Displaying 1-10 of 21 results found.
Primorial base Niven numbers: numbers divisible by their sum of digits in primorial base ( A276150).
+10
27
1, 2, 4, 6, 8, 9, 12, 16, 18, 20, 24, 25, 30, 32, 33, 36, 40, 42, 44, 45, 48, 50, 60, 64, 65, 66, 68, 70, 72, 77, 84, 88, 90, 92, 96, 105, 108, 112, 117, 120, 132, 133, 136, 144, 150, 154, 156, 160, 168, 180, 182, 184, 189, 192, 198, 200, 210, 212, 213, 216, 220
EXAMPLE
1 is a term since A276150(1) = 1 divides 1; 2 is a term since A276150(2) = 1 divides 2;
MATHEMATICA
max = 5; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; sumdig[n_] := Plus @@ IntegerDigits[n, MixedRadix[bases]]; Select[Range[nmax], Divisible[#, sumdig[#]] &]
CROSSREFS
Cf. A005349, A049345, A049445, A064150, A064438, A064481, A118363, A235168, A276150, A328208, A328212, A373834 (characteristic function)
Divisible by the sum of the digits of its base-5 representation.
+10
19
1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 18, 20, 24, 25, 26, 27, 28, 30, 32, 36, 40, 42, 45, 48, 50, 51, 52, 54, 56, 60, 63, 64, 65, 66, 72, 75, 76, 78, 80, 85, 88, 90, 91, 96, 99, 100, 102, 104, 105, 112, 117, 120, 125, 126, 128, 130, 132, 135, 136, 138, 140, 144, 145
EXAMPLE
Base-5 representation of 28 is 103; 1 + 0 + 3 = 4 divides 28.
PROG
(ARIBAS): maxarg := 160; for n := 1 to maxarg do if n mod sum(basearray(n, 5)) = 0 then write(n, " "); end; end; function basearray(n, b: integer): array; var k: integer; stk: stack; begin while n > 0 do k := n mod b; stack_push(stk, k); n := (n - k) div b; end; return stack2array(stk); end; .
(PARI) SumD(x)= { my(s); s=0; while (x>9, s+=x-10*(x\10); x\=10); return(s + x) }
baseE(x, b)= { my(d, e, f); e=0; f=1; while (x>0, d=x-b*(x\b); x\=b; e+=d*f; f*=10); return(e) }
{ n=0; for (m=1, 10^9, if (m%(SumD(baseE(m, 5)))==0, write("b064481.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 15 2009 (PARI) isok(n) = !(n % sumdigits(n, 5)); \\ Michel Marcus, Jun 24 2018
Base phi Niven numbers: numbers divisible by the number of 1's in their base phi representation ( A055778).
+10
19
1, 2, 6, 12, 15, 16, 18, 20, 30, 35, 36, 45, 48, 55, 60, 70, 72, 78, 84, 90, 91, 95, 96, 98, 104, 108, 132, 144, 147, 154, 168, 175, 184, 189, 208, 224, 231, 232, 245, 252, 256, 261, 264, 270, 275, 280, 282, 287, 294, 315, 322, 324, 330, 336, 340, 342, 351, 357
EXAMPLE
6 is a term since its base phi representation is 1010.0001, and the number of 1's is 3, which is a divisor of 6.
MATHEMATICA
phiDigSum[1] = 1; phiDigSum[n_] := Plus @@ RealDigits[n, GoldenRatio, 2*Ceiling[ Log[GoldenRatio, n]] ][[1]]; Select[Range[360], Divisible[#, phiDigSum[#]] &]
Niven numbers in base 3/2: numbers divisible by their sum of digits in fractional base 3/2 ( A244040).
+10
17
1, 2, 6, 9, 14, 21, 40, 42, 56, 72, 84, 108, 110, 120, 126, 130, 143, 154, 156, 162, 165, 168, 169, 176, 180, 182, 189, 198, 220, 225, 231, 243, 252, 280, 288, 297, 306, 308, 320, 322, 330, 336, 348, 350, 364, 390, 423, 430, 432, 459, 460, 462, 480, 490, 504
EXAMPLE
6 is a term since its representation in base 3/2 is 210 and 2 + 1 + 0 = 3 is a divisor of 6.
9 is a term since its representation in base 3/2 is 2100 and 2 + 1 + 0 + 0 = 3 is a divisor of 9.
MATHEMATICA
s[0] = 0; s[n_] := s[n] = s[2*Floor[n/3]] + Mod[n, 3]; q[n_] := Divisible[n, s[n]]; Select[Range[500], q]
CROSSREFS
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).
Niven numbers in base i-1: numbers that are divisible by the sum of their digits in base i-1.
+10
17
1, 2, 3, 4, 5, 6, 7, 10, 12, 15, 16, 18, 20, 24, 25, 30, 32, 33, 35, 36, 40, 42, 44, 45, 48, 50, 54, 60, 64, 65, 66, 70, 77, 80, 88, 90, 96, 99, 100, 110, 112, 120, 124, 125, 126, 130, 140, 144, 145, 147, 150, 156, 160, 168, 170, 180, 182, 184, 185, 186, 190, 192
COMMENTS
Numbers k that are divisible by A066323(k). Equivalently, Niven numbers in base -4, since A066323(k) is also the sum of the digits of k in base -4.
EXAMPLE
2 is a term since its representation in base i-1 is 1100 and 1+1+0+0 = 2 is a divisor of 2.
10 is a term since its representation in base i-1 is 111001100 and 1+1+1+0+0+1+1+0+0 = 5 is a divisor of 10.
MATHEMATICA
v = {{0, 0, 0, 0}, {0, 0, 0, 1}, {1, 1, 0, 0}, {1, 1, 0, 1}}; q[n_] := Divisible[n, Total[Flatten @ v[[1 + Reverse @ Most[Mod[NestWhileList[(# - Mod[#, 4])/-4 &, n, # != 0 &], 4]]]]]]; Select[Range[200], q]
CROSSREFS
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).
Gray-code Niven numbers: numbers divisible by the number of 1's in their binary reflected Gray code ( A005811).
+10
15
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
LINKS
Eric Weisstein's World of Mathematics, Gray Code.
EXAMPLE
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.
MATHEMATICA
gcNivenQ[n_] := Divisible[n, DigitCount[BitXor[n, Floor[n/2]], 2, 1]]; Select[Range[150], gcNivenQ]
CROSSREFS
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).
Lucas-Niven numbers: numbers that are divisible by the number of terms in their minimal (or greedy) representation in terms of the Lucas numbers ( A130310).
+10
13
1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 14, 18, 20, 22, 24, 27, 29, 30, 32, 36, 39, 40, 42, 47, 48, 50, 54, 57, 58, 60, 64, 66, 69, 72, 76, 78, 80, 81, 84, 90, 92, 94, 96, 100, 104, 108, 120, 123, 124, 126, 129, 130, 132, 134, 135, 138, 140, 144, 152, 153, 156, 159, 160
COMMENTS
Numbers k such that A116543(k) | k.
EXAMPLE
6 is a term since its minimal Lucas representation, A130310(6) = 1001, has A116543(6) = 2 1's and 6 is divisible by 2.
MATHEMATICA
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[160], lucasNivenQ]
CROSSREFS
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351719.
Lazy-Lucas-Niven numbers: numbers divisible by the number of terms in their maximal (or lazy) representation in terms of the Lucas numbers ( A130311).
+10
13
1, 2, 4, 6, 9, 12, 16, 20, 25, 40, 42, 54, 60, 66, 78, 84, 91, 96, 104, 112, 120, 126, 144, 154, 161, 168, 175, 176, 180, 182, 184, 192, 203, 210, 216, 217, 224, 232, 234, 240, 243, 264, 270, 280, 288, 304, 306, 310, 315, 320, 322, 328, 336, 344, 350, 360, 378
COMMENTS
Numbers k such that A131343(k) | k.
EXAMPLE
6 is a term since its maximal Lucas representation, A130311(6) = 111, has A131343(6) = 3 1's and 6 is divisible by 3.
MATHEMATICA
lazy = Select[IntegerDigits[Range[3000], 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]]]; Position[Divisible[Range[Length[s]], Plus @@@ IntegerDigits[s]], True] // Flatten
CROSSREFS
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714.
Tribonacci-Niven numbers: numbers that are divisible by the number of terms in their minimal (or greedy) representation in terms of the tribonacci numbers ( A278038).
+10
13
1, 2, 4, 6, 7, 8, 12, 13, 14, 18, 20, 21, 24, 26, 27, 28, 30, 33, 36, 39, 40, 44, 46, 48, 56, 60, 68, 69, 72, 75, 76, 80, 81, 82, 84, 87, 88, 90, 94, 96, 100, 108, 115, 116, 120, 126, 128, 129, 132, 135, 136, 138, 140, 149, 150, 156, 162, 168, 174, 176, 177, 180
COMMENTS
Numbers k such that A278043(k) | k. The positive tribonacci numbers ( A000073) are all terms. If k = A000073( A042964(m)) is an odd tribonacci number, then k+1 is a term. Ray (2005) and Ray and Cooper (2006) called these numbers "3-Zeckendorf Niven numbers" and proved that their asymptotic density is 0. - Amiram Eldar, Sep 06 2024
REFERENCES
Andrew B. Ray, On the natural density of the k-Zeckendorf Niven numbers, Ph.D. dissertation, Central Missouri State University, 2005.
EXAMPLE
6 is a term since its minimal tribonacci representation, A278038(6) = 110, has A278043(6) = 2 1's and 6 is divisible by 2.
MATHEMATICA
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[180], q]
CROSSREFS
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719.
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).
+10
11
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
COMMENTS
Numbers k such that A352104(k) | k.
EXAMPLE
6 is a term since its maximal tribonacci representation, A352103(6) = 110, has A352104(6) = 2 1's and 6 is divisible by 2.
MATHEMATICA
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]
CROSSREFS
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089.
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