A248035 Least positive integer m such that m + n divides phi(m)^2 + phi(n)^2, where phi(.) is Euler's totient function.

1, 3, 2, 1, 15, 14, 3, 8, 9, 30, 30, 14, 7, 6, 5, 9, 3, 8, 55, 60, 9, 4, 83, 28, 25, 71, 9, 1, 24, 4, 43, 32, 1523, 30, 13, 9, 35, 3, 21, 24, 17, 18, 7, 8, 99, 166, 5, 4, 3, 32, 205, 6, 36, 18, 19, 19, 40, 78, 9, 8

Offset

1,2

Comments

Conjecture: a(n) exists for any n > 0. Moreover, a(n) <= n^2 except for n = 33.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Example

a(5) = 15 since 15 + 5 = 20 divides phi(15)^2 + phi(5)^2 = 8^2 + 4^2 = 80.
a(33) = 1523 since 1523 + 33 = 1556 divides phi(1523)^2 + phi(33)^2 = 1522^2 + 20^2 = 2316884 = 1489*1556.

Mathematica

Do[m=1; Label[aa]; If[Mod[EulerPhi[m]^2+EulerPhi[n]^2, m+n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 1, 60}]
lpim[n_]:=Module[{m=1, p2=EulerPhi[n]^2}, While[Mod[p2+EulerPhi[m]^2, m+n]!=0, m++]; m]; Array[lpim, 60] (* Harvey P. Dale, Nov 19 2020 *)

Crossrefs

Cf. A000010, A247975, A248007, A248036.

Sequence in context: A136217 A166884 A136220 * A088956 A106208 A350710

Adjacent sequences: A248032 A248033 A248034 * A248036 A248037 A248038

Keyword

nonn

Author

Zhi-Wei Sun, Sep 29 2014

Status

approved