A337076
Niven numbers (A005349) with a record gap to the next Niven number.
Original entry on oeis.org
1, 10, 12, 63, 72, 90, 288, 378, 558, 2889, 3784, 6480, 19872, 28971, 38772, 297864, 478764, 589860, 989867, 2879865, 9898956, 49989744, 88996914, 689988915, 879987906, 989888823, 2998895823, 6998899824, 8889999624, 8988988866, 9879997824, 18879988824, 286889989806
Offset: 1
10 is a term since it is a Niven number, and the next Niven number is 12, with a gap 12 - 10 = 2, which is a record, since all the numbers below 10 are also Niven numbers.
- Jean-Marie De Koninck and Nicolas Doyon, Large and Small Gaps Between Consecutive Niven Numbers, J. Integer Seqs., Vol. 6, 2003, Article 03.2.5.
- R. E. Kennedy and C. N. Cooper, On the natural density of the Niven numbers, The College Mathematics Journal, Vol. 15, No. 4 (1984), pp. 309-312.
-
nivenQ[n_] := Divisible[n, Plus @@ IntegerDigits[n]]; gapmax = 0; n1 = 1; s = {}; Do[If[nivenQ[n], gap = n - n1; If[gap > gapmax, gapmax = gap; AppendTo[s, n1]]; n1 = n], {n, 2, 10^6}]; s
A347496
Primorial base Niven numbers (A333426) with a record gap to the next primorial base Niven number.
Original entry on oeis.org
1, 2, 9, 12, 25, 50, 120, 344, 400, 770, 1120, 3920, 13566, 13734, 19845, 22748, 148148, 167854, 176220, 889896, 2946216, 3685416, 5072256, 7139280, 8521056, 9058900, 9625336, 17825857, 19392072, 27504848, 76952788, 106691001, 162789696, 198582784, 212847225
Offset: 1
The first 8 primorial base Niven numbers are 1, 2, 4, 6, 8, 9, 12 and 16. The gaps between them are 1, 2, 2, 2, 1, 3 and 4. The record gaps, 1, 2, 3 and 4, occur after the terms 1, 2, 9 and 12.
-
max = 7; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; sumdig[n_] := Plus @@ IntegerDigits[n, MixedRadix[bases]]; primoNivenQ[n_] := Divisible[n, sumdig[n]]; gapmax = 0; n1 = 1; s = {}; Do[If[primoNivenQ[n], gap = n - n1; If[gap > gapmax, gapmax = gap; AppendTo[s, n1]]; n1 = n], {n, 2, nmax}]; s
A347495
Factorial base Niven numbers (A118363) with a record gap to the next factorial base Niven number.
Original entry on oeis.org
1, 2, 9, 12, 30, 40, 60, 192, 224, 318, 550, 640, 1136, 1989, 4875, 4980, 23355, 24272, 24378, 40131, 60192, 63872, 80472, 238680, 280140, 2027340, 2872620, 3622068, 13400475, 21293094, 25399080, 28584626, 111020840, 278690360, 355419734, 398884590, 834592590
Offset: 1
The first 8 factorial base Niven numbers are 1, 2, 4, 6, 8, 9, 12 and 16. The gaps between them are 1, 2, 2, 2, 1, 3 and 4. The record gaps, 1, 2, 3 and 4, occur after the terms 1, 2, 9 and 12.
-
fivenQ[n_] := Module[{s = 0, i = 2, k = n}, While[k > 0, k = Floor[n/i!]; s = s + (i - 1)*k; i++]; Divisible[n, n - s]]; gapmax = 0; n1 = 1; s = {}; Do[If[fivenQ[n], gap = n - n1; If[gap > gapmax, gapmax = gap; AppendTo[s, n1]]; n1 = n], {n, 2, 10^5}]; s (* after Jean-François Alcover at A034968 *)
A376028
Zeckendorf-Niven numbers (A328208) with a record gap to the next Zeckendorf-Niven number.
Original entry on oeis.org
1, 6, 18, 30, 36, 48, 208, 5298, 6132, 6601, 8280, 12228, 17052, 68220, 113990, 120504, 438570, 1015416, 1343232, 1848400, 5338548, 12727143, 83877810, 330963120, 409185360, 418561770, 2428646640, 2834120595, 2876557200, 2940992640, 7218753758, 7306145012, 7609637140
Offset: 1
6 is a term since it is a Zeckendorf-Niven number, and the next Zeckendorf-Niven number is 8, with a gap 8 - 6 = 2, which is a record since all the numbers below 6 are also Zeckendorf-Niven numbers.
- Andrew B. Ray, On the natural density of the k-Zeckendorf Niven numbers, Ph.D. dissertation, Central Missouri State University, 2005.
-
z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; znQ[n_] := Divisible[n, z[n]]; seq[kmax_] := Module[{gapmax = 0, gap, k1 = 1, s = {}}, Do[If[znQ[k], gap = k - k1; If[gap > gapmax, gapmax = gap; AppendTo[s, k1]]; k1 = k], {k, 2, kmax}]; s]; seq[10^4]
A376029
Tribonacci-Niven numbers (A352089) with a record gap to the next tribonacci-Niven number.
Original entry on oeis.org
1, 2, 8, 48, 140, 244, 620, 705, 1395, 6210, 9656, 14322, 52024, 88128, 95589, 119440, 151130, 325105, 407128, 472790, 520971, 686103, 4456608, 7066416, 13207389, 15488160, 23381160, 42317212, 49496700, 53564016, 128163495, 165750096, 387900480, 421730960, 485880806
Offset: 1
The first 7 tribonacci-Niven numbers are 1, 2, 4, 6, 7, 8 and 12. The gaps between them are 1, 2, 2, 1, 1 and 4, and the record gaps, 1, 2 and 4 occur after 1, 2 and 8, the first 3 terms of this sequence.
- Andrew B. Ray, On the natural density of the k-Zeckendorf Niven numbers, Ph.D. dissertation, Central Missouri State University, 2005.
-
t[1] = 1; t[2] = 2; t[3] = 4; t[n_] := t[n] = t[n - 1] + t[n - 2] + t[n - 3]; tnQ[n_] := Module[{s = 0, m = n, k}, While[m > 0, k = 1; While[t[k] <= m, k++]; k--; s++; m -= t[k]; k = 1]; Divisible[n, s]]; seq[kmax_] := Module[{gapmax = 0, gap, k1 = 1, s = {}}, Do[If[tnQ[k], gap = k - k1; If[gap > gapmax, gapmax = gap; AppendTo[s, k1]]; k1 = k], {k, 2, kmax}]; s]; seq[10^4]
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