A000042 -id:A000042 - cp's OEIS frontend

A048611 Find smallest pair (x,y) such that x^2 - y^2 = 11...1 (n times) = (10^n-1)/9; sequence gives value of x.

1, 6, 20, 56, 156, 340, 2444, 4440, 167000, 55556, 267444, 333400, 132687920, 5555556, 10731400, 40938800, 2682647040, 333334000, 555555555555555556, 3334367856, 11034444280, 35595935980, 5555555555555555555556

Offset: 1

Felice Russo

Least solutions for 'Difference between two squares is a repunit of length n'.

			For n=2, 6^2 - 5^2 = 11.

David Wells, "Curious and Interesting Numbers", Revised Ed. 1997, Penguin Books, p. 119. ISBN 0-14-026149-4.

H. Havermann, Repunit Square Differences (gives many more terms)

Cf. A048612, A000042, A002275, A033676, A033677.

s = Flatten[Table[r = (10^i - 1)/9; d = Divisors[r]; p = d[[Length[d]/2]]; Solve[{x - y == p, x + y == r/p}, {y, x}], {i, 2, 56}]]; Prepend[Cases[s, Rule[x, n_] -> n], 1]

from sympy import divisors
def A048611(n):
    d = divisors((10**n-1)//9)
    l = len(d)
    return (d[l//2]+d[(l-1)//2])//2 # Chai Wah Wu, Apr 05 2021

a(n) = (A033677((10^n-1)/9)+A033676((10^n-1)/9))/2. - Chai Wah Wu, Apr 05 2021

Corrected and extended by Patrick De Geest, Jun 15 1999
More terms from Hans Havermann, Jul 02 2000
Offset corrected by Chai Wah Wu, Apr 05 2021

A076697 Indices of record values in A079451, largest prime factor of Lucas numbers A000032.

Original entry on oeis.org

0, 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 26, 31, 37, 41, 47, 53, 61, 68, 71, 76, 79, 86, 113, 136, 164, 172, 178, 202, 218, 229, 262, 278, 284, 307, 313, 328, 353, 373, 436, 443, 458, 487, 503, 557, 577, 586, 613, 617, 746, 751, 758, 863, 914

Offset: 0

Shane Findley, Oct 25 2002

nonn

From  M. F. Hasler, Apr 09 2025: (Start)
Original name: Next-to-largest factor of Lucas(n).
The offset 0 is coherent with the fact that the initial term is a starting value rather than a record value.
When A000032(n) is prime (<=> n is in A001606), it necessarily sets a new record for the largest prime factor, since A000032 is increasing from the second term on. Therefore, A001606 is a subsequence. (End)

Ron Knott, Lucas numbers.
Douglas S. McNeil, in reply to Robert Israel, A076697, SeqFan google group, April 9, 2025.

Cf. A000042 (Lucas numbers, starting with 2), A079451 (largest prime factor of these).

Cf. A001606 (Indices of prime Lucas numbers: a subsequence).

A076697_first(n, m=0)=vector(n,i, i>1 || n=-1; until(mA079451(n++), m), );n) \\ M. F. Hasler, Apr 09 2025

def A076697(n):
    try: terms, M = A076697.terms, A076697.M
    except AttributeError: A076697.terms = terms = [0]; A076697.M = M = 2
    while len(terms) <= n: terms.append(next(i for i in range(terms[-1]+1, 1<<59)
        if M < (M:=max(A079451(i),M)))); A076697.M = M
    return terms[n] # M. F. Hasler, Apr 10 2025

New definition and data corrected and extended by M. F. Hasler, Apr 09 2025

A083813 a(n) = 3*(10^n-1).

Original entry on oeis.org

27, 297, 2997, 29997, 299997, 2999997, 29999997, 299999997, 2999999997, 29999999997, 299999999997, 2999999999997, 29999999999997, 299999999999997

Offset: 1

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

nonn

Original definition: 3n+1 is the digit reversal of n+1.
1. a(n) = 27 + 270 + 2700 + ... up to n terms = sum of n terms of the geometric progression with the first term 27 and common ratio 10.
2. a(n) = 27*A000042(n) (the unary sequence).
Equals A086574 restricted to positive indices. See that entry for many more comments, formulas and references. - M. F. Hasler, Jul 29 2016

Essentially a duplicate of A086574.

Cf. A000042, A083811, A083812.

3(10^Range[20]-1) (* or *) Table[10 FromDigits[PadRight[{2},n,9]]+7,{n,20}] (* Harvey P. Dale, Jan 25 2020 *)

Edited by M. F. Hasler, Jul 29 2016

A093521 Runs of 1's of lengths 1, prime(1), prime(2), prime(3), ... separated by 0's.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0

Offset: 1

Robert G. Wilson v, Mar 29 2004

nonn

Carl Sagan's "Contact" sequence.

Zeros occur at positions given by 1+A110895(k). - Antti Karttunen, Nov 08 2018

W. A. Dembski and J. M. Kushiner, Signs of Intelligence, Baker Book House Co., Grand Rapids, MI, p30-31, 2001,
Carl Sagan, Contact, Simon and Schuster, Chapter 4 "Prime Numbers," pp. 68-82, NY, 1985.

Antti Karttunen, Table of n, a(n) for n = 1..24235 (first 101 runs)
Robin Dougherty, Robert Zemeckis' Contact, Salon.
Alex Kasman, Mathematical Fiction, Contact (1985).
Alex Kasman, How 'Contact' by Carl Sagan Ends.
Index entries for characteristic functions

Cf. A000040, A000042, A005171, A031974, A055976, A056051, A066247, A091247, A110895, A175851, A175856.

a = Table[1, {100}]; Do[ a[[Sum[Prime[i], {i, n}] + n]] = 0, {n, 1, 8}]; a

up_to = 111;
A093521list(up_to) = { my(v=vector(up_to), i=2, j); v[1] = 1; v[2] = 0; forprime(p=2, oo, j=p; while(j, if(i==up_to, return(v), i++; v[i] = 1; j--)); if(i==up_to, return(v), i++; v[i] = 0)); };
v093521 = A093521list(up_to);
A093521(n) = v093521[n];

Data section extended up to n=111 by Antti Karttunen, Nov 08 2018

A272232 Smallest k > 0 such that R_k//n//R_k is prime, where R_k is the repunit A002275(k) of length k and // denotes concatenation; or -1 if no such k exists.

Original entry on oeis.org

1, 9, -1, 1, 2, 1, 10, 3, 1, 1, 3, -1, 2, 3, 33, 1, 2, 1, 1, 21, 1, 2, -1, 1, 7, 48, 292, 4, 3, 1, 1, 2, 1, -1, 135, -1, 1, -1, 1, 34, 3, 3, 40, 2, -1, 1, 3, 1, 1, 32, 61, 1, 2, 1, 137, -1, 3, 1, 2, 42, 1, 14, 1, 262, 2, 22, -1, 3, 9, 2, 33, 73, 1, 3, 1, 2, 3, -1, 2, 2, 1

Offset: 0

Felix Fröhlich, Apr 23 2016

a(2) = -1 (see second comment in A258372).
a(n) = -1 if n > 0 is in A099814 (see fourth comment in A004022).
a(n) = -1 if n is of the form A000042(i)*10^j+A000042(i) for some j > i > 0, since the resulting number is divisible by A002275(k)//A000042(i).
a(n) = -1 if n is a term of A010785 with an even number of digits, since any number of the form 1..1d..d1..1 with an even number of digits d is divisible by 11.
a(n) = 1 if there exists an integer x such that n = (A002275(A004023(x))-A011557(x)-1)/10.
From Chai Wah Wu, Nov 07 2019: (Start)
a(n) = -1 if n has an even number of digits and is a multiple of 11. In particular, a(n) = -1 if n is a term of A056524.
a(n) = -1 if n = (10^k+1)(10^m-1)/9 for some m > 0, k >= 0.
(End)
a(140) > 20000. - Hans Havermann, May 21 2022

			a(0) = 1 since 101 is prime; a(1) refers to the prime 1111111111111111111.
a(124) = -1 because R_k//124//R_k is divisible by 125*10^k-1.

Hans Havermann, Table of n, a(n) for n = 0..139

Cf. A000042, A002275, A056524, A099814, A004023, A258372, A306861.

Table[SelectFirst[Range[10^4], PrimeQ@ FromDigits@ Flatten@ {#, IntegerDigits@ n, #} &@ Table[1, {#}] &], {n, 0, 91}] /. k_ /; MissingQ@ k -> 0 (* Michael De Vlieger, Apr 25 2016, Version 10.2 *)

a(n) = my(k=1); while(!ispseudoprime(eval(Str((10^k-1)/9, n, (10^k-1)/9))), k++); k

a(35)-a(80) from Giovanni Resta, May 01 2016

Escape clausae value changed to -1 by N. J. A. Sloane, May 17 2022

A334131 Numbers that can be written as a product of distinct repunits.

Original entry on oeis.org

0, 1, 11, 111, 1111, 1221, 11111, 12221, 111111, 122221, 123321, 1111111, 1222221, 1233321, 1356531, 11111111, 12222221, 12333321, 12344321, 13566531, 111111111, 122222221, 123333321, 123444321, 135666531, 135787531, 1111111111, 1222222221

Offset: 1

Ilya Gutkovskiy, Apr 14 2020

			13566531 = 11*111*11111. - _David A. Corneth_, Mar 26 2021

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Repunit

Cf. A000042, A002275, A015009, A077488, A083278, A308365.

A355280 Binary numbers (digits in {0, 1}) with no run of digits with length < 2.

Original entry on oeis.org

11, 111, 1100, 1111, 11000, 11100, 11111, 110000, 110011, 111000, 111100, 111111, 1100000, 1100011, 1100111, 1110000, 1110011, 1111000, 1111100, 1111111, 11000000, 11000011, 11000111, 11001100, 11001111, 11100000, 11100011, 11100111, 11110000, 11110011, 11111000, 11111100, 11111111

Offset: 1

M. F. Hasler, Oct 17 2022

This is the binary representation of the terms in A033015.

The sequence can be seen as a table where row r contains the terms with r digits. Then row r+1 is obtained by from the terms of row r by duplicating their last digit, and from those of row r-1 by appending twice the 1's complement of their last digit. This yields the row lengths given in FORMULA.

			There can't be a terms with only 1 digit, so the smallest term is a(1) = 11.
The only 3-digit term is a(2) = 111, since in 100 the digit 1 is alone, and in 101 and 110 the digit 0 is alone.
With four digits we must have either no or two digits 0 and they must be at the end (to avoid isolated '1's), i.e., a(3) = 1100 and a(4) = 1111.

Robert Israel, Table of n, a(n) for n = 1..10945

Cf. A033015 (the same terms converted from base 2 to base 10),

Subsequence of A007088 (the binary numbers); A000042 (numbers in base 1) = A002275 \ {0} (repunits) are subsequences; A061851 is the subsequence of palindromes.

F:= proc(d) option remember;
   local R,i,j, x0;
   R:= NULL;
   for i from d-2 to 2 by -1 do
     x0:= (10^d - 10^i)/9;
     for j from i-2 to 0 by -1 do
        R:= R, op(map(t -> t + x0, procname(j)))
     od
   od;
   sort([R, (10^d-1)/9])
end proc:
F(0):= [0]; F(1):= [];
seq(op(F[i]),i=2..9); # Robert Israel, May 12 2025

{is_A355280(n,d=digits(n))=vecmax(d)==1 && is_A033015(fromdigits(d,2))}
A355280(n)=A007088(A033015(n))
concat(apply( {A355280_row(n)=if(n>2, setunion([x*10+x%10|x<-A355280_row(n-1)],[x*100+11*(1-x%10)|x<-A355280_row(n-2)]), n>1, [11],[])}, [1..8])) \\ "Row" of n-digit terms. For (very) large n one should implement memoization instead of this naive recursion.

def A355280_row(n): return [] if n<2 else [11] if n==2 else sorted(
    [x*10+x%10 for x in A355280_row(n-1)] +
    [x*100+11-x%10*11 for x in A355280_row(n-2)]) # M. F. Hasler, Oct 17 2022

a(n) = A007088(A033015(n)).

The number of terms with n digits is Fibonacci(n-1); the largest such term is A000042(n) = A002275(n).

A357774 Binary expansions of odd numbers with two zeros in their binary expansion.

Original entry on oeis.org

1001, 10011, 10101, 11001, 100111, 101011, 101101, 110011, 110101, 111001, 1001111, 1010111, 1011011, 1011101, 1100111, 1101011, 1101101, 1110011, 1110101, 1111001, 10011111, 10101111, 10110111, 10111011, 10111101, 11001111, 11010111, 11011011, 11011101, 11100111, 11101011

Offset: 1

Bernard Schott, Oct 19 2022

For m >= 4, there are A000217(m-3) terms with m digits.

Cf. A000217, A007088, A357773.
A267524 \ {1, 10, 100} and A267705 \ {1, 10} are two subsequences.
Similar, but with k zeros in their binary expansion: A000042 (k=0), A190619 (k=1).

FromDigits[IntegerDigits[#, 2]] & /@ Select[Range[1, 250, 2], DigitCount[#, 2, 0] == 2 &] (* Amiram Eldar, Oct 19 2022 *)

isok(k) = (k%2) && (#binary(k) == hammingweight(k)+2); \\ A357773
f(n) = fromdigits(binary(n), 10); \\ A007088
lista(nn) = apply(f, select(isok, [1..nn])); \\ Michel Marcus, Oct 19 2022

from itertools import combinations, count, islice
def agen(): # generator of terms
    for d in count(4):
        b, c = 2**d - 1, 2**(d-1)
        for i, j in combinations(range(1, d-1), 2):
            yield int(bin(b - (c >> i) - (c >> j))[2:])
print(list(islice(agen(), 30))) # Michael S. Branicky, Oct 19 2022

from itertools import count, islice
def A357774_gen(): # generator of terms
    for l in count(2):
        m = (10**(l+2)-1)//9
        for i in range(l,0,-1):
            k = m-10**i
            yield from (k-10**j for j in range(i-1,0,-1))
A357774_list = list(islice(A357774_gen(),30)) # Chai Wah Wu, Feb 19 2023

from math import isqrt, comb
from sympy import integer_nthroot
def A357774(n):
    a = (m:=integer_nthroot(6*n, 3)[0])+(n>comb(m+2,3))+3
    b = isqrt((j:=comb(a-1,3)-n+1)<<3)+3>>1
    c = j-comb((r:=isqrt(w:=j<<1))+(w>r*(r+1)),2)
    return (10**a-1)//9-10**b-10**c # Chai Wah Wu, Dec 19 2024

a(n) = A007088(A357773(n)).

A046415 Repunit of length a(n) has exactly 4 prime factors (counted with multiplicity).

Original entry on oeis.org

8, 9, 10, 14, 41, 43, 49, 53, 109, 157, 167, 173, 197, 199, 223, 229, 269, 283, 307, 349

Offset: 1

Patrick De Geest, Jul 15 1998

P. De Geest, Repunits prime factors
Makoto Kamada, Factorizations of 11...11 (Repunit)
Eric Weisstein's World of Mathematics, Repunit

Cf. A000042, A001222, A002275, A004022, A004023, A046053.

Select[Range[350],PrimeOmega[FromDigits[PadRight[{},#,1]]]==4&] (* Harvey P. Dale, Oct 27 2020 *)

isok(n) = bigomega((10^n - 1)/9) == 4; \\ Michel Marcus, Apr 23 2017

More terms from Robert Gerbicz, Nov 22 2010

Offset corrected to 1, a(18)-a(20) added by Ray Chandler, Apr 23 2017

A046416 Repunit of length a(n) has exactly 5 prime factors (counted with multiplicity).

Original entry on oeis.org

6, 25, 29, 62, 89, 134, 137, 142, 151, 179, 239, 257, 271, 277, 289

Offset: 1

Patrick De Geest, Jul 15 1998

P. De Geest, Repunits prime factors
Makoto Kamada, Factorizations of 11...11 (Repunit)
Eric Weisstein's World of Mathematics, Repunit

Cf. A000042, A001222, A002275, A004022, A004023, A046053.

More terms from Robert Gerbicz, Nov 22 2010
a(14) by Bo Gyu Jeong, Jun 30 2012
a(15) from Ray Chandler, Apr 23 2017

cp's OEIS Frontend

A048611 Find smallest pair (x,y) such that x^2 - y^2 = 11...1 (n times) = (10^n-1)/9; sequence gives value of x.

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A076697 Indices of record values in A079451, largest prime factor of Lucas numbers A000032.

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A083813 a(n) = 3*(10^n-1).

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A093521 Runs of 1's of lengths 1, prime(1), prime(2), prime(3), ... separated by 0's.

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A272232 Smallest k > 0 such that R_k//n//R_k is prime, where R_k is the repunit A002275(k) of length k and // denotes concatenation; or -1 if no such k exists.

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A334131 Numbers that can be written as a product of distinct repunits.

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A355280 Binary numbers (digits in {0, 1}) with no run of digits with length < 2.

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Maple

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A357774 Binary expansions of odd numbers with two zeros in their binary expansion.