A259916 - OEIS

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

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OFFSET

1,3

COMMENTS

The conjecture in A259915 implies that a(n) exists for any n > 0.

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.

EXAMPLE

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.

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 *)

CROSSREFS

Cf. A000010, A000203, A000290, A259789, A259915.

Sequence in context: A125549 A104876 A349944 * A050516 A376662 A100670

Adjacent sequences: A259913 A259914 A259915 * A259917 A259918 A259919

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jul 08 2015

STATUS

approved