A244440 Numbers n such that n + phi(n) is a power of 10.

68, 668, 6668, 67744, 72352, 666668, 7143040, 66666752, 71430400, 666666752, 714304000, 6666666668, 7143040000, 71430400000, 666666666668, 666666668032, 714304000000, 714499133440, 7143040000000, 7144991334400

Offset

1,1

Comments

All terms are even.

If p = (5^m*10^n+1)/3 is an odd prime then s = 2^(m+1)*p is in the sequence, because s+phi(s) = 10^(m+n).

For many integers m, p cannot be prime.

Take m=1, we get these terms: 68, 668, 6668, ... .

Take m=5, we get these terms: 666666666666666666666666688,

666666666666666666666666666666666666688, ... .

Take m=7, we get these terms: 66666752, 666666752, ... .

Take m=11, we get these terms: 666666668032, 666666666666666668032, ... .

Take m=13, we get these terms: 6666666666666666672128, 6666666666666666666672128, ... .

Take m=17, we get these terms: 66666666666666754048, 66666666666666666754048, ... .

...

From Robert Israel, Aug 04 2014: (Start)

Since n < n + phi(n) < 2*n, the most significant digit of n must be 5,6,7,8 or 9.

n is not divisible by the square of any prime other than 2 or 5. (End)

a(15) > 10^11. - Hiroaki Yamanouchi, Aug 27 2014

If n is in the sequence and 10 divides n then for every positive integer m, 10^m*n is also in the sequence. Since phi(10^m*n)+10^m*n = 10^m*(phi(n)+n). n=7143040 is the first such number. - Farideh Firoozbakht, Dec 09 2014

Numbers n such that phi(n) equals to 10^ceiling(log(10,n))-n. - Farideh Firoozbakht, Jan 01 2015

a(21) > 10^13. - Giovanni Resta, Jul 13 2015

Links

Table of n, a(n) for n=1..20.

Example

68+phi(68) = 68+32 = 10^2.
72352+phi(72352) = 72352+27648 = 10^5.

Maple

select(proc(n) local t; t:= n + numtheory:-phi(n); t = 10^ilog10(t) end proc, [$1..10^6]); # Robert Israel, Aug 04 2014

Mathematica

a244440[n_Integer] := Flatten[Position[Map[IntegerQ[Log10[# + EulerPhi[#]]] &, Range[n]], True]] (* Michael De Vlieger, Aug 03 2014 *)

Prog

(Python) from sympy import totient
[n for n in range(1, 10**7) if 10**(int(log10(n+totient(n)))) == n+totient(n)] # Chai Wah Wu, Aug 03 2014
(PARI) for(n=1, 10^7, v=digits(eulerphi(n)+n-1); if(vecmin(v)==9, print1(n, ", "))) \\ Derek Orr, Aug 03 2014
(PARI) for(n=1, 10^7, if (ispower(eulerphi(n)+n, , &m) && (m==10), print1(n, ", "))); \\ Michel Marcus, Aug 04 2014
(PARI) for(n=1, 10^9, my(t=eulerphi(n)+n, s=t/10^valuation(t, 10)); if (s==1, print1(n, ", "))); \\ Joerg Arndt, Aug 05 2014
(Magma) [n: n in [1..10^7] | 10^Ilog(10, s) eq s where s is n+EulerPhi(n)]; // Bruno Berselli, Aug 05 2014

Crossrefs

Cf. A000010, A011557.

Sequence in context: A281565 A281883 A282333 * A166398 A200876 A039507

Adjacent sequences: A244437 A244438 A244439 * A244441 A244442 A244443

Keyword

nonn,base,more

Author

Farideh Firoozbakht, Jul 30 2014

Extensions

a(8)-a(14) from Hiroaki Yamanouchi, Aug 27 2014

a(15)-a(20) from Giovanni Resta, Jul 13 2015

Status

approved