725, 744, 777, 804, 824

Offset: 1

Amiram Eldar, Mar 04 2022

Numbers k such that A278043(k) | k and A278043(k+1) | k+1.

The odd tribonacci numbers, A000073(A042964(m)), are all terms.

			6 is a term since 6 and 7 are both tribonacci-Niven numbers: the minimal tribonacci representation of 6, A278038(6) = 110, has 2 1's and 6 is divisible by 2, and the minimal tribonacci representation of 7, A278038(7) = 1000, has one 1 and 7 is divisible by 1.

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

Cf. A000073, A042964, A278038, A278043.
Subsequence of A352089.
Subsequences: A352091, A352092.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720.

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; q[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; Divisible[n, DigitCount[Total[2^(s - 1)], 2, 1]]]; Select[Range[1000], q[#] && q[# + 1] &]

A352108 Numbers k such that k and k+1 are both lazy-tribonacci-Niven numbers (A352107).

Original entry on oeis.org

1, 20, 39, 75, 115, 135, 155, 175, 176, 184, 204, 215, 264, 567, 684, 704, 725, 791, 846, 872, 1089, 1104, 1115, 1134, 1183, 1184, 1211, 1224, 1407, 1575, 1840, 1880, 2064, 2075, 2151, 2191, 2232, 2259, 2260, 2415, 2529, 2583, 2624, 2780, 2820, 2848, 2888, 2988

Offset: 1

Amiram Eldar, Mar 05 2022

			20 is a term since 20 and 21 are both lazy-tribonacci-Niven numbers: the maximal tribonacci representation of 20, A352103(20) = 10111, has 4 1's and 20 is divisible by 4, and the maximal tribonacci representation of 21, A352103(20) = 11001, has 3 1's and 21 is divisible by 3.

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

Cf. A352103, A352104.
Subsequence of A352107.
Subsequences: A352109, A352110.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090.

t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; trib[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; AppendTo[s, k]; m -= t[k]; k = 1]; IntegerDigits[Total[2^(s - 1)], 2]]; q[n_] := Module[{v = trib[n]}, nv = Length[v]; i = 1; While[i <= nv - 3, If[v[[i ;; i + 3]] == {1, 0, 0, 0}, v[[i ;; i + 3]] = {0, 1, 1, 1}; If[i > 3, i -= 4]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, False, Divisible[n, Total[v[[i[[1, 1]] ;; -1]]]]]]; Select[Range[3000], q[#] && q[# + 1] &]

A352321 Numbers k such that k and k+1 are both Pell-Niven numbers (A352320).

Original entry on oeis.org

1, 4, 5, 9, 14, 28, 29, 33, 39, 63, 87, 110, 111, 115, 125, 140, 164, 168, 169, 183, 255, 275, 308, 338, 410, 444, 483, 507, 564, 579, 584, 704, 791, 984, 985, 999, 1004, 1024, 1025, 1115, 1134, 1154, 1164, 1211, 1265, 1308, 1323, 1351, 1395, 1415, 1424, 1491

Offset: 1

Amiram Eldar, Mar 12 2022

All the odd-indexed Pell numbers (A001653) are terms.

			4 is a term since 4 and 5 are both Pell-Niven numbers: the minimal Pell representation of 4, A317204(20) = 20, has the sum of digits 2+0 = 2 and 4 is divisible by 2, and the minimal Pell representation of 5, A317204(5) = 100, has the sum of digits 1+0+0 = 1 and 5 is divisible by 1.

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

Cf. A000129, A265744, A317204.
Subsequence of A352320.
Subsequences: A001653, A352322.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090, A352108.

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; q[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; Divisible[n, Plus @@ IntegerDigits[ Total[3^(s - 1)], 3]]]; Select[Range[1500], q[#] && q[#+1] &]

A352343 Numbers k such that k and k+1 are both lazy-Pell-Niven numbers (A352342).

Original entry on oeis.org

1, 24, 63, 209, 216, 459, 560, 584, 656, 729, 999, 1110, 1269, 1728, 1859, 1989, 2100, 2196, 2197, 2255, 2650, 2651, 2820, 3443, 3497, 4080, 4563, 5291, 5784, 5785, 5837, 5928, 6252, 6383, 7344, 7657, 7812, 8150, 8203, 8459, 8670, 8749, 9251, 9295, 9372, 9464, 9840, 9884

Offset: 1

Amiram Eldar, Mar 12 2022

Numbers k such that A352340(k) | k and A352340(k+1) | k+1.

			24 is a term since 24 and 25 are both lazy-Pell-Niven numbers: the maximal Pell representation of 24, A352339(24) = 1210, has the sum of digits A352340(24) = 1+2+1+0 = 4 and 24 is divisible by 4, and the maximal Pell representation of 25, A352339(25) = 1211, has the sum of digits A352340(25) = 1+2+1+1 = 5 and 25 is divisible by 5.

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

Cf. A352339, A352340.
Subsequence of A352342.
Subsequences: A352344, A352345.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090, A352108, A352321.

pell[1] = 1; pell[2] = 2; pell[n_] := pell[n] = 2*pell[n - 1] + pell[n - 2]; pellp[n_] := Module[{s = {}, m = n, k}, While[m > 0, k = 1; While[pell[k] <= m, k++]; k--; AppendTo[s, k]; m -= pell[k]; k = 1]; IntegerDigits[Total[3^(s - 1)], 3]]; lazyPellNivenQ[n_] := Module[{v = pellp[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] > 0 && v[[i + 1]] == 0 && v[[i + 2]] < 2, v[[i ;; i + 2]] += {-1, 2, 1}; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; Divisible[n, Plus @@ v[[i[[1, 1]] ;; -1]]]]; Select[Range[10^4], lazyPellNivenQ[#] && lazyPellNivenQ[#+1] &]

A352509 Numbers k such that k and k+1 are both Catalan-Niven numbers (A352508).

Original entry on oeis.org

1, 4, 5, 9, 32, 44, 55, 56, 134, 144, 145, 146, 155, 184, 234, 324, 329, 414, 426, 429, 434, 455, 511, 512, 603, 636, 930, 1004, 1014, 1160, 1183, 1215, 1287, 1308, 1448, 1472, 1505, 1562, 1595, 1808, 1854, 1967, 1985, 1995, 2051, 2075, 2096, 2135, 2165, 2255

Offset: 1

Amiram Eldar, Mar 19 2022

			4 is a term since 4 and 5 are both Catalan-Niven numbers: the Catalan representation of 4, A014418(20) = 20, has the sum of digits 2+0 = 2 and 4 is divisible by 2, and the Catalan representation of 5, A014418(5) = 100, has the sum of digits 1+0+0 = 1 and 5 is divisible by 1.

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

Cf. A000108, A014418, A014420.
Subsequence of A352508.
Subsequences: A038003, A352510, A352511.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090, A352108, A352321, A352343.

c[n_] := c[n] = CatalanNumber[n]; q[n_] := Module[{s = {}, m = n, i}, While[m > 0, i = 1; While[c[i] <= m, i++]; i--; m -= c[i]; AppendTo[s, i]]; Divisible[n, Plus @@ IntegerDigits[Total[4^(s - 1)], 4]]]; Select[Range[2300], q[#] && q[#+1] &]

A364217 Numbers k such that k and k+1 are both Jacobsthal-Niven numbers (A364216).

Original entry on oeis.org

1, 2, 3, 8, 11, 14, 15, 27, 32, 42, 43, 44, 45, 51, 56, 75, 86, 87, 92, 95, 99, 104, 125, 128, 135, 144, 155, 171, 176, 182, 183, 195, 204, 264, 267, 275, 287, 305, 344, 363, 375, 387, 428, 444, 455, 474, 497, 512, 524, 535, 544, 545, 552, 555, 581, 605, 623, 639

Offset: 1

Amiram Eldar, Jul 14 2023

A001045(2*n+1) = A007583(n) = (2^(2*n+1) + 1)/3 is a term for n >= 0, since its representation is 2*n 1's, so A364215(A001045(2*n+1)) = 1 divides A001045(2*n+1), and the representation of A001045(2*n+1) + 1 = (2^(2*n+1) + 4)/3 is max(2*n-1, 0) 0's between 2 1's, so A364215(A001045(2*n+1) + 1) = 2 which divides (2^(2*n+1) + 4)/3.

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

Cf. A364215.
Subsequence of A364216.
Subsequences: A364218, A364219, A364220, A364221.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090, A352108, A352321, A352343, A352509.

consecJacobsthalNiven[kmax_, len_] := Module[{m = 1, c = Table[False, {len}], s = {}}, Do[c = Join[Rest[c], {Divisible[k, DigitCount[m, 2, 1]]}]; While[m++; OddQ[IntegerExponent[m, 2]]]; If[And @@ c, AppendTo[s, k - len + 1]], {k, 1, kmax}]; s]; consecJacobsthalNiven[640, 2]

lista(kmax, len) = {my(m = 1, c = vector(len)); for(k = 1, kmax, c = concat(vecextract(c, "^1"), !(k % sumdigits(m, 2))); until(valuation(m, 2)%2 == 0, m++); if(vecsum(c) == len, print1(k-len+1, ", ")));}
lista(640, 2)

A364380 Numbers k such that k and k+1 are both greedy Jacobsthal-Niven numbers (A364379).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 11, 14, 15, 20, 21, 26, 27, 32, 42, 43, 44, 45, 51, 56, 68, 75, 84, 85, 86, 87, 92, 99, 104, 105, 111, 115, 116, 125, 128, 135, 144, 155, 170, 171, 176, 182, 183, 195, 204, 213, 219, 224, 260, 264, 267, 275, 304, 305, 324, 329, 341, 344

Offset: 1

Amiram Eldar, Jul 21 2023

The positive Jacobsthal numbers, A001045(n) for n >= 1, are terms since their representation in Jacobsthal greedy base is one 1 followed by n-1 0's, so A265745(A001045(n)) = 1 divides A001045(n), and the representation of A001045(n) + 1 is 2 if n <= 2 and otherwise n-3 0's between two 1's, so A265745(A001045(n) + 1) = 2 which divides A001045(n) + 1.

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

Cf. A265745, A265747.
Subsequence of A364379.
Subsequences: A364381, A364382, A364383.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715, A351720, A352090, A352108, A352321, A352343, A352509, A364217.

consecGreedyJN[kmax_, len_] := Module[{m = 1, c = Table[False, {len}], s = {}}, Do[c = Join[Rest[c], {greedyJacobNivenQ[k]}]; If[And @@ c, AppendTo[s, k - len + 1]], {k, 1, kmax}]; s]; consecGreedyJN[350, 2] (* using the function greedyJacobNivenQ[n] from A364379 *)

lista(kmax, len) = {my(c = vector(len)); for(k = 1, kmax, c = concat(vecextract(c, "^1"), isA364379(k)); if(vecsum(c) == len, print1(k-len+1, ", ")));} \\ using the function isA364379(n) from A364379
lista(350, 2)

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

Amiram Eldar, Jan 08 2020

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

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

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

Amiram Eldar, Jul 01 2023

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.

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

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.

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

Amiram Eldar, Jul 07 2023

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

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.

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 *)

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)

cp's OEIS Frontend

A352090 Numbers k such that k and k+1 are both tribonacci-Niven numbers (A352089).

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A352108 Numbers k such that k and k+1 are both lazy-tribonacci-Niven numbers (A352107).

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A352321 Numbers k such that k and k+1 are both Pell-Niven numbers (A352320).

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A352343 Numbers k such that k and k+1 are both lazy-Pell-Niven numbers (A352342).

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A352509 Numbers k such that k and k+1 are both Catalan-Niven numbers (A352508).

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A364217 Numbers k such that k and k+1 are both Jacobsthal-Niven numbers (A364216).

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A364380 Numbers k such that k and k+1 are both greedy Jacobsthal-Niven numbers (A364379).

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

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

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