A344342 -id:A344342 - cp's OEIS frontend

A344341 Gray-code Niven numbers: numbers divisible by the number of 1's in their binary reflected Gray code (A005811).

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 15, 16, 20, 24, 27, 28, 30, 31, 32, 33, 36, 39, 40, 42, 44, 45, 48, 51, 52, 56, 57, 60, 62, 63, 64, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 99, 100, 104, 105, 108, 111, 112, 116, 120, 123, 124, 126, 127, 128, 129, 132, 135, 136

Offset: 1

Amiram Eldar, May 15 2021

			2 is a term since its Gray code is 11 and 1+1 = 2 is a divisor of 2.
6 is a term since its Gray code is 101 and 1+0+1 = 2 is a divisor of 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gray Code.
Wikipedia, Gray code.

Cf. A005811, A014550.
Subsequences: A344342, A344343, A344344.
Similar sequences: A005349 (decimal), A049445 (binary), A064150 (ternary), A064438 (quaternary), A064481 (base 5), A118363 (factorial), A328208 (Zeckendorf), A328212 (lazy Fibonacci), A331085 (negaFibonacci), A333426 (primorial), A334308 (base phi), A331728 (negabinary), A342426 (base 3/2), A342726 (base i-1).

gcNivenQ[n_] := Divisible[n, DigitCount[BitXor[n, Floor[n/2]], 2, 1]]; Select[Range[150], gcNivenQ]

A344343 Starts of runs of 3 consecutive Gray-code Niven numbers (A344341).

Original entry on oeis.org

1, 2, 6, 7, 14, 30, 31, 62, 126, 127, 174, 184, 234, 243, 254, 304, 474, 483, 510, 511, 534, 543, 544, 783, 784, 903, 904, 954, 963, 1022, 1134, 1144, 1253, 1264, 1448, 1475, 1504, 1895, 1914, 1923, 1974, 2046, 2047, 2093, 2094, 2104, 2814, 2888, 2944, 3054, 3064

Offset: 1

Amiram Eldar, May 15 2021

			1 is a term since 1, 2 and 3 are all Gray-code Niven numbers.

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

Cf. A005811, A014550.
Subsequence of A344341 and A344342.
Subsequences: A344344.
Similar sequences: A154701 (decimal), A328206 (factorial), A328210 (Zeckendorf), A328214 (lazy Fibonacci), A330932 (binary), A331087 (negaFibonacci), A333428 (primorial), A334310 (base phi), A331822 (negabinary), A342428 (base 3/2).

gcNivenQ[n_] := Divisible[n, DigitCount[BitXor[n, Floor[n/2]], 2, 1]]; Select[Range[3000], AllTrue[# + {0, 1, 2}, gcNivenQ] &]

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] &]

A344344 Starts of runs of 4 consecutive Gray-code Niven numbers (A344341).

Original entry on oeis.org

1, 6, 30, 126, 510, 543, 783, 903, 2046, 2093, 3773, 3903, 7133, 7743, 8190, 8223, 8703, 10087, 12303, 12543, 14343, 14463, 15423, 15903, 16143, 16263, 20167, 22687, 27727, 30247, 30653, 30783, 32766, 35629, 40327, 47509, 47887, 49133, 50407, 57533, 60071, 60487

Offset: 1

Amiram Eldar, May 15 2021

Are there 5 consecutive Gray-code Niven numbers? There are no such numbers below 10^10.

			1 is a term since 1, 2, 3 and 4 are all Gray-code Niven numbers.

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

Cf. A005811, A014550.
Subsequence of A344341, A344342 and A344343.
Similar sequences: A141769 (decimal), A328207 (factorial), A328211 (Zeckendorf), A328215 (lazy Fibonacci), A330933 (binary), A334311 (base phi), A331824 (negabinary), A342429 (base 3/2).

gcNivenQ[n_] := Divisible[n, DigitCount[BitXor[n, Floor[n/2]], 2, 1]]; Select[Range[60000], AllTrue[# + {0, 1, 2, 3}, gcNivenQ] &]

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] &]

cp's OEIS Frontend

A344341 Gray-code Niven numbers: numbers divisible by the number of 1's in their binary reflected Gray code (A005811).

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A344343 Starts of runs of 3 consecutive Gray-code Niven numbers (A344341).

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

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

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

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A344344 Starts of runs of 4 consecutive Gray-code Niven numbers (A344341).

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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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