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 1-6 of 6 results.

A089675 Numbers k such that 10^k - 3 is prime.

Original entry on oeis.org

1, 2, 3, 17, 140, 990, 1887, 3530, 5996, 13820, 21873, 26045, 87720, 232599, 480684, 538640
Offset: 1

Views

Author

Michael Gottlieb (mzrg(AT)verizon.net), Jan 05 2004

Keywords

Comments

Numbers k such that 9*R_k - 2 is a prime number, where R_k = 11...1 is the repunit (A002275) of length k.
If k is in the sequence (10^k-3 is prime) and m=3*(10^k-3) then phi(m)=reversal(m), i.e., m is in the sequence A069215. - Farideh Firoozbakht, Dec 25 2004
No further terms for k <= 407197, see Kamada link.

Examples

			10^2 - 3 = 97 is a prime number (in fact all terms are the largest less than 10^k).

Links

Crossrefs

Cf. A002275, A056662, A069215, A093172, A101700.

Programs

  • Mathematica
    m = 1000; For[n = 1, n < m, If[PrimeQ[10^n - 3], Print[n]]; n++]

Formula

a(n) = A056662(n) + 1.

Extensions

a(8) from Robert G. Wilson v, Jan 14 2004
a(9) and a(10) from Gabriel Cunningham (gcasey(AT)mit.edu), Mar 06 2004
a(11) from Gabriel Cunningham (gcasey(AT)mit.edu), Mar 13 2004
a(12) from Henri Lifchitz.
Edited by Patrick De Geest, Dec 28 2004
Edited by Ray Chandler, Dec 23 2010
a(15) from Paul Bourdelais, Jan 06 2021
a(16) from Paul Bourdelais, Jan 28 2021

A069215 Numbers n such that phi(n) = reversal(n).

Original entry on oeis.org

1, 21, 63, 270, 291, 2991, 6102, 46676013, 69460293, 2346534651, 6313047393, 23400000651, 80050617822, 234065340651, 234659934651, 2340000000651, 2530227348360, 2934000006591
Offset: 1

Views

Author

Joseph L. Pe, Apr 11 2002

Keywords

Comments

If 10^n-3 is prime (n is in the sequence A089765) and m=3*(10^n-3) then m is in this sequence, for example 299999999999999991 is a term of this sequence because 299999999999999991=3*(10^17-3) and 17 is in the sequence A089675. So 3*(10^A089675-3) is a subsequence of this sequence, A101700 is this subsequence. - Farideh Firoozbakht, Dec 26 2004
A072395 is a subsequence of this sequence. If m is in the sequence and 10 doesn't divide m then reversal(m) is in the sequence A085331, so see Comments on A085331. - Farideh Firoozbakht, Jan 09 2005
If p=(79*10^(4n+1)-83)/101 is prime then 3p is in the sequence. The proof is easy. 21, 2346534651 & 3*(79*10^2697-83)/101 are the first three such terms. - Farideh Firoozbakht, Apr 22 2008, Aug 16 2008
a(19) > 10^13. - Giovanni Resta, Aug 07 2019

Examples

			phi(291) = 192.
phi(6102) = 2016 = reversal(6102), so 6102 belongs to the sequence.

Crossrefs

Cf. A101700, A004086, A000010, A085331, A072395, A101700, A102278.

Programs

  • Mathematica
    Do[If[EulerPhi[n] == FromDigits[Reverse[IntegerDigits[n]]], Print[n]], {n, 1, 10^5}]
  • PARI
    for( n=1,1e9, A004086(n)==eulerphi(n) & print1(n","))

Extensions

More terms from Farideh Firoozbakht, Aug 31 2004
One more term from Farideh Firoozbakht, Jan 09 2005
a(11)-a(13) from Donovan Johnson, Feb 03 2012
a(14)-a(15) from Giovanni Resta, Oct 28 2012
a(16)-a(18) from Giovanni Resta, Aug 07 2019

A085331 Numbers n such that phi(rev(n))=n.

Original entry on oeis.org

1, 12, 36, 192, 1992, 2016, 31067664, 39206496, 1564356432, 3937403136, 15600000432, 22871605008, 156043560432, 156439956432, 1560000000432, 1956000004392
Offset: 1

Views

Author

Labos Elemer, Jul 04 2003

Keywords

Comments

rev(2*(10^k-4)) = 3*(10^k-3). If 10^k-3 is prime, then phi(3*(10^k-3)) = 2*(10^k-4), so 2*(10^k-4) is a term. 10^1-3=7 is prime, so 2*(10^1-4)=12 is a term, a(2). 10^2-3=97 is prime, so 2*(10^2-4)=192 is a term, a(4). 10^3-3=997 is prime, so 2*(10^3-4)=1992 is a term, a(5). 10^17-3 is prime, so 2*(10^17-4)=199999999999999992 is a term. 10^140-3 is prime, so 2*(10^140-4) is a term. 10^990-3 is prime, so 2*(10^990-4) is a term. Conjecture: sequence is infinite. - Ray Chandler, Jul 20 2003
Let f(m,n,r,t)=((9).(m).78.(0)(n).21.(9)(m))(r).(9)(t).7 where m, n, r & t are nonnegative integers; dot between numbers means concatenation and "(m)(n)" means number of m's is n. If r*t=0 & p=f(m,n,r,t) is prime then reversal(3*p) = 1.((9)(m).56.(0)(n).43.(9)(m))(r).(9)(t).2 is in the sequence. For example p1=f(0,0,0,0)=7 so reversal(3*p1) = 12 is in the sequence, p2=f(0,0,2,0)=(7821)(2).7=782178217 so reversal(3*p2) = 1.(5643)(2).2 = 1564356432 is in the sequence & p3=f(0,0,674,0) so reversal(3*p3) = 1.(5643)(674).2 is in the sequence. Primes of the form f(m,n,r,t) are a generalized form of primes of the form 10^j-3 that were already related to this sequence by Ray Chandler. For all n, A085331(n) = reversal(A072395(n)). - Farideh Firoozbakht, Jan 08 2005
The list is complete through 2050000000. - Farideh Firoozbakht, Jan 15 2005
a(13) > 10^11. - Donovan Johnson, Feb 03 2012
a(17) > 10^13. - Giovanni Resta, Aug 06 2019

Examples

			phi[{1,21,63,291,2991,6102}] = {1,12,36,192,1992,2016}

Crossrefs

Cf. A000010, A004086, A069215, A101700, A102278.

Programs

  • Mathematica
    v = {1}; Do[ If[ n == EulerPhi[ FromDigits[ Reverse[ IntegerDigits [ n ] ] ] ], v = Append[ v, n ]; Print[ v ], If[ Mod[ n, 1000000 ] == 0, Print[ -n ] ] ], {n, 2, 2050000000, 2} ] (Firoozbakht)

Extensions

The terms 31067664, 39206496, 1564356432 are from Farideh Firoozbakht, Jan 08 2005
a(10)-a(12) from Donovan Johnson, Feb 03 2012
a(13)-a(16) from Giovanni Resta, Aug 06 2019

A086947 Numbers k such that R(k+9) = 3.

Original entry on oeis.org

21, 291, 2991, 29991, 299991, 2999991, 29999991, 299999991, 2999999991, 29999999991, 299999999991, 2999999999991, 29999999999991, 299999999999991, 2999999999999991, 29999999999999991, 299999999999999991, 2999999999999999991, 29999999999999999991, 299999999999999999991
Offset: 1

Views

Author

Ray Chandler, Jul 24 2003

Keywords

Comments

If k is in this sequence then Reverse(k) = (2/3)*k - 2. Also A101703 is the sequence of all numbers k such that Reverse(k) = (2/3)*k - 2. So this sequence is a subsequence of A101703. - Farideh Firoozbakht, Dec 30 2004

Links

Crossrefs

Cf. A004086, A085331, A086948, A101700, A101703, A173833.

Programs

  • Magma
    [3*(10^n-3): n in [1..25] ]; // Vincenzo Librandi, Aug 22 2011
  • Mathematica
    Table[3*(10^n-3), {n, 17}]
Table[FromDigits[PadRight[{3},n,0]],{n,2,20}]-9 (* Harvey P. Dale, Nov 27 2012 *)

Formula

a(n) = 3*(10^n - 3).
R(a(n)) = A086948(n).
From Chai Wah Wu, Aug 01 2020: (Start)
a(n) = 11*a(n-1) - 10*a(n-2) for n > 2.
G.f.: x*(60*x + 21)/((x - 1)*(10*x - 1)). (End)
From Elmo R. Oliveira, May 01 2025: (Start)
E.g.f.: 3*(2 - 3*exp(x) + exp(10*x)).
a(n) = 3*A173833(n). (End)

A101703 Numbers n such that reversal(n) = (2/3)*n - 2.

Original entry on oeis.org

21, 291, 885, 2991, 29991, 234651, 299991, 2340651, 2999991, 8221845, 23400651, 29346591, 29999991, 234000651, 293406591, 299999991, 2340000651, 2346534651, 2934006591, 2993465991, 2999999991, 23400000651, 23465934651, 29340006591, 29934065991, 29999999991, 82277815845, 234000000651
Offset: 1

Views

Author

Farideh Firoozbakht, Dec 31 2004

Keywords

Comments

Numbers of the form 3*(10^n-3) are in the sequence, so A086947 is an infinite subsequence of this sequence. Also A101700 is a subsequence of this sequence.
Let f(r,s,t,z) = 2.(9)(r+s).(34.(0)(t).65)(z).(9)(s).1 where the dot between numbers means concatenation and "(m)(n)" means number of m's is n, for example f(0,2,1,3)= 299340653406534065991, it is interesting that all numbers of the form f(r,s,t,z) where r, s, t & z are nonnegative integers and r*z=0 are in this sequence.
Except for 885 & 8221845 all known terms of this sequence are of the form f(r,s,t,z).
For all r, s & t we have f(r,s,t,0)=f(r,s,0,0)=f(r+2s,0,0,0)=A086947(r+2s+1)= 3*(10^(r+2s+1)-3).
a(1) = 21 = f(0,0,0,0), a(2) = 291 = f(1,0,0,0), a(4) = 2991 = f(2,0,0,0) = f(0,1,0,0), a(5) = 29991 = f(3,0,0,0) = f(1,1,0,0), a(6) = 234651 = f(0,0,0,1), a(7) = 299991 = f(4,0,0,0) = f(0,2,0,0), a(8) = 2340651 = f(0,0,1,1), etc. Next term is greater than 11*10^8.
From David Wasserman, Mar 27 2008: (Start)
234653406534651 is a term that doesn't fit the f(r,s,t,z) format.
We may redefine f so that t is a vector of length z, which must be symmetrical to produce a member. For example f(0,0,[0,1,0],3) = 234653406534651 is a member, but f(0,0,[1,0,0],3) = 234065346534651 is not a member.
23465934651 is another member that doesn't fit the pattern. In general there may be any number of 9's between a 5 and a 3, provided that the 9's are symmetrical. So 2346593465934651 is a member, but 23465993465934651 is not. (End)

Examples

			f(0,1,2,3) = 2934006534006534006591 is in the sequence because reversal(2934006534006534006591) = 1956004356004356004392 = (2/3)*2934006534006534006591-2.

Links

Crossrefs

Cf. A086947, A069215, A101700, A101704.

Programs

  • Mathematica
    Do[If[FromDigits[Reverse[IntegerDigits[n]]] == 2/3*n - 2, Print[n]], {n, 1100000000}]

Extensions

More terms from David Wasserman, Mar 27 2008

A102278 Numbers k such that 78*10^k + 217 is prime.

Original entry on oeis.org

1, 2, 8, 10, 13, 21, 22, 36, 57, 80, 149, 484, 505, 642, 806, 974, 1674, 34177
Offset: 1

Views

Author

Farideh Firoozbakht, Jan 04 2005

Keywords

Comments

If n is a term greater than 2 in this sequence and m = 3*(78*10^n + 217) then phi(m) = reversal(m) (m is in the sequence A069215) because phi(m) = 2*(78*10^n + 216) = 156*10^n + 432 = reversal(234*10^n + 651) = reversal(m).
For example since 8>2 & 8 is in this sequence, for m = 3* (78*10^8 + 217) = 23400000651 phi(m) = reversal(m), so 23400000651 is a term of A069215.
Let f(n,m,r,t) = ((9)(n).78.(0)(m).21.(9)(n))(r).(9)(t).7 where dot between numbers means concatenation and "(m)(n)" means number of m's is n.
In fact I proved that for nonnegative integers n, m, r & t such that r*t = 0 if p = f(n,m,r,t) is prime then phi(3*p) = reversal (3*p). (3*p is in the sequence A069215, some special cases:
Case I, p = f(0,0,0,n-1) = (9)(n-1).7 = 10^n - 3 (see A089675). Case II, p = f(0,n-3,0,0) = 78.(0)(n-3).217 = 78*10^n + 217. Case III, p = f(0,0,n,0) = (7821)(n).7. In this case I found only three such prime p1 = (78217)(0).7 = 7, p2 = (7821)(2).7 = 782178217 & p3 = (7821)(674).7, p3 is a prime with length 2697.
Next term is greater than 8280.
Next term is greater than 24000. - Michael S. Branicky, Mar 22 2023

Examples

			8 is in the sequence because 78.(8-3)(0).217 = 7800000217 is prime.

Crossrefs

Cf. A089675, A069215, A085331, A101700.

Programs

  • Mathematica
    Do[If[PrimeQ[78*10^n + 217], Print[n]], {n, 8280}]
  • PARI
    is(n)=ispseudoprime(78*10^n+217) \\ Charles R Greathouse IV, May 22 2017

Extensions

a(18) from Michael S. Branicky, Oct 15 2024
Showing 1-6 of 6 results.

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