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

Author

Farideh Firoozbakht, Jul 30 2014

Keywords

• nonn
• base
• more

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

Examples

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

Crossrefs

Cf. A000010, A011557.

Programs

Magma

[n: n in [1..10^7] | 10^Ilog(10,s) eq s where s is n+EulerPhi(n)]; // Bruno Berselli, Aug 05 2014

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 *)

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

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

Extensions

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

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