A352108 -id:A352108 - cp's OEIS frontend

A352107 Lazy-tribonacci-Niven numbers: numbers that are divisible by the number of terms in their maximal (or lazy) representation in terms of the tribonacci numbers (A352103).

1, 2, 4, 6, 8, 12, 18, 20, 21, 24, 28, 30, 33, 36, 39, 40, 48, 50, 56, 60, 68, 70, 72, 75, 76, 80, 90, 96, 100, 108, 115, 116, 120, 135, 136, 140, 150, 155, 156, 160, 162, 168, 175, 176, 177, 180, 184, 185, 188, 195, 198, 204, 205, 208, 215, 216, 225, 231, 260

Offset: 1

Amiram Eldar, Mar 05 2022

Numbers k such that A352104(k) | k.

			6 is a term since its maximal tribonacci representation, A352103(6) = 110, has A352104(6) = 2 1's and 6 is divisible by 2.

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

Cf. A352103, A352104.
Subsequences: A352108, A352109, A352110.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089.

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[300], q]

A352109 Starts of runs of 3 consecutive lazy-tribonacci-Niven numbers (A352107).

Original entry on oeis.org

175, 1183, 2259, 5290, 12969, 21130, 51820, 70629, 78090, 79540, 81818, 129648, 160224, 169234, 180908, 228240, 238574, 249494, 278628, 332891, 376335, 383866, 398650, 399644, 454090, 550380, 565200, 683448, 683604, 694274, 728895, 754390, 782110, 809830, 837550

Offset: 1

Amiram Eldar, Mar 05 2022

			175 is a term since 175, 176 and 177 are all divisible by the number of terms in their maximal tribonacci representation:
    k  A352103(k)  A352104(k)  k/A352104(k)
  ---  ----------  ----------  ------------
  175    11111110           7            25
  176    11111111           8            22
  177   100100100           3            59

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

Cf. A352103, A352104.
Subsequence of A352107 and A352108.
A352110 is a subsequence.
Similar sequences: A154701, A328206, A328210, A328214, A330932, A331087, A333428, A334310, A331822, A342428, A344343, A351716, A351721, A352091.

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]]; lazyTriboNivenQ[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]]]]]]; seq[count, nConsec_] := Module[{tri = lazyTriboNivenQ /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ tri, c++; AppendTo[s, k - nConsec]]; tri = Join[Rest[tri], {lazyTriboNivenQ[k]}]; k++]; s]; seq[30, 3]

A352110 Starts of runs of 4 consecutive lazy-tribonacci-Niven numbers (A352107).

Original entry on oeis.org

1081455, 1976895, 2894175, 5886255, 6906912, 15604110, 16588752, 19291479, 20387232, 25919439, 32394942, 34801557, 35654175, 36813582, 36907899, 39117219, 41407392, 43520832, 46181055, 47954499, 52145952, 54524319, 54815397, 56733639, 57775102, 58942959, 59292177

Offset: 1

Amiram Eldar, Mar 05 2022

Conjecture: There are no runs of 5 consecutive lazy-tribonacci-Niven numbers (checked up to 6*10^9).

			1081455 is a term since 1081455, 1081456, 1081457 and 1081458 are all divisible by the number of terms in their maximal tribonacci representation:
        k               A352103(k)   A352104(k)    k/A352104(k)
  -------  -----------------------   ----------    ------------
  1081455  10101011011110110011110           15           72097
  1081456  10101011011110110011111           16           67591
  1081457  10101011011110110100100           13           83189
  1081458  10101011011110110100101           14           77247

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

Cf. A352103, A352104.
Subsequence of A352107, A352108 and A352109.
Similar sequences: A141769, A328211, A328207, A328215, A330933, A331824, A334311, A342429, A344344, A352092.

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)

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

A352107 Lazy-tribonacci-Niven numbers: numbers that are divisible by the number of terms in their maximal (or lazy) representation in terms of the tribonacci numbers (A352103).

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A352109 Starts of runs of 3 consecutive lazy-tribonacci-Niven numbers (A352107).

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A352110 Starts of runs of 4 consecutive 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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A364007 Numbers k such that k and k+1 are both Wythoff-Niven numbers (A364006).

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