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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 *)

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

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.

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Zhi-Wei Sun, <a href="/A248035/b248035.txt">Table of n, a(n) for n = 1..10000</a>

a(5) = 15 since 15 + 5 = 20 divides phi(15)^2 + phi(5)^2 = 8^2 + 4^2 = 80.

Cf. A000010, A247975, A248007, A248036.

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

Conjecture: a(n) exists for any n > 0.

Zhi-Wei Sun, <a href="http://arxiv.org/abs/1409.5685">A new theorem on the prime-counting function</a>, arXiv:1409.5685, 2014.

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}]

Cf. A000010, A247975, A248036.

allocated for Zhi-Wei Sun

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

1,2

Conjecture: a(n) exists for any n > 0.

Zhi-Wei Sun, <a href="http://arxiv.org/abs/1409.5685">A new theorem on the prime-counting function</a>, arXiv:1409.5685, 2014.

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}]

Cf. A000010, A247975.

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Zhi-Wei Sun, Sep 29 2014

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allocated for Zhi-Wei Sun

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