A061601 -id:A061601 - cp's OEIS frontend

A097326 Largest integer m such that m*n has the same decimal digit length as n.

9, 4, 3, 2, 1, 1, 1, 1, 1, 9, 9, 8, 7, 7, 6, 6, 5, 5, 5, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 9, 9, 9, 9, 9

Offset: 1

Rick L. Shepherd, Aug 04 2004

For any positive base B >= 2 the corresponding sequence contains only terms from 1 to B-1 inclusive so the corresponding sequence for binary is all 1's (A000012).

			a(12)=8 as 12 and 8*12=96 both have two decimal digits while 9*12=108 has three.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A061601 (analog for decimal m+n), A035327 (analog for binary m+n), A097327.

Cf. A055642.

limn[n_]:=Module[{k=9,len=IntegerLength[n]},While[IntegerLength[k*n] > len, k--];k]; Array[limn,110] (* Harvey P. Dale, Apr 28 2018 *)
Table[Ceiling[10^IntegerLength[n]/n] - 1, {n, 100}] (* Paolo Xausa, Nov 06 2024 *)

a(n) = my(m=1, sn=#Str(n)); while (#Str(m*n) <= sn, m++); m-1; \\ Michel Marcus, Oct 05 2021

def a(n): return (10**len(str(n))-1)//n
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Oct 05 2021

a(n) = A097327(n) - 1.

a(n) = floor((10^A055642(n) - 1)/n). - Michael S. Branicky, Oct 05 2021

A192817 Numbers that are coprime to their 9's complement.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 13, 14, 16, 17, 19, 20, 23, 25, 26, 28, 29, 31, 32, 34, 35, 37, 38, 40, 41, 43, 46, 47, 49, 50, 52, 53, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 74, 76, 79, 80, 82, 83, 85, 86, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 104, 106

Offset: 1

Alonso del Arte, Dec 01 2011

If an integer is in this sequence, its 9's complement is in the sequence as well. No multiple of 3 is in this sequence. Multiples of 11 are in the sequence if they have an odd number of digits and they are not also multiples of 3.

			25 is in the sequence because its 9's complement is 74 and gcd(25, 74) = 1.

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

Cf. A061601 (9's complement of n), A201462 (complement).

a192817 n = a192817_list !! (n-1)
a192817_list = [x | x <- [1..], gcd x (a061601 x) == 1]
-- Reinhard Zumkeller, Dec 03 2011

[n: n in [1..106] | Gcd(10^#Intseq(n)-1,n) eq 1]; // Bruno Berselli, Dec 02 2011

with(numtheory): P:=proc(q) local k,n; for n from 1 to q do for k from 1 to q do
if type(((n-k)*10^(ilog10(n+k)+1)+n+k)/n,integer) then break; fi; od;
if k=n then print(n); fi; od; end: P(10^4); # Paolo P. Lava, Nov 03 2014

(* First run the program for A061601 to define nineComplement *) Select[Range[100], GCD[#, nineComplement[#]] == 1 &]

A228628 9's complement of prime(n).

Original entry on oeis.org

7, 6, 4, 2, 88, 86, 82, 80, 76, 70, 68, 62, 58, 56, 52, 46, 40, 38, 32, 28, 26, 20, 16, 10, 2, 898, 896, 892, 890, 886, 872, 868, 862, 860, 850, 848, 842, 836, 832, 826, 820, 818, 808, 806, 802, 800, 788, 776, 772, 770, 766, 760, 758, 748, 742, 736, 730, 728

Offset: 1

Michel Lagneau, Aug 28 2013

a(n) = 10^k - 1 - prime(n) where k is the number of digits in prime(n). If d is d digit in prime(n) replace it with 9 - d.

			a(6) = 86 because prime(6) = 13 and 9 - 1 = 8, 9 - 6 = 3.

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

Cf. A000040, A061601, A160668.

a:= n-> (p-> 10^length(p)-p-1)(ithprime(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Dec 08 2017

nineComplement[n_] := FromDigits[Table[9, {Length[IntegerDigits[Prime[n]]]}] - IntegerDigits[Prime[n]]]; Table[nineComplement[n], {n, 1, 71}]

a(n)=my(p=prime(n));10^#Str(p)-p-1 \\ Charles R Greathouse IV, Aug 29 2013

from sympy import primerange
def nc(n): return 10**len(str(n)) - 1 - n
def auptop(limit): return [nc(p) for p in primerange(1, limit+1)]
print(auptop(271)) # Michael S. Branicky, Jul 06 2021

a(n) = A061601(A000040(n)). - Charles R Greathouse IV, Aug 29 2013

a(n) = A160668(n) - 1. Alois P. Heinz, Dec 08 2017

A378139 Smallest prime number such that the number of distinct prime factors with multiplicity of its 9's complement is equal to n. If no such number exists, return -1.

Original entry on oeis.org

2, 3, 23, 11, 19, 263, 167, 103, 487, 1039, 7951, 5903, 28319, 107071, 67231, 590399, 180799, 344639, 1480319, 12181759, 4757119, 10871039, 1611391, 140167679, 203082239, 228248063, 530237951, 1812718591, 5302379519, 13295347711, 12758476799, 132953477119, 1410065407

Offset: 1

Jean-Marc Rebert, Jan 08 2025

			2 is prime, 9-2 = 7 and bigomega(7) = 1.

Michael S. Branicky, Table of n, a(n) for n = 1..1000 (terms 1..37 from Jean-Marc Rebert)

Cf. A000040, A001221, A001222, A061601. A377471,

nc(n) = my(e=length(Str(n))); 10^e-1 - n; \\ A061601
a(n) = my(p=2); while (bigomega(nc(p)) != n, p = nextprime(p+1)); p; \\ Michel Marcus, Jan 08 2025

A084019 a(n) = 9's complement of n-th palindrome (A002113).

Original entry on oeis.org

9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 88, 77, 66, 55, 44, 33, 22, 11, 0, 898, 888, 878, 868, 858, 848, 838, 828, 818, 808, 797, 787, 777, 767, 757, 747, 737, 727, 717, 707, 696, 686, 676, 666, 656, 646, 636, 626, 616

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 23 2003

Leading zeros in the 9's complement are omitted.

For palindromes of the form 10^k-1 the corresponding entry is zero. Apart from this the entries are distinct.

			The 20th palindromic number A002113(20) is 101 having 9's complement 898 (999 - 101 = 898). So a(20) = 898. - _Indranil Ghosh_, Jan 30 2017

Indranil Ghosh, Table of n, a(n) for n = 1..25000

Cf. A002113, A061601, A084020.

# Program for generating the b-file
def a(n):
    return 10**len(str(n))-n-1
i=0
j=1
while j<=250:
    if i==int(str(i)[::-1]):
        print(str(j)+" "+str(a(i)))
        j+=1
    i+=1 # Indranil Ghosh, Jan 30 2017

def A084019(n):
    if n == 1: return 9
    y = 10*(x:=10**(len(str(n>>1))-1))
    if nChai Wah Wu, Jun 13 2024

A109910 a(n) = 9's complement of digit reversal of n.

Original entry on oeis.org

9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 8, 88, 78, 68, 58, 48, 38, 28, 18, 8, 7, 87, 77, 67, 57, 47, 37, 27, 17, 7, 6, 86, 76, 66, 56, 46, 36, 26, 16, 6, 5, 85, 75, 65, 55, 45, 35, 25, 15, 5, 4, 84, 74, 64, 54, 44, 34, 24, 14, 4, 3, 83, 73, 63, 53, 43, 33, 23, 13, 3, 2

Offset: 0

Amarnath Murthy, Jul 16 2005

			a(17) = 28. Digit reversal of 17 = 71, 9's complement of 71 is 99-71 = 28.

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A004086, A061601, A109911.

R := proc(n) local nn, nnn: nn:=convert(n, base, 10): add(nn[nops(nn)+1-j]*10^(j-1), j=1..nops(nn)): end:
A109910 := proc(n) return 10^length(max(R(n),1)) - R(n) - 1: end:
seq(A109910(n),n=0..70); # Nathaniel Johnston, Apr 28 2011

Table[10^If[# == 0, 1, IntegerLength@ #] - 1 - # &@ FromDigits@ Reverse@ IntegerDigits@ n, {n, 0, 70}] (* Michael De Vlieger, Feb 01 2017 *)

def A109910(n):
    x=int(str(n)[::-1])
    return 10**len(str(x))-1-x # Indranil Ghosh, Jan 30 2017

a(n) = A061601(A004086(n)). - Indranil Ghosh, Jan 30 2017

A111708 a(n) = n concatenated with 9's complement of n.

Original entry on oeis.org

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 1089, 1188, 1287, 1386, 1485, 1584, 1683, 1782, 1881, 1980, 2079, 2178, 2277, 2376, 2475, 2574, 2673, 2772, 2871, 2970, 3069, 3168, 3267, 3366, 3465, 3564, 3663, 3762, 3861, 3960, 4059, 4158, 4257, 4356, 4455, 4554

Offset: 0

Amarnath Murthy, Aug 22 2005

If n has k digits a(n) is a multiple of 10^k -1.
The sequence contains 10^(k) multiples of 10^k -1.
a(n)/{10^k -1} gives all natural numbers.
All terms are multiples of 9: A007953(a(n))=9*A055642(n); A010888(a(n))=9. - Reinhard Zumkeller, Sep 17 2015

			a(90) = 9009.

Reinhard Zumkeller, Table of n, a(n) for n = 0..9999

Cf. A061601, A010888, A007953, A055642, subsequence of A262277.

a111708 0 = 9
a111708 n = f [] n where
   f ys 0 = foldl (\v d -> 10 * v + d) 0 $ ys ++ map (9 -) ys
   f ys x = f (d : ys) x' where (x', d) = divMod x 10
-- Reinhard Zumkeller, Sep 17 2015

More terms from Joshua Zucker, May 08 2006

Definition cleared and offset changed by Reinhard Zumkeller, Sep 17 2015

A201462 Numbers that are not coprime to their 9's complement.

Original entry on oeis.org

3, 6, 9, 11, 12, 15, 18, 21, 22, 24, 27, 30, 33, 36, 39, 42, 44, 45, 48, 51, 54, 55, 57, 60, 63, 66, 69, 72, 75, 77, 78, 81, 84, 87, 88, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 148, 150, 153, 156, 159

Offset: 1

Alonso del Arte, Dec 01 2011

All multiples of 3 are in this sequence. Multiples of 11 are in this sequence if they have an even number of digits or if they are also multiples of 3.

			22 is in the sequence because its 9's complement is 77 and gcd(22, 77) = 11 > 1.

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

Cf. A061601 (9's complement of n), A192817 (complement).

a201462 n = a201462_list !! (n-1)
a201462_list = [x | x <- [1..], gcd x (a061601 x) > 1]
-- Reinhard Zumkeller, Dec 03 2011

[n: n in [3..159] | Gcd(10^#Intseq(n)-1,n) gt 1]; // Bruno Berselli, Dec 02 2011

(* First run the program for A061601 to define nineComplement *) Select[Range[200], GCD[#, nineComplement[#]] > 1 &]

A262277 Numbers having in decimal representation the same distinct decimal digits as their 9's complement.

Original entry on oeis.org

18, 27, 36, 45, 54, 63, 72, 81, 90, 118, 181, 188, 227, 272, 277, 336, 363, 366, 445, 454, 455, 544, 545, 554, 633, 636, 663, 722, 727, 772, 811, 818, 881, 900, 909, 990, 1089, 1098, 1118, 1181, 1188, 1278, 1287, 1368, 1386, 1458, 1485, 1548, 1584, 1638

Offset: 1

Reinhard Zumkeller, Sep 17 2015

If d is a digit of any term then also 9 - d;

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

Cf. A061601, A227362, subsequences: A111708, A050278, A171102.

import Data.List (nub, sort)
a262277 n = a262277_list !! (n-1)
a262277_list = filter f [1..] where
   f x = sort ds' == sort (map (9 -) ds') where
     ds' = nub $ ds x
     ds 0 = []; ds z = d : ds z' where (z', d) = divMod z 10

isok(m) = my(d=digits(m), c=apply(x->9-x, d)); Set(d) == Set(c); \\ Michel Marcus, Jan 22 2022

A227362(A061601(a(n))) = A227362(a(n)).

A284811 Fixed points of the transform A267193.

Original entry on oeis.org

18, 27, 36, 45, 54, 63, 72, 81, 90, 1098, 1188, 1278, 1368, 1458, 1548, 1638, 1728, 1818, 1908, 2097, 2187, 2277, 2367, 2457, 2547, 2637, 2727, 2817, 2907, 3096, 3186, 3276, 3366, 3456, 3546, 3636, 3726, 3816, 3906, 4095, 4185, 4275, 4365, 4455, 4545, 4635, 4725

Offset: 1

Paolo P. Lava, Apr 05 2017

These numbers are called antipalindromic in base 10 by Dvorakova et al. - Michel Marcus, Aug 18 2020

			1278 is a term of the sequence because its complement in base 10 is 8721 and the digit reversal is again 1278.

Rémy Sigrist, Table of n, a(n) for n = 1..9999
Lubomira Dvorakova, Stanislav Kruml, and David Ryzak, Antipalindromic numbers, arXiv:2008.06864 [math.CO], 2020.

Cf. A035928, A061601, A267193.

Subsequence of A008591 (multiples of 9).

P:=proc(q,h) local a,b,k,n; for n from 1 to q do a:=convert(n,base,h); b:=0;
for k from 1 to nops(a) do a[k]:=h-1-a[k]; b:=h*b+a[k]; od; if b=n then print(n); fi; od; end: P(10^2,10);

isok(m) = {my(d=digits(m)); for (j=1, #d, if (d[j] + d[#d+1-j] != 9, return(0));); return (1);} \\ Michel Marcus, Aug 18 2020

a(n) = my (d=digits(n)); n*10^#d + fromdigits(apply (t -> 9-t, Vecrev(d))) \\ Rémy Sigrist, Aug 18 2020

cp's OEIS Frontend

A097326 Largest integer m such that m*n has the same decimal digit length as n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A192817 Numbers that are coprime to their 9's complement.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

A228628 9's complement of prime(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A378139 Smallest prime number such that the number of distinct prime factors with multiplicity of its 9's complement is equal to n. If no such number exists, return -1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A084019 a(n) = 9's complement of n-th palindrome (A002113).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Python

A109910 a(n) = 9's complement of digit reversal of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A111708 a(n) = n concatenated with 9's complement of n.

Views

Author

Keywords

Comments

Examples

Links