A331820 -id:A331820 - cp's OEIS frontend

A342427 Numbers k such that k and k + 1 are both Niven numbers in base 3/2 (A342426).

1, 168, 459, 1817, 2196, 2197, 2655, 3128, 3280, 3699, 4199, 4575, 4927, 5184, 5795, 6600, 7215, 7259, 7656, 7657, 8448, 9636, 11304, 11339, 12492, 14160, 14175, 14424, 14805, 15624, 15625, 16335, 16336, 16925, 17802, 19170, 20349, 20811, 21624, 21735, 22197

Offset: 1

Amiram Eldar, Mar 11 2021

			168 is a term since both 168 and 169 are Niven numbers in base 3/2. 168 in base 3/2 is 2120220210 and 2+1+2+0+2+2+0+2+1+0 = 12 is a divisor of 168. 169 in base 3/2 is 2120220211 and 2+1+2+0+2+2+0+2+1+1 = 13 is a divisor of 169.

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

Subsequence of A342426.
Subsequences: A342428 and A342429.
Similar sequences: A330927 (decimal), A328205 (factorial), A328209 (Zeckendorf), A328213 (lazy Fibonacci), A330931 (binary), A331086 (negaFibonacci), A333427 (primorial), A334309 (base phi), A331820 (negabinary).

s[0] = 0; s[n_] := s[n] = s[2*Floor[n/3]] + Mod[n, 3]; q[n_] := Divisible[n, s[n]]; Select[Range[22000], q[#] && q[# + 1] &]

A344342 Numbers k such that k and k + 1 are both Gray-code Niven numbers (A344341).

Original entry on oeis.org

1, 2, 3, 6, 7, 8, 14, 15, 27, 30, 31, 32, 39, 44, 51, 56, 62, 63, 75, 99, 104, 111, 123, 126, 127, 128, 135, 144, 155, 159, 174, 175, 184, 185, 195, 204, 207, 215, 224, 231, 234, 235, 243, 244, 248, 254, 255, 264, 275, 284, 294, 300, 304, 305, 315, 335, 354, 375

Offset: 1

Amiram Eldar, May 15 2021

			1 is a term since 1 and 2 are both Gray-code Niven numbers.

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

Cf. A005811, A014550.
Subsequence of: A344341.
Subsequences: A344343 and A344344.
Similar sequences: A330927 (decimal), A328205 (factorial), A328209 (Zeckendorf), A328213 (lazy Fibonacci), A330931 (binary), A331086 (negaFibonacci), A333427 (primorial), A334309 (base phi), A331820 (negabinary), A342427 (base 3/2).

gcNivenQ[n_] := Divisible[n, DigitCount[BitXor[n, Floor[n/2]], 2, 1]]; Select[Range[400], And @@ gcNivenQ[# + {0, 1}] &]

A351720 Numbers k such that k and k + 1 are both lazy-Lucas-Niven numbers (A351719).

Original entry on oeis.org

1, 175, 216, 399, 656, 729, 737, 759, 1000, 1991, 2716, 2820, 2925, 3970, 4068, 4224, 4499, 4641, 5316, 5819, 6565, 6720, 6902, 7890, 9840, 10751, 11843, 12194, 12614, 13034, 13272, 14909, 15483, 15495, 16029, 17234, 17444, 17731, 18074, 18885, 19305, 19669, 20188

Offset: 1

Amiram Eldar, Feb 17 2022

			175 is a term since 175 and 176 are both lazy-Lucas-Niven numbers: the maximal Lucas representation of 175, A130311(175) = 1110110101, has 7 1's and 175 is divisible by 5, and the maximal Lucas representation of 176, A130311(7) = 1110110111, has 8 1's and 176 is divisible by 8.

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

Cf. A000032, A130311, A131343.
Subsequence of A351719.
A351721 is a subsequence.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351715.

lazy = Select[IntegerDigits[Range[10^6], 2], SequenceCount[#, {0, 0}] == 0 &]; t = Total[#*Reverse@LucasL[Range[0, Length[#] - 1]]] & /@ lazy; s = FromDigits /@ lazy[[TakeWhile[Flatten[FirstPosition[t, #] & /@ Range[Max[t]]], NumberQ]]]; SequencePosition[Divisible[Range[Length[s]], Plus @@@ IntegerDigits[s]], {True, True}][[;; , 1]]

A351715 Numbers k such that k and k + 1 are both Lucas-Niven numbers (A351714).

Original entry on oeis.org

1, 2, 3, 6, 7, 10, 11, 29, 39, 47, 57, 80, 123, 129, 134, 152, 159, 170, 176, 199, 206, 245, 279, 326, 384, 387, 398, 404, 521, 531, 543, 560, 579, 615, 644, 651, 684, 755, 843, 849, 854, 872, 879, 890, 896, 944, 1024, 1052, 1064, 1070, 1071, 1095, 1350, 1382

Offset: 1

Amiram Eldar, Feb 17 2022

			6 is a term since 6 and 7 are both Lucas-Niven numbers: the minimal Lucas representation of 6, A130310(6) = 1001, has 2 1's and 6 is divisible by 2, and the minimal Lucas representation of 7, A130310(7) = 10000, has one 1 and 7 is divisible by 1.

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

Subsequence of A351714.
A351716 is a subsequence.
Similar sequences: A330927, A328205, A328209, A328213, A330931, A331086, A333427, A334309, A331820, A342427, A344342, A351720.

lucasNivenQ[n_] := Module[{s = {}, m = n, k = 1}, While[m > 0, If[m == 1, k = 1; AppendTo[s, k]; m = 0, If[m == 2, k = 0; AppendTo[s, k]; m = 0, While[LucasL[k] <= m, k++]; k--; AppendTo[s, k]; m -= LucasL[k]; k = 1]]]; Divisible[n, Plus @@ IntegerDigits[Total[2^s], 2]]]; Select[Range[1400], And @@ lucasNivenQ/@{#, #+1} &]

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

Original entry on oeis.org

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)

cp's OEIS Frontend

A342427 Numbers k such that k and k + 1 are both Niven numbers in base 3/2 (A342426).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A344342 Numbers k such that k and k + 1 are both Gray-code Niven numbers (A344341).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A351720 Numbers k such that k and k + 1 are both lazy-Lucas-Niven numbers (A351719).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A351715 Numbers k such that k and k + 1 are both Lucas-Niven numbers (A351714).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords