A000042 -id:A000042 - cp's OEIS frontend

A096044 Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^10-M)/9, where M is the infinite lower Pascal's triangle matrix, 1<=k<=n.

1, 11, 2, 111, 33, 3, 1111, 444, 66, 4, 11111, 5555, 1110, 110, 5, 111111, 66666, 16665, 2220, 165, 6, 1111111, 777777, 233331, 38885, 3885, 231, 7, 11111111, 8888888, 3111108, 622216, 77770, 6216, 308, 8, 111111111, 99999999, 39999996, 9333324, 1399986, 139986, 9324, 396, 9

Offset: 1

Gary W. Adamson, Jun 17 2004

			Triangle T(n,k) begins:
       1;
      11,     2;
     111,    33,     3;
    1111,   444,    66,    4;
   11111,  5555,  1110,  110,   5;
  111111, 66666, 16665, 2220, 165, 6;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A007318. First column gives A000042. Row sums give A016135.

Cf. A096034, A096035, A096039, A096040, A096041, A096042, A096043.

P:= proc(n) option remember; local M; M:= Matrix(n, (i, j)-> binomial(i-1, j-1)); (M^10-M)/9 end: T:= (n, k)-> P(n+1)[n+1, k]: seq(seq(T(n, k), k=1..n), n=1..11);  # Alois P. Heinz, Oct 07 2009

P[n_] := P[n] = With[{M = Array[Binomial[#1-1, #2-1]&, {n, n}]}, (MatrixPower[M, 10] - M)/9]; T[n_, k_] := P[n+1][[n+1, k]]; Table[ Table[T[n, k], {k, 1, n}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

Edited and more terms from Alois P. Heinz, Oct 07 2009

A100563 Number of bases less than sqrt(n) in which n is a palindrome.

Original entry on oeis.org

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

Offset: 1

Gordon Hamilton, Nov 29 2004

Is there a number m such that a(n) > 0 for all n > m? I call the set of numbers for which a(n)=0 "unkempt" for refusing to use a mirror in any base. Is there an infinite number of unkempt numbers? a(n) can be arbitrarily large.
The sequence A123586 gives the values of n where a(n)=0. - Robert G. Wilson v, Nov 01 2014
Is there a closed-form formula for this function? - Robert G. Wilson v, Nov 01 2014
From Robert G. Wilson v, Nov 26 2014: (Start)
The first occurrence, beginning at 0, of n is: 1, 5, 17, 65, 121, 562, 1432, 1477, 4369, 36582, 35101, 86677, 83161, 360361, 291721, 720721, 887041, 1496881, 1670761, 3931201, 3341521, 5654881, 7207201, 7761601,...
Positions where a(n)=k:
k = 0:  A123586;
k = 1:  5, 7, 9, 10, 13, 15, 16, 20, 23, 25, 27, 28, 29, 33, 34, 36, 37, 38, 40, ...;
k = 2:  17, 21, 26, 31, 46, 51, 52, 55, 57, 63, 67, 73, 78, 80, 82, 91, 92, 93, 98, ...;
k = 3:  65, 85, 100, 130, 154, 164, 170, 178, 191, 195, 203, 209, 242, 282, 292, ...;
k = 4:  121, 235, 255, 257, 273, 300, 325, 341, 343, 373, 400, 495, 601, 610, 626, 666, ...;
k = 5:  562, 676, 771, 819, 1009, 1111, 1220, 1333, 1365, 1441, 1543, 1978, 1981, 2000, ...;
k = 6:  1432, 2380, 2666, 2925, 3280, 4035, 4095, 4161, 4225, 4401, 4525, 4561, 4681, ...;
k = 7:  1477, 4097, 4591, 7141, 7993, 8191, 9640, 10081, 10297, 10626, 10858, 11761, ...; etc.
(End)

			100 is a palindrome in bases 3, 7 and 9, so a(100) = 3.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
Wikipedia, Base 1

Cf. A060873, A060874, A060875, A060876, A060877, A060878, A060879, A060947, A060948, A060949, A123586.

Cf. A000042.

f[n_] := Module[{p}, Table[ p = IntegerDigits[n, b]; If[p == Reverse@ p, {b, p}, Sequence @@ {}], {b, 2, Sqrt@ n}]]; Array[ Length@ f@# &, 105] (* Robert G. Wilson v, Nov 01 2014 *)

a(n) = {my(nb = 0); for (b=2, sqrt(n), d = digits(n, b); nb+= (Vecrev(d) == d);); nb;} \\ Michel Marcus, Nov 05 2014

a(n) = A135551(n) - A033831(n). - Robert G. Wilson v, Nov 01 2014

a(58) from Robert G. Wilson v, Nov 05 2014

A131664 A string of n 1's repeated n times.

Original entry on oeis.org

1, 11, 11, 111, 111, 111, 1111, 1111, 1111, 1111, 11111, 11111, 11111, 11111, 11111, 111111, 111111, 111111, 111111, 111111, 111111, 1111111, 1111111, 1111111, 1111111, 1111111, 1111111, 1111111, 11111111, 11111111, 11111111, 11111111

Offset: 1

Paul Curtz, Oct 03 2007

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000042, A002024.

seq(((10^n-1)/9)$n, n=1..10); # Robert Israel, Jul 08 2019

from math import isqrt
def A131664(n): return (10**(isqrt(n<<3)+1>>1)-1)//9 # Chai Wah Wu, Oct 17 2022

G.f.: z/(1-z) * Sum_{n>=0} 10^n*z^(n*(n+1)/2).

A137318 Concatenation of segments of the digit sequence 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3....

Original entry on oeis.org

1, 31, 313, 1313, 13131, 313131, 3131313, 13131313, 131313131, 3131313131, 31313131313, 131313131313, 1313131313131, 31313131313131, 313131313131313, 1313131313131313, 13131313131313131, 313131313131313131

Offset: 1

Ctibor O. Zizka, Apr 06 2008

A000042 is 1,11,111,1111,11111,... concatenation of 111111111111111....
A002276 is 2,22,222,2222,22222,... concatenation of 222222222222222....
A133013 is 2,35,71113,... concatenation of 2 3 5 7 11 13 17 19 23 29,...

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

Cf. A000040, A000042.

Module[{nn=20},FromDigits/@TakeList[PadRight[{},(nn(nn+1))/2,{1,3}],Range[ nn]]] (* Harvey P. Dale, Aug 04 2021 *)

O.g.f.: x*(100x^4 + 200x^3 + 83x^2 + 20x + 1)/((10x-1)(100x^2+1)(x-1)(x^2+1)). - R. J. Mathar, Apr 09 2008

More terms from R. J. Mathar, Apr 09 2008

A246757 Largest n-digit number divisible by the product of its decimal digits.

Original entry on oeis.org

9, 36, 816, 9612, 93744, 973728, 9939915, 99221112, 997711344, 9993393711, 99934212672, 999641938176, 9999121936392, 99996414731136, 999994123418112, 9999982411646976, 99999318116613312, 999991631331122112, 9999944111773994112, 99999911232931433472, 999999832211912282112

Offset: 1

Max Alekseyev, Sep 02 2014

The smallest such numbers are given by repunits A000042 or A002275.

Max Alekseyev, Table of n, a(n) for n = 1..30

Subsequence of A007602.

{ A246757(n) = my(m,d,p,q); m=n\2; forstep(k=10^m-1,(10^m-1)/9,-1, d=digits(k); q=prod(i=1,#d,d[i]); if(q==0,next); forstep(s=(((k+1)*10^(n-m))\q)*q,k*10^(n-m),-q,  d=digits(s); p=prod(i=1,#d,d[i]); if(p==0 || s%p,next); return(s) )) }

from operator import mul
from functools import reduce
def A246757(n):
    for i in range(10**n-1,int('1'*n)-1,-1):
        pd = reduce(mul,(int(d) for d in str(i)))
        if pd and not i % pd:
            return i # Chai Wah Wu, Sep 08 2014

A360574 Binary expansions of odd numbers with three zeros in their binary expansion.

Original entry on oeis.org

10001, 100011, 100101, 101001, 110001, 1000111, 1001011, 1001101, 1010011, 1010101, 1011001, 1100011, 1100101, 1101001, 1110001, 10001111, 10010111, 10011011, 10011101, 10100111, 10101011, 10101101, 10110011, 10110101, 10111001, 11000111, 11001011, 11001101, 11010011, 11010101, 11011001, 11100011

Offset: 1

Bernard Schott, Feb 18 2023

For m >= 5, there are A000292(m-4) terms with m digits.

			1010101 has three digits 0 and is the binary expansion of the odd integer 85, so 1010101 is a term.

Cf. A000292, A007088, A360573.

Similar, but with k zeros in their binary expansion: A000042 (k=0), A190619 (k=1), A357774 (k=2).

FromDigits[IntegerDigits[#, 2]] & /@ Select[Range[1, 250, 2], DigitCount[#, 2, 0] == 3 &] (* Amiram Eldar, Feb 18 2023 *)

from itertools import count, islice
from sympy.utilities.iterables import multiset_permutations
def A360574_gen(): # generator of terms
    yield from (int('1'+''.join(d)+'1') for l in count(0) for d in  multiset_permutations('000'+'1'*l))
A360574_list = list(islice(A360574_gen(),30)) # Chai Wah Wu, Feb 18 2023

a(n) = A007088(A360573(n)).

A046421 Index of smallest repunit having exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

1, 2, 3, 13, 8, 6, 15, 12, 28, 18, 24, 32, 36, 30, 54, 42, 78, 100, 72, 176, 60, 208, 84, 132, 160, 198, 120, 204, 216, 308, 168, 280, 306, 180, 210, 264, 270, 252, 378, 336, 300

Offset: 0

Patrick De Geest, Jul 15 1998

a(40) = 300; all other subsequent terms are > 322. - Ray Chandler, Apr 23 2017

a(41) <= 684, a(42) <= 546, a(43) <= 528, a(44) <= 462, a(45) = 360, a(46) <= 576, a(47) <= 624, a(48) <= 768. - Daniel Suteu, Jan 21 2023

			For n = 5: R_6 = 111111 = 3*7*11*13*37 is the smallest repunit with five prime factors, so a(5) = 6.

Patrick De Geest, Repunits prime factors
Eric Weisstein's World of Mathematics, Repunit

Cf. A000042, A001222, A002275, A004022, A046053.
Initial terms of A004023, A046413, A046414, A046415, A046416, A046417, A046418, A046419.
Cf. A086565 (equivalent with distinct prime factors).

a(n) = my(k=1); while(bigomega((10^k - 1)/9) !=n, k++); k; \\ Michel Marcus, Apr 23 2017

a(1) = 2 inserted and a(19)-a(37) added by Ray Chandler, Apr 23 2017
a(38)-a(40) from Jinyuan Wang, Apr 17 2020
Name corrected by Felix Fröhlich, Jun 04 2022

A057303 Numbers n such that the number of distinct digits in n is a digit of n.

Original entry on oeis.org

1, 11, 12, 20, 21, 23, 24, 25, 26, 27, 28, 29, 32, 42, 52, 62, 72, 82, 92, 103, 111, 112, 121, 122, 123, 130, 132, 134, 135, 136, 137, 138, 139, 143, 153, 163, 173, 183, 193, 200, 202, 203, 211, 212, 213, 220, 221, 223, 224, 225, 226, 227, 228, 229, 230, 231

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk), Aug 25 2000

The repunits (A000042) are a subsequence. Analogous in construction to the refactorable numbers (A033950).

			103 has 3 distinct digits in base 10 and 3 is a base 10 digit of 103.

S. Colton, Unpublished PhD Thesis, University of Edinburgh, 2000.

S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics

Cf. A000042.

isok(n) = {my(d = vecsort(digits(n),,8)); vecsearch(d, #d);} \\ Michel Marcus, Feb 23 2016

A075244 Least number requiring the base n to produce a prime by base reversal.

Original entry on oeis.org

2, 3, 15, 8, 109, 9, 119, 16, 27, 70, 2197, 36, 1265, 158, 213, 178, 4205, 126, 14189, 260, 273, 304, 4865, 120, 1295, 78, 81, 532, 44323, 150, 47317, 952, 771, 102, 16705, 492, 6209, 114, 1209, 2020, 132743, 294, 22945, 2834, 2721, 2276, 66455, 144

Offset: 1

Robert G. Wilson v, Sep 09 2002

Question, Is every base necessary to convert the natural numbers into primes?

			a(1) = 2 because two = 11 in unary (A000042) and its reversal 11 = 2. a(2) = 3 because three = 11 in base 2 (A007088) and its reversal 11 in base 2 = 3. a(3) = 15 because fifteen = 120 in base 3 (A007089) and its reversal 21 in base 3 = 7. a(4) = 8 -> 2. a(7) = 119 because 119 base 7 = 230 in base 7 (A007093) and its reversal 32 base 7 = 161.

Cf. A075241 & A075242.

f[n_] := Block[{b = 2}, While[b < n && !PrimeQ[ FromDigits[ Reverse[ IntegerDigits[n, b]], b]], b++ ]; If[b != n, b, 0]]; a = Table[0, {70}]; Do[b = f[n]; If[b < 76 && a[[b]] == 0, a[[b]] = n], {n, 2, 133000}]

A083808 Smallest prime == 1 (mod n-th unary number U(n) = (10^n-1)/9).

Original entry on oeis.org

2, 23, 223, 24443, 199999, 666667, 19999999, 22222223, 666666667, 44444444441, 22222222223, 2444444444443, 17777777777777, 88888888888889, 1333333333333333, 64444444444444439, 88888888888888889

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 08 2003

			a(4) = 24443 ==1 (mod 1111).

Cf. A000042.

with(numtheory): for n from 1 to 30 do u := (10^n-1)/9: for j from u+1 to 1000*u+1 by u do if isprime(j) then printf(`%d,`,j); break; fi: od:od:

Corrected and extended by James Sellers, May 19 2003

cp's OEIS Frontend

A096044 Triangle read by rows: T(n,k) = (n+1,k)-th element of (M^10-M)/9, where M is the infinite lower Pascal's triangle matrix, 1<=k<=n.

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A100563 Number of bases less than sqrt(n) in which n is a palindrome.

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A131664 A string of n 1's repeated n times.

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A137318 Concatenation of segments of the digit sequence 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3....

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A246757 Largest n-digit number divisible by the product of its decimal digits.

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A360574 Binary expansions of odd numbers with three zeros in their binary expansion.

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Python

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A046421 Index of smallest repunit having exactly n prime factors (counted with multiplicity).

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PARI

Extensions

A057303 Numbers n such that the number of distinct digits in n is a digit of n.

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