A033175 -id:A033175 - cp's OEIS frontend

A098207 a(n) is the square of near-repdigit number A033175(n).

1, 961, 109561, 11095561, 1110955561, 111109555561, 11111095555561, 1111110955555561, 111111109555555561, 11111111095555555561, 1111111110955555555561, 111111111109555555555561

Offset: 0

Labos Elemer, Oct 20 2004

While repunit-squares are palindromic, squares of near repdigits provide other curious digit-patterns.

Index entries for linear recurrences with constant coefficients, signature (111, -1110, 1000).

Cf. A033175, A055557.

a(n) = A033175(n)^2.
From Chai Wah Wu, Nov 09 2018: (Start)
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
G.f.: (-4000*x^2 - 850*x - 1)/((x - 1)*(10*x - 1)*(100*x - 1)). (End)

A098208 4th powers of A033175(n) near repdigit numbers.

Original entry on oeis.org

1, 923521, 12003612721, 123111473904721, 1234222258516824721, 12345333336962946024721, 123456444444807407238024721, 1234567555555591851850158024721, 12345678666666670296296279358024721

Offset: 0

Labos Elemer, Oct 20 2004

Display peculiar digit patterns.

Cf. A033175, A055557, A098207.

(FromDigits/@Table[Join[PadLeft[{},n,3],{1}],{n,0,20}])^4 (* Harvey P. Dale, Oct 20 2011 *)

From Chai Wah Wu, Nov 09 2018: (Start)
a(n) = 11111*a(n-1) - 11222110*a(n-2) + 1122211000*a(n-3) - 11111000000*a(n-4) + 10000000000*a(n-5) for n > 4.
G.f.: (-160000000000*x^4 - 102065000000*x^3 - 1753593000*x^2 - 912410*x - 1)/((x - 1)*(10*x - 1)*(100*x - 1)*(1000*x - 1)*(10000*x - 1)). (End)

A051200 Except for initial term, primes of form "n 3's followed by 1".

Original entry on oeis.org

3, 31, 331, 3331, 33331, 333331, 3333331, 33333331, 333333333333333331, 3333333333333333333333333333333333333331, 33333333333333333333333333333333333333333333333331

Offset: 1

N. J. A. Sloane

"A remarkable pattern that is entirely accidental and leads nowhere" - M. Gardner, referring to the first 8 terms.
a(2)*a(3)*a(4) = 34179391, a Zeisel number (A051015) with coefficients (10,21).
a(2)*a(3)*a(4)*a(5) = 1139233281421, a Zeisel number with coefficients (10,21).
a(2)*a(3)*..*a(6) = 379741768929343351, a Zeisel number with coefficients (10,21).
a(2)*a(3)*..*a(7) = 1265805010367017001532181, a Zeisel number with coefficients (10,21).
a(2)*a(3)*..*a(8) = 42193497392022209194699696424911, a Zeisel number with coefficients (10,21).
Besides first 3, primes of the form (10^n-7)/3, n>1. See A123568. - Vincenzo Librandi, Aug 06 2010
The integer lengths of the terms of the sequence are 1, 2, 3, 4, 5, 6, 7, 8, 18, 40, 50, 60, 78, 101, 151, 319, 382, etc. - Harvey P. Dale, Dec 01 2018

Martin Gardner, The Last Recreations, Chapter 12: Strong Laws of Small Primes, Springer-Verlag, 1997, pp. 191-205, especially p. 194.
W. Sierpiński, 200 Zadan z Elementarnej Teorii Liczb, Warsaw, 1964; Problem 88 [in English: 200 Problems from the Elementary Theory of Numbers]
W. Sierpiński, 250 Problems in Elementary Number Theory. New York: American Elsevier, Warsaw, 1970, pp. 8, 56-57.
F. Smarandache, Properties of numbers, University of Craiova, 1973

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, 3.

Cf. A055520, A089017, A089018, A093671, A056698, A105427, A105428, A033175, A123568.

Join[{3},Select[Rest[FromDigits/@Table[PadLeft[{1},n,3], {n,50}]], PrimeQ]]  (* Harvey P. Dale, Apr 20 2011 *)

Union of 3 and A123568.

More terms from James Sellers

Cross reference added by Harvey P. Dale, May 21 2014

A055557 Numbers k such that 3*R_k - 2 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 18, 40, 50, 60, 78, 101, 151, 319, 382, 784, 1732, 1918, 8855, 11245, 11960, 12130, 18533, 22718, 23365, 24253, 24549, 25324, 30178, 53718, 380976, 424861, 563535, 666903

Offset: 1

Labos Elemer, Jul 10 2000

nonn

Also numbers k such that (10^k-7)/3 is prime.
Sierpiński attributes the primes for k = 2,...,8 to A. Makowski.
The history of the discovery of these numbers may be as follows: a(1)-a(7), Makowski; a(8)-a(18), Caldwell; a(19), Earls; a(20)-a(31), Kamada. (Corrections to this account will be welcomed.)
Concerning certifying primes, see the references by Goldwasser et al., Atkin et al. and Morain. - Labos
No more than 14 consecutive exponents can provide primes because for exponents 15m+2, 16m+9, 18m+12, 22m+21, terms are divisible by 31, 17, 19, 23 respectively. Here 7 of possible 14 is realized. - Labos Elemer, Jan 19 2005
(10^(15m+2)-7)/3 == 0 (mod 31). So 15m+2 isn't a term for m > 0. - Seiichi Manyama, Nov 05 2016

C. Caldwell, The near repdigit primes 333...331, J.Recreational Math. 21:4 (1989) 299-304.
S. Goldwasser and J. Kilian, Almost All Primes Can Be Quickly Certified. in Proc. 18th STOC, 1986, pp. 316-329.
W. Sierpiński, 200 Zadan z Elementarnej Teorii Liczb [200 Problems from the Elementary Theory of Numbers], Warszawa, 1964; Problem 88.

A. Atkin and F. Morain, Elliptic Curves and Primality Proving, Math. Comp. 61:29-68, 1993.
Makoto Kamada, Prime numbers of the form 33...331.
Mathematics.StackExchange.com, 31,331,3331, 33331,333331,3333331,33333331 are prime
F. Morain, Implementation of the Atkin-Goldwasser-Kilian Primality Testing Algorithm, INRIA Research Report, # 911, October 1988.
Dave Rusin, Primes in exponential sequences [Broken link]
Dave Rusin, Primes in exponential sequences [Cached copy]
Index entries for primes involving repunits

Cf. A051200, A033175, A055520.

Do[ If[ PrimeQ[(10^n - 7)/3], Print[n]], {n, 50410}]
One may run the prime certificate program as follows <True]}, {n, 1, 16}] (* Labos Elemer *)

for(n=1,2000, if(isprime((10^n-7)/3),print(n)))

a(n) = A055520(n) + 1.

Corrected and extended by Jason Earls, Sep 22 2001
a(20)-a(31) were found by Makoto Kamada (see links for details). At present they correspond only to probable primes.
a(32)-a(33) from Leonid Durman, Jan 09-10 2012
a(34)-a(35) from Kamada data by Tyler Busby, Apr 14 2024

A086578 a(n) = 7*(10^n - 1).

Original entry on oeis.org

0, 63, 693, 6993, 69993, 699993, 6999993, 69999993, 699999993, 6999999993, 69999999993, 699999999993, 6999999999993, 69999999999993, 699999999999993, 6999999999999993, 69999999999999993, 699999999999999993, 6999999999999999993, 69999999999999999993, 699999999999999999993

Offset: 0

Ray Chandler, Jul 22 2003

Original definition: a(n) = k where R(k+7) = 7.

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A002275, A004086 (R(n)).
One of family of sequences of form a(n) = k, where R(k+m) = m, m=1 to 9; m=1: A002283, m=2: A086573, m=3: A086574, m=4: A086575, m=5: A086576, m=6: A086577, m=7: A086578, m=8: A086579, m=9: A086580.
Sequences of the form m*10^n - 7: 3*A033175 (m=1, 10), A086943 (m=3), 3*A185127 (m=4), this sequence (m=7), A100412 (m=8).

[7*(10^n -1): n in [0..20]]; // G. C. Greubel, Apr 14 2023

LinearRecurrence[{11,-10}, {0,63}, 31] (* G. C. Greubel, Apr 14 2023 *)

[7*(10^n -1) for n in range(21)] # G. C. Greubel, Apr 14 2023

a(n) = 7*9*A002275(n) = 7*A002283(n).
R(a(n)) = A086575(n).
From Chai Wah Wu, Jul 08 2016: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 1.
G.f.: 63*x/((1 - x)*(1 - 10*x)). (End)
E.g.f.: 7*(exp(10*x) - exp(x)). - G. C. Greubel, Apr 14 2023

Edited by Jinyuan Wang, Aug 04 2021

A100412 a(n) = 8*10^n - 7.

Original entry on oeis.org

1, 73, 793, 7993, 79993, 799993, 7999993, 79999993, 799999993, 7999999993, 79999999993, 799999999993, 7999999999993, 79999999999993, 799999999999993, 7999999999999993, 79999999999999993, 799999999999999993

Offset: 0

Farideh Firoozbakht, Dec 08 2004

Also: Numbers n such that n is reversal(n)-th odd number. (This was the original definition. - Ed.)

All semiprimes in this sequence (n = 2, 4, 7, 9, 11, 16, 18, 23, 31, 32, 40, ...) are in A136543. - M. F. Hasler, Nov 03 2012

			793 is in the sequence because 793 is 397th odd number.
1 is in the sequence because 1 is the 1st odd number. - _M. F. Hasler_, Nov 03 2012

G. C. Greubel, Table of n, a(n) for n = 0..500
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Cf. A100413, A136543.

Sequences of the form m*10^n - 7: 3*A033175 (m=1, 10), A086943 (m=3), 3*A185127 (m=4), A086578 (m=7), this sequence (m=8).

[8*10^n -7: n in [0..20]]; // G. C. Greubel, Apr 14 2023

Mathematica
```
Table[8*10^n-7, {n,0,20}]
```

A100412(n):=8*10^n-7$
makelist(A100412(n),n,0,17); /* Martin Ettl, Nov 08 2012 */

Vec((1+62*x)/((1-x)*(1-10*x)) + O(x^100)) \\ Colin Barker, Oct 14 2014

[8*10^n -7 for n in range(21)] # G. C. Greubel, Apr 14 2023

From Colin Barker, Oct 14 2014: (Start)
a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3).
G.f.: (1+62*x)/((1-x)*(1-10*x)). (End)
E.g.f.: 8*exp(10*x) - 7*exp(x). - G. C. Greubel, Apr 14 2023

Edited and extended to offset 0 by M. F. Hasler, Nov 03 2012

A104484 Number of distinct prime divisors of 33...331 (with n 3s).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 2, 2, 3, 1, 3, 4, 3, 2, 4, 3, 3, 2, 4, 3, 3, 2, 3, 4, 6, 2, 2, 3, 3, 4, 4, 1, 3, 3, 6, 6, 3, 2, 3, 5, 3, 1, 3, 3, 4, 6, 3, 3, 5, 3, 5, 1, 3, 6, 5, 4, 3, 5, 3, 3, 5, 4, 6, 6, 3, 6, 2, 2, 4, 1, 5, 5, 5, 4, 4, 7, 2, 2, 5, 6

Offset: 0

Parthasarathy Nambi, Apr 18 2005

Interestingly, the first seven members in this sequence are all primes.

			Number of distinct prime divisors of 31 is 1 (prime).
Number of distinct prime divisors of 331 is 1 (prime).
Number of distinct prime divisors of 3331 is 1 (prime).
Number of distinct prime divisors of 33331 is 1 (prime).
Number of distinct prime divisors of 333331 is 1 (prime).
Number of distinct prime divisors of 3333331 is 1 (prime).
Number of distinct prime divisors of 33333331 is 1 (prime).

Amiram Eldar, Table of n, a(n) for n = 0..203

Cf. A001221, A033175, A051200, A105267.

Table[Length[FactorInteger[(10^(n + 1) - 7)/3]], {n, 1, 50}] (* Stefan Steinerberger *)

a(n) = omega((10^(n + 1) - 7)/3); \\ Michel Marcus, May 13 2020

a(n) = A001221(A033175(n+1)). - Amiram Eldar, Jan 24 2020, corrected, May 13 2020

More terms from Amiram Eldar, Jan 24 2020

a(0) inserted by Amiram Eldar, May 13 2020

A055520 Numbers k such that 30*R_k + 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 17, 39, 49, 59, 77, 100, 150, 318, 381, 783, 1731, 1917, 8854, 11244, 11959, 12129, 18532, 22717, 23364, 24252, 24548, 25323, 30177, 53717, 380975, 424860

Offset: 1

Eric W. Weisstein

Also numbers k such that (10^(k+1)-7)/3 is prime.

Martin Gardner, Strong Laws of Small Primes, in The Last Recreations, p. 194 (1997).

Makoto Kamada, Prime numbers of the form 33...331.
Eric Weisstein's World of Mathematics, 3
Index entries for primes involving repunits

Cf. A055557.

Indices of A033175 that are primes. Cf. A051200, A055557.

Do[ If[ PrimeQ[30*(10^n - 1)/9 + 1], Print[n]], {n, 0, 50410}]

a(n) = A055557(n) - 1. - Robert Price, Jan 30 2015

a(32)-a(33) from Leonid Durman, Jan 09-10 2012

A105267 a(n) = the number of divisors of 33...31, with n 3s.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 8, 8, 16, 4, 4, 8, 2, 8, 24, 8, 4, 16, 8, 8, 4, 16, 8, 8, 6, 8, 16, 64, 4, 4, 8, 8, 16, 16, 2, 8, 8, 64, 64, 8, 4, 8, 32, 8, 2, 8, 8, 16, 64, 8, 8, 32, 8, 32, 2, 8, 64, 32, 16, 8, 32, 8, 8, 32, 16, 64, 64, 8, 64, 4, 4, 16, 2

Offset: 0

Parthasarathy Nambi, Apr 29 2005

nonn

The first seven 33...31 numbers are prime, so those terms are 2. - Don Reble, Oct 26 2006

Amiram Eldar, Table of n, a(n) for n = 0..203

Cf. A000005, A051200, A033175, A104484.

a[n_] := DivisorSigma[0, (10^(n + 1) - 7)/3]; Array[a, 30, 0] (* Amiram Eldar, May 13 2020 *)

a(n) = numdiv((10^(n + 1) - 7)/3); \\ Michel Marcus, May 13 2020

a(n) = A000005(A033175(n)). - Amiram Eldar, May 13 2020

More terms from Don Reble, Oct 26 2006

A123568 Prime numbers of the form (10^n - 7)/3.

Original entry on oeis.org

31, 331, 3331, 33331, 333331, 3333331, 33333331, 333333333333333331, 3333333333333333333333333333333333333331, 33333333333333333333333333333333333333333333333331

Offset: 1

Artur Jasinski, Nov 12 2006

nonn

The number of initial 3s is n - 1.

Note that each n from 2 to 8 gives primes, but after that the n that correspond to primes are progressively further apart. Singh (1997) gives this as an example of why mathematicians don't trust a preponderance of evidence as proof: in the 17th century, when factoring numbers with as few as eight digits wasn't as easy as it is today, the pattern suggested that all numbers of this form are prime. - Alonso del Arte, Nov 11 2012

			a(7) = 33333331 because that is the seventh number of the specified form to be prime.
333333331 is not in the sequence because it is composite, being the product of 17 and 19607843.

Simon Singh, Fermat's Enigma. New York: Walker & Company (1997) p. 159.

Alois P. Heinz, Table of n, a(n) for n = 1..17

Cf. A055557, A051200, A033175, A065586, A068104.

Do[If[PrimeQ[(10^n - 7)/3], Print[(10^n - 7)/3]], {n, 1, 100}] (* Jasinski *)
Select[(10^Range[50] - 7)/3, PrimeQ[#] &] (* Alonso del Arte, Nov 11 2012 *)
Select[Table[FromDigits[PadLeft[{1},n,3]],{n,50}],PrimeQ] (* Harvey P. Dale, Dec 05 2018 *)

select(ispseudoprime, vector(20, n, (10^n-7)/3)) \\ Charles R Greathouse IV, Nov 12 2012

cp's OEIS Frontend

A098207 a(n) is the square of near-repdigit number A033175(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A098208 4th powers of A033175(n) near repdigit numbers.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A051200 Except for initial term, primes of form "n 3's followed by 1".

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A055557 Numbers k such that 3*R_k - 2 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A086578 a(n) = 7*(10^n - 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A100412 a(n) = 8*10^n - 7.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Maxima

PARI

SageMath

Formula

Extensions

A104484 Number of distinct prime divisors of 33...331 (with n 3s).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs