A014261 -id:A014261 - cp's OEIS frontend

A351894 Numbers that contain only odd digits in their factorial-base representation.

1, 3, 9, 21, 33, 45, 81, 93, 153, 165, 201, 213, 393, 405, 441, 453, 633, 645, 681, 693, 873, 885, 921, 933, 1113, 1125, 1161, 1173, 1353, 1365, 1401, 1413, 2313, 2325, 2361, 2373, 2553, 2565, 2601, 2613, 2793, 2805, 2841, 2853, 3753, 3765, 3801, 3813, 3993, 4005

Offset: 1

Amiram Eldar, Feb 24 2022

All the terms above 1 are odd multiples of 3.

			3 is a term since its factorial-base presentation, 11, has only odd digits.
21 is a term since its factorial-base presentation, 311, has only odd digits.

Amiram Eldar, Table of n, a(n) for n = 1..18056 (terms below 11!)
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation.

Cf. A007623, A016945, A351893.
Subsequence: A007489
Similar sequences: A003462 \ {0} (ternary), A014261 (decimal), A032911 (base 4), A032912 (base 5), A033032 (base 6), A033033 (base 7), A033034 (base 8), A033035 (base 9), A033036 (base 11), A033037 (base 12), A033038 (base 13), A033039 (base 14), A033040 (base 15), A033041 (base 16), A126646 (binary).

max = 7; fctBaseDigits[n_] := IntegerDigits[n, MixedRadix[Range[max, 2, -1]]]; Select[Range[1, max!, 2], AllTrue[fctBaseDigits[#], OddQ] &]

A007928 Numbers containing an even digit.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 32, 34, 36, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 54, 56, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 76, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92, 94, 96, 98, 100

Offset: 1

R. Muller

Or, numbers whose product of digits is even.

Complement of A014261; A196563(a(n)) > 0. - Reinhard Zumkeller, Oct 04 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
F. Smarandache, Only Problems, Not Solutions!, Xiquan Publ., Phoenix-Chicago, 1993.
Index entries for 10-automatic sequences.

import Data.List (findIndices)
a007928 n = a007928_list !! (n-1)
a007928_list = findIndices (> 0) a196563_list
-- Reinhard Zumkeller, Oct 04 2011

[ n : n in [0..129] | IsEven(&*Intseq(n,10)) ];

is(n)=vecmin(Set(digits(n)%2))==0 \\ Charles R Greathouse IV, Feb 14 2017

a(n) ~ n. - Charles R Greathouse IV, Sep 07 2012

A033033 Numbers all of whose base 7 digits are odd.

Original entry on oeis.org

1, 3, 5, 8, 10, 12, 22, 24, 26, 36, 38, 40, 57, 59, 61, 71, 73, 75, 85, 87, 89, 155, 157, 159, 169, 171, 173, 183, 185, 187, 253, 255, 257, 267, 269, 271, 281, 283, 285, 400, 402, 404, 414, 416, 418, 428, 430, 432, 498, 500, 502, 512

Offset: 1

Clark Kimberling

			38 in base 7 is 53_7. All the digits of 38 in base 7; 5 and 3; are odd. So 38 is in the sequence. - _David A. Corneth_, Aug 24 2019

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A007093, A014261.

[m:m in [1..600]| Intseq(m,7) subset {1,3,5}]; // Marius A. Burtea, Aug 24 2019

Select[Range[600],AllTrue[IntegerDigits[#,7],OddQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 28 2014 *)

is(n) = {my(d = Set(digits(n, 7))); for(i = 1, #d, if(d[i]%2 == 0, return(0))); 1} \\ David A. Corneth, Aug 24 2019

A061808 a(n) is the smallest number with all digits odd that is divisible by 2n-1.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 315, 115, 75, 135, 319, 31, 33, 35, 37, 39, 533, 559, 135, 517, 539, 51, 53, 55, 57, 59, 793, 315, 195, 335, 759, 71, 73, 75, 77, 79, 1377, 913, 595, 957, 979, 91, 93, 95, 97, 99, 1111, 515, 315, 535, 1199, 111, 113, 115, 117, 119, 1331, 1353, 375, 1397, 1935

Offset: 1

Amarnath Murthy, May 28 2001

From Yang Haoran, Dec 02 2017, edited by M. F. Hasler, Mar 05 2025: (Start)
Record value for a(n) = (2n-1) * A296009(n):
   (1, 3, 5, ..., 19) * 1 = (1, 3, 5, ..., 19)
   21 *  15 =    315
   29 *  11 =    319
   41 *  13 =    533
   43 *  13 =    559
   61 *  13 =    793
   81 *  17 =   1377
  127 *  11 =   1397
  129 *  15 =   1935
  149 *  13 =   1937
  167 *  19 =   3173
  169 *  33 =   5577
  201 * 155 =  31155
  299 * 105 =  31395
  401 * 133 =  53333
  601 * 119 =  71519
  633 * 283 = 179139
  (complete up to here)
  ...
  990001 * 12121113 = 11999913991113 (the first A296009(n) > 2n-1).
(End)
All terms must be odd. - M. F. Hasler, Mar 05 2025

Robert Israel, Table of n, a(n) for n = 1..10000
Mathematical Excalibur, Problem 300, Vol. 1 No. 3 (2008), p. 3.

Cf. A061807, A014261.

Equals A296009 * (2n-1).

a:=[]; for n in [1..120 by 2] do k:=1; while not Set(Intseq(n*k)) subset {1,3,5,7,9} do k:=k+2; end while; Append(~a,k*n); end for; a; // Marius A. Burtea, Sep 20 2019

Ad[1]:= [1,3,5,7,9]:
for n from 2 to 9 do Ad[n]:= map(t -> seq(10*t+j,j=[1,3,5,7,9]), Ad[n-1]) od:
Aod:= [seq(op(Ad[i]),i=1..9)]:
f:= proc(n) local k;
   for k from 1 to nops(Aod) do
       if Aod[k] mod (2*n-1) = 0 then return(Aod[k]) fi
     od;
     NotFound
end proc:
map(f, [$1..100]); # Robert Israel, Feb 15 2017

Table[Block[{k = 2 n - 1}, While[Nand[AllTrue[IntegerDigits@ k, OddQ], Divisible[k, 2 n - 1]], k += 2]; k], {n, 59}] (* Michael De Vlieger, Dec 02 2017 *)

isoddd(n) = #select(x->((x%2) == 0), digits(n)) == 0;
a(n) = {my(m = 2*n-1, k = 1); while(!isoddd(k*m), k++); k*m;} \\ Michel Marcus, Sep 20 2019

apply( {A061808(n)=forstep(k=n*2-1,oo,n*4-2,vecmin(digits(k)%2)&& return(k))}, [1..99])

A061808 = lambda n: next(m for m in range(2*n-1,9<<99,4*n-2) if all(int(d)&1 for d in str(m))) # M. F. Hasler, Mar 05 2025

From  M. F. Hasler, Mar 05 2025: (Start)
a(n) = (2n-1)*A296009(n).
a(n) == 1 (mod 2) for all n. (End)

Corrected and extended by Larry Reeves (larryr(AT)acm.org), May 30 2001

A325114 Integers k such that no nonzero subsequence of the decimal representation of k is divisible by 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 18, 19, 20, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33, 34, 36, 38, 39, 40, 41, 43, 44, 45, 46, 48, 50, 51, 52, 53, 54, 55, 58, 59, 60, 61, 62, 64, 65, 66, 68, 69, 80, 81, 82, 83, 85, 86, 88, 89, 90, 92, 93, 94, 95, 96, 99, 100, 101, 102, 103, 104, 106, 108, 109, 110, 111, 113, 115, 116, 118, 120

Offset: 1

Jonathan Kal-El Peréz, Mar 27 2019

Does not contain 114 (helps to distinguish this from related sequences).
From David A. Corneth, Sep 10 2024: (Start)
Any term greater than 10^6 must have a digit 0. Proof: Any term between 10^6 and 10^7 has a 0.
Proof via induction and contradiction; any 7 digital number term has a digit 0. Suppose some number with k with q > 7 digits has no digit 0. Then floor(k/10) is a term and has no digit 0 and q - 1 digits. But there is no such number. A contradiction. Therefore any term with at least 7 digits has a digit 0. (End)

David A. Corneth, Table of n, a(n) for n = 1..10000
David A. Corneth, PARI program

Cf. A014261 (for 2), A325112 (for 3), A325113 (for 4), A261189 (for 5).

See A376046 for complement.

With[{k = 7}, Select[Range@ 100, NoneTrue[DeleteCases[FromDigits /@ Rest@ Subsequences[IntegerDigits@ #], 0], Mod[#, k] == 0 &] &]] (* Michael De Vlieger, Mar 31 2019 *)

PARI
```
\\ See Corneth link
```

More than the usual number of terms are shown in order to distinguish this from a new sequence arising from the game of "buzz" (cf. A092433). - N. J. A. Sloane, Sep 09 2024

A349247 Least n-digit number k with only odd digits such that the k-th triangular number also has only odd digits.

Original entry on oeis.org

1, 13, 177, 1777, 15173, 135173, 3397973, 13535137, 135157537, 1193111377, 11979759377, 119595919137, 1195991117973, 11979931335173, 119777591993777, 1199999593111377, 11977793913551137, 119593573333335733, 1195935733333335733, 11977593393931151137, 119759371717733717537

Offset: 1

M. F. Hasler, Nov 23 2021

It appears that all a(n), n > 9, have initial digits "119".

It also appears that the sequence of digits of the terms converges to a limit, (1, 1, 9, 3, 1, ...). Can this be proved or disproved?

S. S. Gupta, Can You Find (CYF) no. 55, Nov 11 2021.
A. Zimmermann, Al Zimmermann's Programming Contests: Oddly Triangular, Sep. 7-8, 2022

Cf. A000217 (triangular numbers), A014261 (numbers with only odd digits), A117960 (triangular numbers with only odd digits), A349243 (indices of the former), A347475 (such indices with only odd digits), A355277 (largest such k-digit term).

apply( A349247(n)=A347475_next(10^n\9), [1..15]) \\ Edited (moved function body to A347475) by M. F. Hasler, Sep 13 2022

from itertools import product
def A349247(n):
    for a in product('13579',repeat=n):
        if set(str((m:=int(''.join(a)))*(m+1)>>1)) <= {'1', '3', '5', '7', '9'}:
            return m # Chai Wah Wu, Sep 08 2022

A349247 = lambda n: next_A347475(10**n//9) # M. F. Hasler, Sep 10 2022

a(n) = min { k in A347475 | k >= 10^(n-1) }.

A350536 a(n) is the smallest proper multiple of 2n+1 which contains only odd digits, or -1 if no such multiple exists.

Original entry on oeis.org

3, 9, 15, 35, 99, 33, 39, 75, 51, 57, 315, 115, 75, 135, 319, 93, 99, 175, 111, 117, 533, 559, 135, 517, 539, 153, 159, 715, 171, 177, 793, 315, 195, 335, 759, 355, 511, 375, 539, 395, 1377, 913, 595, 957, 979, 1911, 1395, 1995, 3395, 9999, 1111, 515, 315, 535, 1199, 333

Offset: 0

Bernard Schott, Jan 04 2022

Generalization of the problem 1/2 of International Mathematical Talent Search, round 2 (see link and 2nd example).

If the escape clause is used, it will be necessarily for terms coming from n = 12 + 25*k, k >= 0.

			a(10) = 315 = 21 * 15 is the smallest multiple of 21 which contains only odd digits.
a(4998) = 33339995 = 9997 * 3335 is the smallest multiple of 9997 which contains only odd digits, so this is the answer to the IMTS problem.

Chai Wah Wu, Table of n, a(n) for n = 0..10000
International Mathematical Talent Search, Problem 1/2, Round 2.
Index to sequences related to Olympiads and other Mathematical Competitions.

Cf. A061810, A061829, A078221, A296009, A350538.

Terms belong to A014261.

a[n_] := Module[{m = 2*n + 1, k}, k = 3*m; While[!AllTrue[IntegerDigits[k], OddQ], k += 2*m]; k]; Array[a, 50, 0] (* Amiram Eldar, Jan 04 2022 *)

isok(k) = my(d=digits(k)); #d == #select(x->((x%2)==1), d);
a(n) = my(k=6*n+3); while (!isok(k), k+=4*n+2); k; \\ Michel Marcus, Jan 04 2022

from itertools import product, count
def A350536(n):
    m = 2*n+1
    for l in count(len(str(m))):
        for s in product('13579',repeat=l):
            k = int(''.join(s))
            if k > m and k % m == 0:
                return k # Chai Wah Wu, Jan 11 2022

More terms from Michel Marcus, Jan 04 2022

A355277 Largest n-digit number k with only odd digits such that the k-th triangular number also has only odd digits.

Original entry on oeis.org

5, 17, 177, 5573, 79137, 791377, 7913777, 79971937, 557335733, 5995957537, 59995599137, 599591791137, 7991739957973, 79971739957537, 799739357539937, 7991713197753777, 79991971791119137, 799999173991317537, 7997391313911797973

Offset: 1

M. F. Hasler, Sep 07 2022

It appears that all a(n), n > 12, have initial digits "799".
The first digit of a(n) is never 9. - Chai Wah Wu, Sep 08 2022
As in A347475, all terms with more than 2 digits end in 33, 37, 73 or 77. - M. F. Hasler, Sep 12 2022

			T(5) = A000217(5) = 5*6/2 = 5*3 = 15 has only odd digits, and neither T(7) nor T(9) have this property, therefore a(1) = 5.

M. F. Hasler, Table of n, a(n) for n = 1..24, Sep 08 2022
S. S. Gupta, Can You Find (CYF) no. 55, Nov 11 2021.

Cf. A000217 (triangular numbers), A014261 (numbers with only odd digits), A117960 (triangular numbers with only odd digits), A349243 (indices of the former), A347475 (such indices with only odd digits), A349247 (least k-digit term).

apply( A355277(n)=A347475_prec(10^n), [1..15]) \\ M. F. Hasler, Sep 08 2022

from itertools import product
def A355277(n):
    for a in '7531':
        for b in product('97531',repeat=n-1):
            m = int(a+''.join(b))
            if set(str(m*(m+1)>>1)) <= {'1', '3', '5', '7', '9'}:
                return m # Chai Wah Wu, Sep 08 2022

a(n) = max { k in A347475 | k < 10^n }.

A066640 Numbers such that all divisors have only odd digits.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 31, 33, 35, 37, 39, 51, 53, 55, 57, 59, 71, 73, 77, 79, 91, 93, 95, 97, 99, 111, 113, 117, 119, 131, 133, 137, 139, 151, 153, 155, 157, 159, 171, 173, 177, 179, 191, 193, 197, 199

Offset: 1

Amarnath Murthy, Dec 28 2001

Is this sequence infinite? - Charles R Greathouse IV, Sep 07 2012

			77 = 11 * 7 belongs to this sequence but 75 does not as 25 (with a 2) divides 75.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A014261. A030096 is a subsequence.

Select[Range[250], And@@OddQ/@Flatten[IntegerDigits/@Divisors[ # ]]&]

f(n)=vecmin(Vec(Vecsmall(Str(n)))%2)
is(n)=fordiv(n,d,if(!f(d),return(0)));1 \\ Charles R Greathouse IV, Sep 07 2012

from itertools import islice, count
from sympy import divisors
def A066640(): return filter(lambda n: all(set(str(m)) <= {'1','3','5','7','9'} for m in divisors(n,generator=True)), count(1,2))
A066640_list = list(islice(A066640(),20)) # Chai Wah Wu, Nov 22 2021

Corrected and extended by Harvey P. Dale, Jan 01 2002

A076704 Numbers k of the form p^q where both p and q are prime and all digits of k are odd.

Original entry on oeis.org

9, 1331, 357911, 5177717, 5735339, 9393931, 17171515157399, 335571975137771, 7979737131773191, 13337513771953951, 13137917533317175739371379, 33159599371999557199755557, 1593395573971551557179777111133, 131755773357537951113179771515713, 315113377779977515359339551539771

Offset: 1

Zak Seidov, Oct 26 2002

Up to 10^17, there are only 10 odd-digit prime powers of prime numbers. a(1) = 3^2, a(2) = 11^3, a(3) = 71^3, a(4) = 173^3, a(5) = 179^3, a(6) = 211^3, a(7) = 25799^3, a(8) = 69491^3, a(9) = 199831^3, and a(10) = 237151^3.
The only candidates for even-digit prime powers of prime numbers are of the form 2^n, and below 2^10000 there are only 2, 4, 8, 64, and 2048, two of which are not raised to prime powers.
a(11) <= 13137917533317175739371379 and a(12) <= 33159599371999557199755557. - Jinyuan Wang, Mar 02 2020

Giovanni Resta, Table of n, a(n) for n = 1..36 (terms < 10^57)

Cf. A014261, A053810, A075308.

pp = Sort[ Flatten[ Table[ Prime[n]^Prime[i], {n, 1, PrimePi[ Sqrt[10^17]]}, {i, 1, PrimePi[ Floor[ Log[ Prime[n], 10^17]]]}]]]; Do[ If[ Union[ OddQ[ IntegerDigits[ pp[[n]]]]] == {True}, Print[ pp[[n]]]], {n, 1, Length[pp]}]

lista(nn) = {my(k, v=List([])); forprime(p=2, nn, forprime(q=2, logint(nn, p), if(Set(digits(k=p^q)%2)==[1], listput(v, k)))); Set(v); } \\ Jinyuan Wang, Mar 02 2020

Edited and extended by Robert G. Wilson v, Oct 31 2002
Corrected and edited by Elliott Line, Jul 11 2013
Better definition from Jon E. Schoenfield, Nov 19 2018
Terms a(11) and beyond from Giovanni Resta, Mar 03 2020

cp's OEIS Frontend

A351894 Numbers that contain only odd digits in their factorial-base representation.

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A007928 Numbers containing an even digit.

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A033033 Numbers all of whose base 7 digits are odd.

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A061808 a(n) is the smallest number with all digits odd that is divisible by 2n-1.

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A325114 Integers k such that no nonzero subsequence of the decimal representation of k is divisible by 7.

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A349247 Least n-digit number k with only odd digits such that the k-th triangular number also has only odd digits.

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PARI

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A350536 a(n) is the smallest proper multiple of 2n+1 which contains only odd digits, or -1 if no such multiple exists.

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