A097647 -id:A097647 - cp's OEIS frontend

A281880 Non-palindromic numbers k such that phi(k) | phi(R(k)), where R(k) is the digits reversal of k.

12, 15, 18, 19, 36, 37, 56, 124, 126, 132, 165, 168, 178, 189, 190, 192, 198, 199, 219, 234, 238, 298, 308, 348, 387, 396, 418, 427, 429, 468, 506, 518, 724, 756, 924, 1004, 1066, 1078, 1089, 1094, 1107, 1143, 1209, 1212, 1314, 1332, 1358, 1364, 1386, 1445, 1452

Offset: 1

Paolo P. Lava, Feb 01 2017

			a(1) = 12 because phi(21) / phi(12) = 12 / 4 = 3;
a(2) = 15 because phi(51) / phi(15) = 32 / 8 = 4;
a(3) = 18 because phi(81) / phi(18) = 54 / 6 = 9.

Robert Israel, Table of n, a(n) for n = 1..1500
Paolo P. Lava, First 250 terms with the ratio phi(R(k))/phi(k)

Cf. A000010, A004086, A097647, A281879.

with(numtheory): T:=proc(w) local x, y, z; x:=w; y:=0;
for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end:
P:=proc(q) local n; for n from 1 to q do
if n<>T(n) then if type(phi(T(n))/phi(n),integer) then print(n); fi; fi; od; end: P(10^6);

Select[Range@ 1500, Function[k, And[Reverse@ # != #, Divisible[EulerPhi[FromDigits@ Reverse@ #], EulerPhi@ k]] &@ IntegerDigits@ k]] (* Michael De Vlieger, Feb 04 2017 *)

A097648 a(n) is the least non-palindromic number m such that phi(m)=phi(reversal(m))=4*10^(n+2), or 0 if no such number exists.

Original entry on oeis.org

10040, 110440, 1014040, 11154440, 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, 0, 0, 0, 0, 0, 0

Offset: 1

Farideh Firoozbakht, Sep 04 2004

It seems that 10 divides all terms of this sequence.

			a(4)=11154440 because phi(11154440)=phi(04445111)=4000000 and 11154440 is the earliest non-palindromic number with this property.

C. Rivera, f(p)=f(p') , puzzle 282

Subsequence of A097647.

a[n_]:=(For[m=4*10^(n+2), !(m!=FromDigits[Reverse[IntegerDigits[m]]] &&EulerPhi[m]==EulerPhi[FromDigits[Reverse[IntegerDigits [m]]]]==4*10^(n+2)), m++ ];m);Do[Print[a[n]], {n, 4}]

a[n_]:=(For[m=4*10^(n+2), !(m!=FromDigits[Reverse[IntegerDigits[m]]] &&EulerPhi[m]==EulerPhi[FromDigits[Reverse[IntegerDigits [m]]]]==4*10^(n+2)), m++ ];m)

Better definition and more terms from David Wasserman, Dec 28 2007

a(27)-a(49) from Max Alekseyev, Oct 17 2008; Aug 15 2013; Jun 14 2022

A229903 a(n) = (190/99)*(100^A001651(n)-1).

Original entry on oeis.org

190, 19190, 191919190, 19191919190, 191919191919190, 19191919191919190, 191919191919191919190, 19191919191919191919190, 191919191919191919191919190, 19191919191919191919191919190, 191919191919191919191919191919190

Offset: 1

Jahangeer Kholdi, Oct 17 2013

This sequence is a subsequence of A097647. Because if 3 does not divide m then gcd(100^m-1,19*91)=1 and we have phi(190*(100^m-1)/99)=phi(190)*phi((100^m-1)/9)=phi(91)*phi((100^m-1)/99)=phi(91*(100^m-1)/99)=phi(reversal(190*(100^m-1)/99)).

Index entries for linear recurrences with constant coefficients, signature (1,1000000,-1000000).

Cf. A000010, A001651, A004086, A097647.

Table[190/99*(100^Floor[(3n-1)/2]-1),{n,11}]

Vec(190*x*(10000*x^2+100*x+1)/((x-1)*(1000*x-1)*(1000*x+1)) + O(x^100)) \\ Colin Barker, Nov 01 2013

a(n) = (100/99)*(100^floor((3n-1)/2)-1).

G.f.: 190*x*(10000*x^2+100*x+1) / ((x-1)*(1000*x-1)*(1000*x+1)). - Colin Barker, Nov 01 2013

A259077 Non-palindromic composite numbers such that n' = [Rev(n)]', where n' is the arithmetic derivative of n.

Original entry on oeis.org

366, 663, 3245, 3685, 5423, 5863, 8178, 8718, 14269, 15167, 16237, 18449, 18977, 36679, 73261, 76151, 77981, 94481, 96241, 97663, 140941, 149041, 150251, 152051, 196891, 198691, 302363, 308459, 319853, 335148, 358913, 363203, 841533, 921239, 932129, 954803, 958099, 990859

Offset: 1

Paolo P. Lava, Jun 18 2015

			366' = 311 = 663';
3245' = 999 = 5423'; etc.

Cf. A003415, A004086, A085329, A097647.

with(numtheory): T:=proc(w) local x,y,z; x:=w; y:=0;
for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10);
od; y; end: P:=proc(q) local a,b,p,n;
for n from 1 to q do if not isprime(n) then if n<>T(n) then a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
b:=T(n)*add(op(2,p)/op(1,p),p=ifactors(T(n))[2]);
if a=b then print(n); fi; fi; fi; od; end: P(10^9);

Solutions to A003415(n) = A003415(A004086(n)), with A004086(n) <> n.

A259365 Numbers n such that phi(Rev(n)) - phi(n) = n.

Original entry on oeis.org

28, 139872, 2764928, 34141176, 329774256

Offset: 1

Paolo P. Lava, Jun 25 2015

a(6) > 3*10^10. - Giovanni Resta, Jun 26 2015

			phi(82) - phi(28) = 40 - 12 = 28;
phi(278931) - phi(139872) = 184032 - 44160 = 139872; etc.

Cf. A000010, A004086, A097647.

with(numtheory): T:=proc(w) local x,y,z; x:=w; y:=0;
for z from 1 to ilog10(x)+1 do
y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end:
P:=proc(q) local a,j,k,n; for n from 1 to q do
if phi(T(n))-phi(n)=n then print(n); fi; od; end: P(10^9);

Solutions of the equation A000010( A004086(n) ) - A000010(n) = n.

cp's OEIS Frontend

A281880 Non-palindromic numbers k such that phi(k) | phi(R(k)), where R(k) is the digits reversal of k.

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A097648 a(n) is the least non-palindromic number m such that phi(m)=phi(reversal(m))=4*10^(n+2), or 0 if no such number exists.

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A229903 a(n) = (190/99)*(100^A001651(n)-1).

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A259077 Non-palindromic composite numbers such that n' = [Rev(n)]', where n' is the arithmetic derivative of n.

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A259365 Numbers n such that phi(Rev(n)) - phi(n) = n.

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