cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 14 results. Next

A105323 Numbers of the form 41*(2*10^n+1) where (2*10^n+1)/3 is prime (n is in the sequence A096507).

Original entry on oeis.org

861, 8241, 82000041, 8200000041, 82000000041, 8200000000041, 8200000000000000000041, 8200000000000000000000041, 8200000000000000000000000000000000000000041
Offset: 1

Views

Author

Farideh Firoozbakht, Apr 16 2005

Keywords

Comments

A105323=41*A093170=41*(2*10^A096507+1)=41*(2*10^(A056657+1)+1). If m is in the sequence then d(m)*reversal(m)=sigma(m) (see A104907). So this sequence is a subsequence of A104907.

Examples

			861 is in the sequence because 861=41*(2*10^1+1); (2*10^1+1)/3=7 and 7 is prime.

Crossrefs

Cf. A104907, A056657, A096507, A093170.

Programs

  • Mathematica
    Do[If[PrimeQ[(2*10^n + 1)/3], Print[41*(2*10^n + 1)]], {n, 63}]

A096508 Numbers k for which 8*R_k + 1 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

2, 14, 17, 35, 4175, 4472, 9812, 12260, 12341, 13760, 14576, 53411, 144683, 148328
Offset: 1

Views

Author

Labos Elemer, Jul 12 2004

Keywords

Comments

Also numbers k such that (8*10^k + 1)/9 is prime.
a(15) > 2*10^5. - Robert Price, Sep 06 2014

Examples

			35 is a term because 88888888888888888888888888888888889 (34 8's) is a prime number.

Links

Crossrefs

Cf. A002275, A056663, A096503-A096507.

Programs

  • Maple
    select(n -> isprime((8*10^n+1)/9), [$1..10000]); # Robert Israel, Sep 07 2014
  • Mathematica
    Do[ If[ PrimeQ[ 8(10^n - 1)/9 + 1], Print[n]], {n, 0, 30000}] (* Robert G. Wilson v, Oct 15 2004 *)
  • PARI
    for(n=1,10^4,if(ispseudoprime(8*(10^n-1)/9+1),print1(n,", "))) \\ Derek Orr, Sep 06 2014

Formula

a(n) = A056663(n) + 1.

Extensions

Four missing terms (9812, 12260, 12341, 13760) added, and a(12)-a(14) added from Kamada data, by Robert Price, Sep 06 2014

A266145 Number of n-digit primes in which n-1 of the digits are 6's.

Original entry on oeis.org

4, 2, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Michael De Vlieger and Robert G. Wilson v, Dec 21 2015

Keywords

Comments

The leading digits must be 6's and only the trailing digit can vary.
For n large a(n) is usually zero.

Examples

			a(2) = 2 since 61 and 67 are prime.
a(3) = 1 since 661 is the only prime.

Links

Crossrefs

Cf. A265733, A266141, A266142, A266143, A266144, A266146, A266147, A266148, A266149, A098088, A096507.

Programs

  • Mathematica
    d = 6; Array[Length@ Select[d (10^# - 1)/9 + (Range[0, 9] - d), PrimeQ] &, 100]
Join[{4},Table[Count[Table[10FromDigits[PadRight[{},k,6]]+n,{n,{1,3,7,9}}], ?PrimeQ],{k,110}]] (* _Harvey P. Dale, Dec 23 2017 *)
  • Python
    from _future_ import division
from sympy import isprime
def A266145(n):
    return 4 if n==1 else sum(1 for d in [-5,-3,1,3] if isprime(2*(10**n-1)//3+d)) # Chai Wah Wu, Dec 27 2015

A093170 Primes of the form 60*R_k + 7, where R_k is the repunit (A002275) of length k.

Original entry on oeis.org

7, 67, 666667, 66666667, 666666667, 66666666667, 66666666666666666667, 66666666666666666666667, 66666666666666666666666666666666666666667, 666666666666666666666666666666666666666666666666666666666666667
Offset: 1

Views

Author

Rick L. Shepherd, Mar 26 2004

Keywords

Comments

Primes of the form (2*10^k + 1)/3. - Vincenzo Librandi, Nov 16 2010
Occur in the factorization of some of the numbers of the form 13...3 not in A093671, cf. second Kamada link. - M. F. Hasler, Sep 14 2014

Links

Crossrefs

Cf. A002275, A056657 (corresponding k), A093671, A096507.

Programs

  • Maple
    A093170:=n->`if`(isprime((2*10^n+1)/3),(2*10^n+1)/3,NULL): seq(A093170(n), n=1..70); # Wesley Ivan Hurt, Sep 14 2014
  • Mathematica
    Select[Table[FromDigits[PadLeft[{7},n,6]],{n,70}],PrimeQ] (* Harvey P. Dale, Jan 26 2013 *)

Formula

a(n) = (20*10^A056657(n)+1)/3 = (2*10^A096507(n)+1)/3.

Extensions

Edited by Ray Chandler, Feb 23 2012

A096846 Numbers n for which 8*R_n - 1 is prime, where R_n = 11...1 is the repunit (A002275) of length n.

Original entry on oeis.org

1, 3, 4, 6, 9, 12, 72, 118, 124, 190, 244, 304, 357, 1422, 2691, 5538, 7581, 21906, 32176, 44358, 120552, 137073, 152260
Offset: 1

Views

Author

Labos Elemer, Jul 15 2004

Keywords

Comments

Also numbers n such that (8*10^n-17)/9 is prime.
The numbers corresponding to a(1)-a(15) are certified prime, the numbers corresponding to a(16)-a(20) are probable primes. a(21) > 10^5. - Robert Price, May 20 2014

Examples

			n=6: a(4)=888887 which is prime.

Links

Crossrefs

Cf. A002275, A056695, A093171, A096506, A096507, A096508, A096845.

Programs

  • Mathematica
    Do[ If[ PrimeQ[ 8(10^n - 1)/9 - 1], Print[n]], {n, 0, 5000}] (* Robert G. Wilson v, Oct 15 2004; corrected by Derek Orr, Sep 06 2014 *)
  • PARI
    for(n=1,10^4,if(ispseudoprime(8*(10^n-1)/9-1),print1(n,", "))) \\ Derek Orr, Sep 06 2014

Formula

a(n) = A056695(n) + 1. - Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 01 2008

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 01 2008
a(18)-a(20) discovered and reported to Makoto Kamada by Erik Branger; added to OEIS by Robert Price, May 20 2014
a(21)-a(23) from Kamada data by Tyler Busby, Apr 23 2024

A056657 Numbers k such that 60*R_k + 7 is prime, where R_k = 11...1 is the repunit (A002275) of length k.

Original entry on oeis.org

0, 1, 5, 7, 8, 10, 19, 22, 40, 62, 65, 118, 121, 148, 251, 283, 304, 591, 745, 874, 1203, 1363, 2239, 2402, 5105, 5775, 5812, 12455, 14234, 39605, 55543, 84238, 275921
Offset: 1

Views

Author

Robert G. Wilson v, Aug 09 2000

Keywords

Comments

Also numbers k such that (20*10^k+1)/3 is prime.

Examples

			7, 67, 666667, 66666667, 666666667, 66666666667, etc. are primes.

Links

Crossrefs

Cf. A002275, A093170, A096507.

Programs

  • Mathematica
    Do[ If[ PrimeQ[ 60*(10^n - 1)/9 + 7 ], Print[n]], {n, 25556}]

Formula

a(n) = A096507(n) - 1.

Extensions

More terms from Robert G. Wilson v, Oct 22 2003
2239,2402,5105,5775 from Farideh Firoozbakht, Dec 23 2003
39605 and 55543 from Serge Batalov, Jun 2009
84238 from Serge Batalov, Jul 06 2009 confirmed as next term by Ray Chandler, Feb 23 2012
a(33) derived from A096507 by Robert Price, Jul 07 2024

A096506 Numbers n for which 2*R_n + 1 is a prime, where R_n = 11...1 is the repunit (A002275) of length n.

Original entry on oeis.org

1, 2, 3, 8, 11, 36, 95, 101, 128, 260, 351, 467, 645, 1011, 1178, 1217, 2442, 3761, 3806, 15617, 26459, 63117, 88545, 93497
Offset: 1

Views

Author

Labos Elemer, Jul 12 2004

Keywords

Comments

Also numbers n such that (2*10^n + 7)/9 is prime.
Per Kamada link, 181457, 202059, 262874 are also terms, found by Rytis Slatkevicius. - Michael S. Branicky, Sep 13 2024

Examples

			n=36: 222222222222222222222222222222222223 is a prime number.

Links

Crossrefs

Cf. A056656, A093162, A096503, A096504, A096505, A096507, A096508.

Programs

  • Mathematica
    Do[ If[ PrimeQ[ 2(10^n - 1)/9 + 1], Print[n]], {n, 7000}] (* Robert G. Wilson v, Oct 14 2004 *)

Formula

a(n) = A056656(n) + 1.

Extensions

a(20)-a(24) from Kamada link by Ray Chandler, Feb 27 2012

A105324 Numbers n such that 2*reversal(n)=sigma(n).

Original entry on oeis.org

6, 73, 483, 4074, 4473, 4623, 7993, 42813, 69855, 253782, 799993, 7999993, 46000023, 426000213, 4600000023, 6718967838, 42600000213, 46000000023, 79999999993, 426000000213
Offset: 1

Views

Author

Farideh Firoozbakht, Apr 16 2005

Keywords

Comments

I. If p=8*10^n-7 is a prime then p is in the sequence because reversal(p)=4*10^n-3 & sigma(p)=8*10^n-6 so 2*reversal(p) =sigma(p). 73,7993,799993 & 7999993 are such terms.
II. If q=(2*10^n+1)/3 is a prime then (a): 69*q is in the sequence because 69*q=46*10^n+23; reversal (69*q)=32*10^n+64 & sigma(69*q)=96*q+96=64*10^n+128 so 2*reversal (69*q)=sigma(69*q). 483,4623 & 46000023 are such terms. (b): 639*q is in the sequence because 639*q=426*10^n+213; reversal (639*q)=312*10^n+624 & sigma(639*q)=936*q+936=624*10^n+1248 so 2*reversal(639*q)=sigma(639*q). 42813 & 426000213 are such terms.
a(21) > 10^12. - Giovanni Resta, Oct 28 2012

Examples

			253782 is in the sequence because reversal(253782)=287352; sigma(253782)=574704 & 2*287352=574704.

Crossrefs

Cf. A093170, A096507, A099190, A105322, A105323, A105325, A105326.

Programs

  • Mathematica
    reversal[n_]:= FromDigits[Reverse[IntegerDigits[n]]]; Do[If[2* reversal[n]== DivisorSigma[1, n], Print[n]], {n, 1000000000}]
Select[Range[8*10^6],2*IntegerReverse[#]==DivisorSigma[1,#]&] (* The program generates the first 12 terms of the sequence. *) (* Harvey P. Dale, Oct 29 2022 *)

Extensions

a(15)-a(19) from Donovan Johnson, Dec 21 2008
a(20) from Giovanni Resta, Oct 28 2012

A104907 Numbers n such that d(n)*reversal(n)=sigma(n), where d(n) is number of positive divisors of n.

Original entry on oeis.org

1, 73, 861, 7993, 8241, 799993, 7999993, 44908500, 82000041, 293884500, 6279090751, 8200000041, 62698513951, 79999999993, 82000000041, 374665576800, 597921764310, 7999999999993, 8200000000041
Offset: 1

Views

Author

Farideh Firoozbakht, Apr 16 2005

Keywords

Comments

All primes of the form 8*10^n-7 are in the sequence, so 8*10^A099190-3 is a subsequence of this sequence. A105322 is this subsequence. Also if p=(2*10^n+1)/3 is prime then 123*p is in the sequence, so 123*A093170 is a subsequence of this sequence. A105323 is this subsequence.
a(20) > 10^13. - Giovanni Resta, Jul 13 2015

Examples

			Let p=8*10^n-7 be a prime so d(p)=2; reversal(p)=4*10^n-3 and sigma(p)
=8*10^n-6 hence d(p)*reversal(p)=sigma(p) and this shows that p
is in the sequence. 73,7993,799993 and 7999993 are such terms.
Also let q=(2*10^n+1)/3 be a prime, so 123*q=82*10^n+41; reversal
(123*q)=14*10^n+28; d(123*q)=8 and sigma(123*q)=168*q+168=112*10^n
+224 hence d(123*q)*reversal(123*q)=sigma(123*q) and this shows
that 123*q is in the sequence. 861,8241 and 82000041 are such terms.

Crossrefs

Cf. A056657, A093170, A096507, A099190, A105322, A105323, A105324.

Programs

  • Mathematica
    reversal[n_]:= FromDigits[Reverse[IntegerDigits[n]]]; Do[If[DivisorSigma[0, n]*reversal[n] == DivisorSigma[1, n], Print[n]], {n, 1125000000}]
Select[Range[8*10^6],DivisorSigma[0,#]IntegerReverse[#]==DivisorSigma[1,#]&] (* The program generates the first 7 terms of the sequence. *) (* Harvey P. Dale, Jan 31 2023 *)

Extensions

a(11)-a(15) from Donovan Johnson, Feb 06 2010
a(16) from Giovanni Resta, Feb 06 2014
a(17)-a(19) from Giovanni Resta, Jul 13 2015

A233467 Numbers n such that sigma(n) = reversal(3n).

Original entry on oeis.org

14, 134, 164, 1333334, 133333334, 929247534, 1333333334, 133333333334
Offset: 1

Views

Author

Farideh Firoozbakht, Mar 14 2014

Keywords

Comments

If m is a natural number and p=(2*10^m+1)/3 is prime then 2*p is in the sequence (see A096507).
a(9) > 10^12. - Giovanni Resta, Mar 15 2014

Examples

			sigma(14)=24=reversal(42)=reversal(3*14).

Crossrefs

Cf. A000203, A004086, A233465.

Programs

  • Mathematica
    rev[n_] := FromDigits@ Reverse@ IntegerDigits@n; Select[Range[2*10^6], DivisorSigma[1, #] == rev[3*#] &] (* Giovanni Resta, Mar 15 2014 *)

Extensions

a(6)-a(8) from Giovanni Resta, Mar 15 2014
Showing 1-10 of 14 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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