A007095 -id:A007095 - cp's OEIS frontend

A043283 Maximal run length in base-9 representation of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Clark Kimberling

Antti Karttunen, Table of n, a(n) for n = 1..66430

Cf. A007095.
Cf. A043276-A043290 for base-2 to base-16 analogs.
Cf. A002452 (gives the positions of records, the first occurrence of each n).
Cf. also A044940.

A043283[n_]:=Max[Map[Length,Split[IntegerDigits[n,9]]]];Array[A043283,100] (* Paolo Xausa, Sep 27 2023 *)

A043283(n, b=9)={my(m, c=1); while(n>0, n%b==(n\=b)%b && c++ && next; m=max(m, c); c=1); m} \\ M. F. Hasler, Jul 23 2013

A073796 Replace 9^k with (-9)^k in base 9 expansion of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, -9, -8, -7, -6, -5, -4, -3, -2, -1, -18, -17, -16, -15, -14, -13, -12, -11, -10, -27, -26, -25, -24, -23, -22, -21, -20, -19, -36, -35, -34, -33, -32, -31, -30, -29, -28, -45, -44, -43, -42, -41, -40, -39, -38, -37, -54, -53, -52, -51, -50, -49, -48, -47, -46

Offset: 0

Robert G. Wilson v, Aug 12 2002

Base 9 representation for n (in lexicographic order) converted from base -9 to base 10.

Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020

Cf. A007095, A073790, A053985, A065369, A073791, A073792, A073793, A073794, A073795 & A073835.

f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 9]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 9]], {n, 1, 80}]; b

a(9*k+m) = -9*a(k)+m for 0 <= m < 9. - Chai Wah Wu, Jan 16 2020

A037385 Numbers k such that every base-3 digit of k is a base-9 digit of k.

Original entry on oeis.org

1, 2, 9, 13, 18, 26, 81, 82, 83, 84, 85, 90, 99, 108, 117, 121, 162, 163, 164, 168, 170, 171, 180, 216, 234, 242, 244, 252, 325, 333, 488, 504, 650, 666, 729, 730, 731, 732, 733, 738, 739, 740, 741, 742, 747, 748, 749, 750, 751

Offset: 1

Clark Kimberling

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007089, A007095.

Cf. A037380, A037381, A037382, A037383, A037384, A037386.

import Data.List ((\\), nub)
a037385 n = a037385_list !! (n-1)
a037385_list = filter f [1..] where
   f x = null $ nub (ds 3 x) \\ nub (ds 9 x)
   ds b x = if x > 0 then d : ds b x' else []  where (x', d) = divMod x b
-- Reinhard Zumkeller, May 30 2013

Select[Range[1000],SubsetQ[IntegerDigits[#,9],IntegerDigits[#,3]]&] (* Harvey P. Dale, Dec 19 2015 *)

A037396 Numbers k such that every base-5 digit of k is a base-9 digit of k.

Original entry on oeis.org

1, 2, 3, 4, 11, 22, 93, 121, 124, 126, 156, 181, 199, 317, 362, 598, 750, 751, 752, 755, 756, 758, 768, 770, 771, 775, 776, 780, 781, 785, 796, 812, 831, 841, 843, 849, 859, 895, 900, 906, 907, 911, 912, 918, 922, 927, 931, 932

Offset: 1

Clark Kimberling

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007091, A007095.

Cf. A037393, A037394, A037395, A037397.

import Data.List ((\\), nub)
a037396 n = a037396_list !! (n-1)
a037396_list = filter f [1..] where
   f x = null $ nub (ds 5 x) \\ nub (ds 9 x)
   ds b x = if x > 0 then d : ds b x' else []  where (x', d) = divMod x b
-- Reinhard Zumkeller, May 30 2013

is(n)=#setminus(Set(digits(n,5)), Set(digits(n,9)))==0 \\ Charles R Greathouse IV, Feb 11 2017

A110607 Numbers n whose base 9 representations, interpreted as base 10 integers, are semiprimes.

Original entry on oeis.org

4, 6, 9, 13, 14, 19, 20, 23, 24, 30, 31, 32, 35, 42, 46, 50, 52, 53, 56, 59, 67, 70, 74, 77, 78, 79, 87, 91, 95, 98, 100, 101, 102, 111, 112, 118, 119, 120, 122, 123, 131, 134, 136, 141, 151, 152, 156, 158, 160, 163, 164, 165, 167, 168, 174, 175, 176, 178, 179, 181

Offset: 1

Jonathan Vos Post, Jul 30 2005

A108873 is the equivalent using base 3. A110602 is the equivalent using base 4. A110603 is the equivalent using base 5. A110604 is the equivalent using base 6. A110605 is the equivalent using base 7. A110606 is the equivalent using base 8.

			a(1) = 4 because 4 (base 9) = 4 (base 10) = 2 * 2, a semiprime (A001358).
a(2) = 6 because 6 (base 9) = 6 (base 10) = 2 * 3.
a(3) = 9 because 9 (base 9) = 10 and 10 (base 10) = 2 * 5.
a(4) = 13 because 13 (base 9) = 14 and 14 (base 10) = 2 * 7.
a(5) = 14 because 14 (base 9) = 15 and 15 (base 10) = 3 * 5.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001358, A007095, A108873, A110602, A110603, A110604, A110605, A110607.

Select[Range[181], Plus @@ Last /@ FactorInteger[FromDigits[IntegerDigits[ #, 9]]] == 2 &] (* Ray Chandler, Aug 05 2005 *)
Select[Range[200],PrimeOmega[FromDigits[IntegerDigits[#,9]]]==2&] (* Harvey P. Dale, Dec 02 2018 *)

Extended by Ray Chandler, Aug 05 2005

A262154 Pseudoprimes to base 9, written in base 9.

Original entry on oeis.org

4, 8, 31, 57, 111, 144, 247, 347, 444, 627, 651, 754, 825, 854, 861, 1261, 1264, 1371, 1457, 1681, 1811, 2102, 2331, 2531, 3338, 3378, 3581, 3631, 3757, 3774, 4011, 4161, 4445, 4551, 5127, 6002, 6321, 6722, 7311, 7547, 8651, 10044, 10101, 10637, 11111, 11762, 12464, 12831, 12885, 13141, 13201, 15461, 16084, 16451

Offset: 1

Abdul Gaffar Khan, Sep 13 2015

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

Cf. A007095 (numbers in base 9), A020138 (pseudoprimes to base 9).

Cf. A258189, A262101, A262102, A262103, A262104, A262105.

base = 9; t = {}; n = 1;
While[Length[t] < 80, n++;
If[! PrimeQ[n] && PowerMod[base, n - 1, n] == 1,
  AppendTo[t, FromDigits@IntegerDigits[n, 9]]]]; t

lista(nn, b=9) = {for (n=1, nn, if (Mod(b, n)^(n-1)==1 && !ispseudoprime(n) && n>1, print1(subst(Pol(digits(n,b), x), x, 10), ", ");););} \\ Michel Marcus, Sep 30 2015

a(n) = A007095(A020138(n)).

A293662 Base-9 circular primes that are not base-9 repunits.

Original entry on oeis.org

11, 13, 17, 19, 23, 37, 43, 47, 67, 71, 73, 79, 101, 149, 173, 181, 211, 233, 347, 421, 443, 613, 641, 647, 673, 719, 727, 971, 1123, 1361, 1429, 1609, 1697, 2153, 2179, 3371, 3547, 3833, 4019, 4091, 4099, 4229, 5227, 5261, 5281, 5683, 5689, 5741, 5749, 5821

Offset: 1

Felix Fröhlich, Dec 30 2017

Conjecture: The sequence is finite.

			101 written in base 9 is 122. The base-9 numbers 122, 221, 212 written in base 10 are 101, 181, 173, respectively and all those numbers are prime, so 101, 173 and 181 are terms of the sequence.

Cf. A007095, A293142.

Cf. base-b nonrepunit circular primes: A293657 (b=4), A293658 (b=5), A293659 (b=6), A293660 (b=7), A293661 (b=8), A293663 (b=10).

With[{b = 9}, Select[Prime@ Range[PrimePi@ b + 1, 10^3], Function[w, And[AllTrue[Array[FromDigits[RotateRight[w, #], b] &, Length@ w - 1], PrimeQ], Union@ w != {1} ]]@ IntegerDigits[#, b] &]] (* Michael De Vlieger, Dec 30 2017 *)

rot(n) = if(#Str(n)==1, v=vector(1), v=vector(#n-1)); for(i=2, #n, v[i-1]=n[i]); u=vector(#n); for(i=1, #n, u[i]=n[i]); v=concat(v, u[1]); v
decimal(v, base) = my(w=[]); for(k=0, #v-1, w=concat(w, v[#v-k]*base^k)); sum(i=1, #w, w[i])
is_circularprime(p, base) = my(db=digits(p, base), r=rot(db), i=0); if(vecmin(db)==0, return(0), while(1, dec=decimal(r, base); if(!ispseudoprime(dec), return(0)); r=rot(r); if(r==db, return(1))))
forprime(p=1, , if(vecmin(digits(p, 9))!=vecmax(digits(p, 9)), if(is_circularprime(p, 9), print1(p, ", "))))

A337536 Numbers k for which there are only 2 bases b (2 and k+1) where the digits of k contain the digit b-1.

Original entry on oeis.org

2, 3, 4, 10, 36, 40, 82, 256

Offset: 1

Michel Marcus, Aug 31 2020

These could be called "nine-free numbers".
From David A. Corneth, Aug 31 2020: (Start)
This sequence has density 0. Conjecture: this sequence is finite and full. a(9) > 10^100 if it exists.
Suppose we want to see if 22792 = 1011021011_3 is a term. Since it has a digit of 2 in base 3, we can see that it is not. The next number that does not have the digit 2 in base 3 is 1011100000_3 = 22842, so we can proceed from there. In a similar way we can skip numbers based on bases b > 3. (End)
All terms of this sequence increased by 1 (except a(2)=3) are prime. - François Marques, Aug 31 2020
From Devansh Singh, Sep 19 2020: (Start)
If n is one less than an odd prime and we are interested in bases 3 <= b <= n-1 such that n in base b contains the digit b-1, then divisor of b (except 1) -1 cannot be the last digit since divisor of b divides n+1, which is not possible as n+1 is an odd prime.
If the last digit is 1, then b is odd as 1 = 2-1 and 2 cannot divide b as n+1 is an odd prime.
If the last digit is 0, then b-1 is the last digit of n-1 in base b.
b <= n/2 for even n,b <= (n+1)/2 for odd n.
This sequence is equivalent to the existence of only one prime generating polynomial = F(x) (having positive integer coefficients >=0 and <=b-1 for F(b)) such that F(2) = p.
There is no other prime generating polynomial = G(x) (having positive integer coefficients >=0 and <= b-1 for G(b)) that generates p for 2 < x = b <= (p-1)/2.
(End)

			2 is a term because 2 = 10_2 = 2_3, so both have the digit b-1, and there are no other bases where this happens.
4 is a term because 4 = 100_2 = 4_5, so both have the digit b-1, and there are no other bases where this happens.

David A. Corneth, PARI program

Cf. A005836, A007095, A023717, A020654, A020657, A037465, A037474, A037477, A337496, A337535.

Subsequence of A005836 U {2} and of A068499.

isok(n, b) = vecmax(digits(n, b)) == b-1;
b(n) = if (n==1, return (1)); my(b=3); while(!isok(n, b), b++); b; \\ A337535
is(n) = b(n) == n+1;

\\ See Corneth link \\ David A. Corneth, Aug 31 2020

A353117 Base-9 representation of A000422(n).

Original entry on oeis.org

1, 23, 386, 5831, 82456, 1206503, 15355671, 202838110, 2484401020, 31322180131, 3835578242783, 525732558043736, 70648887825106821, 10365384555277543376, 1340704717754261643863, 173544168713353406577421

Offset: 1

Seiichi Manyama, Apr 24 2022

Cf. A000422, A353110, A353111, A353112, A353113, A353114, A353115, A353116.

Cf. A007095, A353107.

def A(k, n)
  (1..n).map{|i| (1..i).to_a.reverse.join.to_i.to_s(k).to_i}
end
p A(9, 20)

a(n) = A007095(A000422(n)).

A004053 For m=2,3,..., write m in bases 2,3,..,m.

Original entry on oeis.org

10, 11, 10, 100, 11, 10, 101, 12, 11, 10, 110, 20, 12, 11, 10, 111, 21, 13, 12, 11, 10, 1000, 22, 20, 13, 12, 11, 10, 1001, 100, 21, 14, 13, 12, 11, 10, 1010, 101, 22, 20, 14, 13, 12, 11, 10, 1011, 102, 23, 21, 15, 14, 13, 12, 11, 10, 1100, 110, 30, 22, 20, 15, 14, 13, 12, 11, 10

Offset: 2

Johan Boye (johbo(AT)ida.liu.se)

			Triangle begins:
    10;
    11,  10;
   100,  11, 10;
   101,  12, 11, 10;
   110,  20, 12, 11, 10;
   111,  21, 13, 12, 11, 10;
  1000,  22, 20, 13, 12, 11, 10;
  1001, 100, 21, 14, 13, 12, 11, 10;
  ...

Alois P. Heinz, Rows n = 2..20, flattened (row 21 would require digits > 9)

Cf. A007088, A007090, A007091, A007092, A007093, A007094, A007095.

FromDigits/@Flatten[Table[IntegerDigits[m,b],{m,2,20},{b,2,m}],1] (* Harvey P. Dale, Dec 01 2024 *)

T(n, k) = fromdigits(digits(n, k), 10);
tabl(nn) = for (n=2, nn, for (b=2, n, print1(T(n, b), ", "))); \\ Michel Marcus, Aug 30 2019

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A043283 Maximal run length in base-9 representation of n.

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A073796 Replace 9^k with (-9)^k in base 9 expansion of n.

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A037385 Numbers k such that every base-3 digit of k is a base-9 digit of k.

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A037396 Numbers k such that every base-5 digit of k is a base-9 digit of k.

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Haskell

PARI

A110607 Numbers n whose base 9 representations, interpreted as base 10 integers, are semiprimes.

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Extensions

A262154 Pseudoprimes to base 9, written in base 9.

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A293662 Base-9 circular primes that are not base-9 repunits.

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A337536 Numbers k for which there are only 2 bases b (2 and k+1) where the digits of k contain the digit b-1.

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