A259916 Least positive integer k such that sigma(k) and phi(k*n) are both squares, where sigma(k) is the sum of all positive divisors of k, and phi(.) is Euler's totient function.
1, 1, 210, 3, 1, 170, 81, 1, 70, 1, 400, 1, 210, 81, 357, 3, 1, 119, 3, 3, 3, 651, 1990, 170, 66, 70, 210, 884, 3810, 357, 1066, 1, 217, 1, 81, 3, 1, 3, 70, 1, 22, 3, 1624, 217, 119, 3383, 11510, 1, 364, 22, 210, 81, 8743, 170, 510, 81, 1, 1270, 2902, 1, 385, 1155, 1, 3, 357, 217, 966, 3, 4179, 81
Offset: 1
Keywords
Examples
a(3) = 210 since sigma(210) = 576 =24^2 and phi(210*3) = 144 = 12^2. a(719) = 42862647 since sigma(42862647) = 58003456 = 7616^2 and phi(42862627*719) = phi(30818243193) = 20210602896 = 142164^2.
References
- 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.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
- Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
Programs
-
Mathematica
SQ[n_]:=IntegerQ[Sqrt[n]] sigma[n_]:=DivisorSigma[1, n] Do[k=0; Label[aa]; k=k+1; If[SQ[sigma[k]]&&SQ[EulerPhi[k*n]], Goto[bb], Goto[aa]]; Label[bb]; Print[n, " ", k]; Continue, {n, 1, 70}] lpi[n_]:=Module[{k=1},While[!IntegerQ[Sqrt[DivisorSigma[1,k]]]|| !IntegerQ[ Sqrt[ EulerPhi[ n*k]]],k++];k]; Array[lpi,70] (* Harvey P. Dale, Jul 17 2020 *)
Comments