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.
%I A244440 #53 Sep 08 2022 08:46:08 %S A244440 68,668,6668,67744,72352,666668,7143040,66666752,71430400,666666752, %T A244440 714304000,6666666668,7143040000,71430400000,666666666668, %U A244440 666666668032,714304000000,714499133440,7143040000000,7144991334400 %N A244440 Numbers n such that n + phi(n) is a power of 10. %C A244440 All terms are even. %C A244440 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). %C A244440 For many integers m, p cannot be prime. %C A244440 Take m=1, we get these terms: 68, 668, 6668, ... . %C A244440 Take m=5, we get these terms: 666666666666666666666666688, %C A244440 666666666666666666666666666666666666688, ... . %C A244440 Take m=7, we get these terms: 66666752, 666666752, ... . %C A244440 Take m=11, we get these terms: 666666668032, 666666666666666668032, ... . %C A244440 Take m=13, we get these terms: 6666666666666666672128, 6666666666666666666672128, ... . %C A244440 Take m=17, we get these terms: 66666666666666754048, 66666666666666666754048, ... . %C A244440 ... %C A244440 From _Robert Israel_, Aug 04 2014: (Start) %C A244440 Since n < n + phi(n) < 2*n, the most significant digit of n must be 5,6,7,8 or 9. %C A244440 n is not divisible by the square of any prime other than 2 or 5. (End) %C A244440 a(15) > 10^11. - _Hiroaki Yamanouchi_, Aug 27 2014 %C A244440 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 %C A244440 Numbers n such that phi(n) equals to 10^ceiling(log(10,n))-n. - _Farideh Firoozbakht_, Jan 01 2015 %C A244440 a(21) > 10^13. - _Giovanni Resta_, Jul 13 2015 %e A244440 68+phi(68) = 68+32 = 10^2. %e A244440 72352+phi(72352) = 72352+27648 = 10^5. %p A244440 select(proc(n) local t; t:= n + numtheory:-phi(n); t = 10^ilog10(t) end proc, [$1..10^6]); # _Robert Israel_, Aug 04 2014 %t A244440 a244440[n_Integer] := Flatten[Position[Map[IntegerQ[Log10[# + EulerPhi[#]]] &, Range[n]], True]] (* _Michael De Vlieger_, Aug 03 2014 *) %o A244440 (Python) from sympy import totient %o A244440 [n for n in range(1,10**7) if 10**(int(log10(n+totient(n)))) == n+totient(n)] # _Chai Wah Wu_, Aug 03 2014 %o A244440 (PARI) for(n=1,10^7,v=digits(eulerphi(n)+n-1);if(vecmin(v)==9,print1(n,", "))) \\ _Derek Orr_, Aug 03 2014 %o A244440 (PARI) for(n=1,10^7,if (ispower(eulerphi(n)+n,,&m) && (m==10), print1(n, ", "))); \\ _Michel Marcus_, Aug 04 2014 %o A244440 (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 %o A244440 (Magma) [n: n in [1..10^7] | 10^Ilog(10,s) eq s where s is n+EulerPhi(n)]; // _Bruno Berselli_, Aug 05 2014 %Y A244440 Cf. A000010, A011557. %K A244440 nonn,base,more %O A244440 1,1 %A A244440 _Farideh Firoozbakht_, Jul 30 2014 %E A244440 a(8)-a(14) from _Hiroaki Yamanouchi_, Aug 27 2014 %E A244440 a(15)-a(20) from _Giovanni Resta_, Jul 13 2015