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A248006 Least positive integer m such that m + n divides phi(m*n), where phi(.) is Euler's totient function. 3
3, 4, 3, 6, 5, 8, 9, 6, 9, 4, 11, 7, 5, 16, 7, 9, 5, 12, 7, 18, 21, 8, 15, 13, 27, 14, 11, 10, 14, 32, 7, 14, 5, 12, 35, 10, 13, 24, 7, 14, 13, 11, 9, 42, 45, 16, 11, 30, 13, 12, 19, 27, 33, 8, 15, 22, 28, 4, 35, 28, 18, 64, 7, 14, 21, 28, 19, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,1

COMMENTS

Conjecture: For any n > 2, a(n) exists and a(n) <= n.

See also A248007 and A248008 for similar conjectures. - Zhi-Wei Sun, Sep 29 2014

The conjecture is true: One can show that 2*n divides phi(n^2) for all n > 2. So, a(n) is at most n. - Derek Orr, Sep 29 2014

a(n) >= 3 for all n. - Robert Israel, Sep 29 2014

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 3..10000

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

EXAMPLE

a(5) = 3 since 3 + 5 divides phi(3*5) = 8.

MAPLE

f:= proc(n)

local m;

for m from 3 do

  if numtheory:-phi(m*n) mod (m+n) = 0 then return m fi

od

end proc;

seq(f(n), n=3..100); # Robert Israel, Sep 29 2014

MATHEMATICA

Do[m=1; Label[aa]; If[Mod[EulerPhi[m*n], m+n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 3, 70}]

PROG

(PARI)

a(n)=m=1; while(eulerphi(m*n)%(m+n), m++); m

vector(100, n, a(n+2)) \\ Derek Orr, Sep 29 2014

CROSSREFS

Cf. A000010, A248004, A248007, A248008.

Sequence in context: A176058 A110738 A175028 * A197699 A005092 A136195

Adjacent sequences:  A248003 A248004 A248005 * A248007 A248008 A248009

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Sep 29 2014

STATUS

approved

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Last modified January 22 01:28 EST 2022. Contains 350481 sequences. (Running on oeis4.)