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 1-4 of 4 results.

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

A082060 Numbers n such that n and phi(n) have the same distinct decimal digits.

Original entry on oeis.org

1, 21, 63, 101, 233, 291, 502, 677, 1021, 1031, 1051, 1061, 1091, 1201, 1226, 1301, 1601, 1801, 1901, 2011, 2201, 2333, 2383, 2393, 2518, 2633, 2677, 2700, 2767, 2817, 2833, 2991, 3011, 3023, 3122, 3203, 3253, 3323, 3623, 3677, 3767, 3823, 3923, 3989
Offset: 1

Views

Author

Labos Elemer, Apr 04 2003

Keywords

Comments

Contains A113781 as a subsequence. - M. F. Hasler, Nov 28 2007
Numbers n such that n and phi(n) have the same decimal digits = A115921. - Jaroslav Krizek, Nov 13 2014

Examples

			n=502 is a member since phi[502]=250

Links

Crossrefs

Cf. A000005, A000010, A082059, A115921, A113781, A113622, A069215, A072395, A113781.

Programs

  • Mathematica
    Select[Range[4000],Union[IntegerDigits[#]]==Union[IntegerDigits[ EulerPhi[ #]]]&] (* Harvey P. Dale, Jan 31 2022 *)
  • PARI
    for(n=1,10^4,if(Set(Vec(Str(n)))==Set(Vec(Str(eulerphi(n)))),print1(n", "))) \\ M. F. Hasler, Nov 28 2007
  • Python
    from sympy import totient
A082060_list = [n for n in range(1,10**4) if set(str(totient(n))) == set(str(n))] # Chai Wah Wu, Dec 13 2015

Extensions

Definition and comment corrected by Jaroslav Krizek, Nov 13 2014

A251808 Numbers n such that if m = reverse(phi(n)) then n = reverse(phi(m)).

Original entry on oeis.org

1, 21, 63, 291, 2744, 2991, 6102, 6711, 46676013, 69460293, 272543398, 896172631
Offset: 1

Views

Author

Paolo P. Lava, Dec 09 2014

Keywords

Comments

Fixed points of the transform n -> reverse(phi(reverse(phi(n)))).
A072395 is a subset of this sequence.
No further terms up to 10^9. - Felix Fröhlich, Dec 30 2014

Examples

			phi(2744) = 1176 and reverse(1176) = 6711;
phi(6711) = 4472 and reverse(4472) = 2744;

Crossrefs

Cf. A000010, A072395.

Programs

  • Maple
    with(numtheory):T:=proc(w) local x,y,z; x:=0; y:=w;
for z from 1 to ilog10(w)+1 do x:=10*x+(y mod 10); y:=trunc(y/10); od; x; end:
P:=proc(q) local k,n; for n from 1 to q do if n=T(phi(T(phi(n))))
then print(n); fi; od; end: P(10^12);
  • PARI
    for(n=1, 1e9, m=eval(concat(Vecrev(Str(eulerphi(n))))); if(n==eval(concat(Vecrev(Str(eulerphi(m))))), print1(n, ", "))) \\ Felix Fröhlich, Dec 30 2014

Extensions

a(9)-a(12) from Felix Fröhlich, Dec 30 2014
Showing 1-4 of 4 results.

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