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A259915
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Least positive integer k such that phi(k) and sigma(k*n) are both squares, where phi(.) is Euler's totient function and sigma(m) is the sum of all positive divisors of m.
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5
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1, 85, 1, 273, 34, 85, 10, 364, 250, 17, 2, 2223, 204, 5, 34, 546, 10, 60, 680, 60, 10, 1, 5, 364, 48, 34, 40, 451, 136, 17, 10, 273, 2, 5, 2, 5089, 10570, 1020, 451, 10, 60, 5, 1970, 114, 114, 17, 2, 4446, 185, 8, 10, 17, 5, 546, 17, 285, 63, 204, 8, 540, 816, 5, 57, 147744, 2761, 1, 505, 451, 5, 1
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OFFSET
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1,2
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COMMENTS
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Conjecture: a(n) exists for any n > 0. In general, every positive rational number r can be written as m/n, where m and n are positive integers with phi(m) and sigma(n) both squares of integers.
For example, 4/5 = 136/170 with phi(136) = 8^2 and sigma(170) = 18^2, and 5/4 = 1365/1092 with phi(1365) = 24^2 and sigma(1092) = 56^2.
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REFERENCES
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Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
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LINKS
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EXAMPLE
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a(2) = 85 since phi(85) = 64 = 8^2 and sigma(85*2) = 324 = 18^2.
a(673) = 3451030792 since phi(3451030792) = 1564993600 = 39560^2 and sigma(3451030792*673) = sigma(2322543723016) = 4768807737600 = 2183760^2.
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MATHEMATICA
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SQ[n_]:=IntegerQ[Sqrt[n]]
sigma[n_]:=DivisorSigma[1, n]
Do[k=0; Label[aa]; k=k+1; If[SQ[EulerPhi[k]]&&SQ[sigma[k*n]], Goto[bb], Goto[aa]]; Label[bb]; Print[n, " ", k]; Continue, {n, 1, 70}]
(* Second program: *)
Table[k = 1; While[Times @@ Boole@ Map[IntegerQ@ Sqrt@ # &, {EulerPhi@ k, DivisorSigma[1, k n]}] < 1, k++]; k, {n, 70}] (* Michael De Vlieger, May 04 2017 *)
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PROG
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(Perl) use ntheory ":all"; for my $n (1..100) { my $k = 1; $k++ until is_power(euler_phi($k), 2) && is_power(divisor_sum($k*$n), 2); say "$n $k" } # Dana Jacobsen, May 04 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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