A118363 -id:A118363 - cp's OEIS frontend

A364216 Jacobsthal-Niven numbers: numbers that are divisible by the sum of the digits in their Jacobsthal representation (A280049).

1, 2, 3, 4, 6, 8, 9, 11, 12, 14, 15, 16, 20, 22, 24, 27, 28, 32, 33, 36, 40, 42, 43, 44, 45, 46, 48, 51, 52, 54, 56, 57, 60, 68, 72, 75, 76, 84, 86, 87, 88, 92, 93, 95, 96, 99, 100, 104, 105, 108, 112, 115, 117, 120, 125, 126, 128, 129, 132, 135, 136, 138, 140

Offset: 1

Amiram Eldar, Jul 14 2023

Numbers k such that A364215(k) | k.

A007583 is a subsequence since A364215(A007583(n)) = 1 for n >= 0.

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

Cf. A280049, A364215.
Subsequences: A007583, A364217, A364218, A364219, A364220, A364221.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352342, A352508.

seq[kmax_] := Module[{m = 1, s = {}}, Do[If[Divisible[k, DigitCount[m, 2, 1]], AppendTo[s, k]]; While[m++; OddQ[IntegerExponent[m, 2]]], {k, 1, kmax}]; s]; seq[140]

lista(kmax) = {my(m = 1); for(k = 1, kmax, if( !(k % sumdigits(m, 2)), print1(k,", ")); until(valuation(m, 2)%2 == 0, m++));}

A364379 Greedy Jacobsthal-Niven numbers: numbers that are divisible by the sum of the digits in their representation in Jacobsthal greedy base (A265747).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 20, 21, 22, 24, 26, 27, 28, 32, 33, 36, 40, 42, 43, 44, 45, 46, 48, 51, 52, 54, 56, 57, 60, 64, 68, 69, 72, 75, 76, 80, 84, 85, 86, 87, 88, 90, 92, 93, 96, 99, 100, 104, 105, 106, 108, 111, 112, 115, 116, 117, 120

Offset: 1

Amiram Eldar, Jul 21 2023

Numbers k such that A265745(k) | k.

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).

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

Cf. A265745, A265747.
Subsequences: A364380, A364381, A364382, A364383.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352342, A352508, A364216.

greedyJacobNivenQ[n_] := Divisible[n, A265745[n]]; Select[Range[120], greedyJacobNivenQ] (* using A265745[n] *)

isA364379(n) = !(n % A265745(n)); \\ using A265745(n)

A226169 Niven numbers when expressed in bases 1 through 10.

Original entry on oeis.org

1, 2, 4, 6, 24, 40, 48, 72, 120, 144, 180, 216, 252, 288, 324, 336, 360, 432, 504, 576, 648, 720, 756, 780, 840, 960, 1008, 1056, 1080, 1092, 1200, 1260, 1296, 1344, 1380, 1440, 1512, 1584, 1620, 1680, 1728, 1764, 1800, 1944, 2016, 2196, 2304, 2352, 2448

Offset: 1

Sergio Pimentel, May 29 2013

The first 10 odd terms greater than 1 are a(1151) = 543375, 5329233, 18640125, 19178775, 23186625, 30131535, 35026425, 36797775, 46101825, 51856875. - Giovanni Resta, Jun 01 2013

			Example: 336 is in the sequence because the sum of digits of 336 when expressed in bases 1 through 10 is: 336, 3, 4, 3, 8, 6, 12, 7, 8, 12; and 336 is divisible by all these numbers.  In this particular example 336 keeps this property in bases 11, 12 and 13, but not 14.

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

Cf. A049445, A118363, A005349, A144261, A144262.

Select[Range[10^4], Catch[Do[If[Mod[#, Total@IntegerDigits[#, b]] > 0, Throw@ False], {b, 2, 10}]; True] &] (* Giovanni Resta, May 29 2013 *)
t = Table[b = 2; While[s = Total[IntegerDigits[n, b]]; s < n && Mod[n, s] == 0, b++]; If[s == n, b = 0]; b, {n, 2000}]; Flatten[Position[t, ?(# == 0 || # > 10 &)]] (* _T. D. Noe, May 30 2013 *)

Missing a(17) and a(35)-a(49) from Giovanni Resta, May 29 2013

A286590 Numbers that are divisible by the product of their factorial base digits (A208575).

Original entry on oeis.org

1, 3, 9, 21, 33, 45, 81, 153, 165, 201, 393, 405, 441, 645, 873, 885, 921, 1113, 1125, 1161, 1365, 2313, 2565, 3765, 4005, 5913, 5925, 5961, 6153, 6165, 6201, 6405, 7353, 7641, 8805, 9045, 15993, 16281, 17433, 26085, 26325, 36393, 36645, 46233, 46245, 46281, 46473, 46485, 46521, 46725, 47673

Offset: 1

Antti Karttunen, Jun 18 2017

After the initial 1, all terms are multiples of three.

Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..120 from Antti Karttunen)
Index entries for sequences related to factorial base representation

Cf. A007489 (a subsequence), A208575, A118363.

Cf. A007602 (for base-10 analog).

max = 8; Select[Range[max!], FreeQ[(d = IntegerDigits[#, MixedRadix[Range[max, 2, -1]]]), 0] && Divisible[#, Times @@ d] &] (* Amiram Eldar, Feb 16 2021 *)

A286604 a(n) = n mod sum of digits of n in factorial base.

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 3, 0, 1, 2, 3, 0, 2, 0, 3, 0, 1, 2, 5, 0, 1, 0, 0, 1, 1, 0, 1, 2, 1, 2, 0, 0, 1, 2, 4, 0, 5, 2, 3, 4, 3, 4, 5, 0, 1, 2, 3, 0, 3, 0, 3, 0, 2, 3, 5, 0, 1, 2, 3, 4, 2, 1, 1, 2, 6, 0, 7, 0, 1, 2, 0, 1, 5, 2, 4, 0, 3, 4, 6, 4, 1, 2, 3, 4, 1, 0, 0, 1, 5, 6, 5, 0, 2, 3, 3, 4, 3, 2, 1, 2, 0, 1, 3, 0, 4, 5, 7, 0, 5, 2, 3, 4, 0, 1, 9, 0

Offset: 1

Antti Karttunen, Jun 18 2017

Antti Karttunen, Table of n, a(n) for n = 1..10080
Index entries for sequences related to factorial base representation.

Cf. A034968, A286606.

Cf. A118363 (positions of zeros), A286607 (of nonzeros).

a[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, s += r; m++]; Mod[n, s]]; Array[a, 100] (* Amiram Eldar, Feb 21 2024 *)

def a007623(n, p=2): return n if nIndranil Ghosh, Jun 21 2017

(define (A286604 n) (modulo n (A034968 n)))

a(n) = n mod A034968(n).

A286607 Numbers that are not divisible by the sum of their factorial base digits (A034968).

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 28, 29, 31, 32, 33, 34, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 97, 98, 99, 100, 101, 102, 103, 104, 106, 107

Offset: 1

Antti Karttunen, Jun 18 2017

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to factorial base representation.

Cf. A034968, A118363 (complement), A286604.

q[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, s += r; m++]; !Divisible[n, s]]; Select[Range[120], q] (* Amiram Eldar, Feb 21 2024 *)

def a007623(n, p=2): return n if nIndranil Ghosh, Jun 21 2017

A358976 Numbers that are coprime to the sum of their factorial base digits (A034968).

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 25, 28, 29, 31, 32, 33, 37, 39, 41, 43, 44, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 76, 77, 79, 83, 84, 85, 87, 88, 89, 92, 93, 95, 97, 98, 101, 102, 103, 106, 107, 109, 110

Offset: 1

Amiram Eldar, Dec 07 2022

Numbers k such that gcd(k, A034968(k)) = 1.
The factorial numbers (A000142) are terms. These are also the only factorial base Niven numbers (A118363) in this sequence.
Includes all the prime numbers.
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 7, 59, 601, 6064, 60729, 607567, 6083420, 60827602, 607643918, 6079478119, ... . Conjecture: The asymptotic density of this sequence exists and equals 6/Pi^2 = 0.607927... (A059956), the same as the density of A094387.

			3 is a term since A034968(3) = 2, and gcd(3, 2) = 1.

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

Cf. A034968, A059956, A118363.
Subsequences: A000040, A000142.
Similar sequences: A094387, A339076, A358975, A358977, A358978.

q[n_] := Module[{k = 2, s = 0, m = n, r}, While[m > 0, r=Mod[m,k]; s+=r; m=(m-r)/k; k++]; CoprimeQ[n, s]]; Select[Range[120], q]

is(n)={my(k=2, s=0, m=n); while(m>0, s+=m%k; m\=k; k++); gcd(s,n)==1;}

A364006 Wythoff-Niven numbers: numbers that are divisible by the number of 1's in their Wythoff representation.

Original entry on oeis.org

1, 3, 4, 6, 7, 8, 10, 12, 15, 18, 20, 21, 24, 26, 28, 32, 35, 39, 40, 42, 45, 47, 51, 52, 54, 55, 56, 60, 68, 72, 76, 80, 84, 86, 88, 90, 91, 98, 100, 102, 105, 117, 120, 123, 125, 135, 136, 138, 141, 143, 144, 156, 164, 168, 172, 174, 176, 178, 180, 188, 192

Offset: 1

Amiram Eldar, Jul 01 2023

Numbers k such that A135818(k) | k.

Includes all the positive even-indexed Fibonacci numbers (A001906), since the Wythoff representation of Fibonacci(2*n), for n >= 1, is 1 followed by n-1 0's.

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

Cf. A135818, A189921.
Subsequences: A364007, A364008, A364009.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352508.

wnQ[n_] := (s = Total[w[n]]) > 0 && Divisible[n, s] (* using the function w[n] from A364005 *)

A364123 Stolarsky-Niven numbers: numbers that are divisible by the number of 1's in their Stolarsky representation (A364121).

Original entry on oeis.org

2, 4, 6, 8, 9, 12, 14, 16, 20, 22, 24, 27, 30, 36, 38, 40, 42, 44, 48, 54, 56, 57, 60, 65, 69, 72, 75, 80, 84, 85, 90, 92, 96, 98, 100, 102, 104, 108, 112, 116, 120, 124, 126, 132, 136, 138, 145, 147, 150, 153, 155, 159, 160, 175, 180, 185, 190, 195, 196, 205

Offset: 1

Amiram Eldar, Jul 07 2023

Numbers k such that A200649(k) | k.

Fibonacci(k) + 1 is a term if k !== 3 (mod 6) (i.e., k is in A047263).

			4 is a term since its Stolarsky representation, A364121(4) = 10, has one 1 and 4 is divisible by 1.
6 is a term since its Stolarsky representation, A364121(6) = 101, has 2 1's and 6 is divisible by 2.

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

Cf. A047263, A200649, A364121.
Subsequences: A364124, A364125, A364126.
Similar sequences: A005349, A049445, A064150, A064438, A064481, A118363, A328208, A328212, A331085, A333426, A342726, A334308, A331728, A342426, A344341, A351714, A351719, A352089, A352107, A352320, A352342, A352508.

stol[n_] := stol[n] = If[n == 1, {}, If[n != Round[Round[n/GoldenRatio]*GoldenRatio], Join[stol[Floor[n/GoldenRatio^2] + 1], {0}], Join[stol[Round[n/GoldenRatio]], {1}]]];
stolNivQ[n_] := n > 1 && Divisible[n, Total[stol[n]]];
Select[Range[200], stolNivQ]

stol(n) = {my(phi=quadgen(5)); if(n==1, [], if(n != round(round(n/phi)*phi), concat(stol(floor(n/phi^2) + 1), [0]), concat(stol(round(n/phi)), [1])));}
isA364123(n) = n > 1 && !(n % vecsum(stol(n)));

A377384 a(n) is the number of iterations that n requires to reach a noninteger or a factorial number under the map x -> x / f(x), where f(k) = A034968(k) is the sum of digits in the factorial-base representation of k; a(n) = 0 if n is a factorial number.

Original entry on oeis.org

0, 0, 1, 1, 1, 0, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 0, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Oct 27 2024

The factorial numbers are fixed points of the map, since f(k!) = 1 for all k >= 0. Therefore they are arbitrarily assigned the value a(k!) = 0.

Each number n starts a chain of a(n) integers: n, n/f(n), (n/f(n))/f(n/f(n)), ..., of them the first a(n)-1 integers are factorial-base Niven numbers (A118363).

			a(8) = 2 since 8/f(8) = 4 and 4/f(4) = 2 is a factorial number that is reached after 2 iterations.
a(27) = 3 since 27/f(27) = 9, 9/f(9) = 3 and 3/f(3) = 3/2 is a noninteger that is reached after 3 iterations.

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

Cf. A000005, A000142, A034968, A118363, A286607, A377385, A377386.

Analogous sequences: A376615 (binary), A377208 (Zeckendorf).

fdigsum[n_] := Module[{k = n, m = 2, r, s = 0}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, s += r; m++]; s]; a[n_] := a[n] = Module[{s = fdigsum[n]}, If[s == 1, 0, If[!Divisible[n, s], 1, 1 + a[n/s]]]]; Array[a, 100]

fdigsum(n) = {my(k = n, m = 2, r, s = 0); while([k, r] = divrem(k, m); k != 0 || r != 0, s += r; m++); s;}
a(n) = {my(f = fdigsum(n)); if(f == 1, 0, if(n % f, 1, 1 + a(n/f)));}

def f(n, p=2): return n if nMichael S. Branicky, Oct 27 2024

a(n) = 0 if and only if n is in A000142 (by definition).
a(n) = 1 if and only if n is in A286607.
a(n) >= 2 if and only if n is in A118363 \ A000142 (i.e., n is a factorial-base Niven number that is not a factorial number).
a(n) >= 3 if and only if n is in A377385 \ A000142.
a(n) >= 4 if and only if n is in A377386 \ A000142.
a(n) < A000005(n).

cp's OEIS Frontend

A364216 Jacobsthal-Niven numbers: numbers that are divisible by the sum of the digits in their Jacobsthal representation (A280049).

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A364379 Greedy Jacobsthal-Niven numbers: numbers that are divisible by the sum of the digits in their representation in Jacobsthal greedy base (A265747).

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A226169 Niven numbers when expressed in bases 1 through 10.

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A286590 Numbers that are divisible by the product of their factorial base digits (A208575).

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A286604 a(n) = n mod sum of digits of n in factorial base.

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A286607 Numbers that are not divisible by the sum of their factorial base digits (A034968).

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A358976 Numbers that are coprime to the sum of their factorial base digits (A034968).

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A364006 Wythoff-Niven numbers: numbers that are divisible by the number of 1's in their Wythoff representation.

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