A259916 -id:A259916 - cp's OEIS frontend

A259915 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.

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

Offset: 1

Zhi-Wei Sun, Jul 08 2015

nonn

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.

			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.

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.

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..150
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000010, A000203, A000290, A259789, A259916.

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 *)

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

A334337 Least positive integer m such that phi(m*n) is a cube, where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 1, 5, 4, 3, 4, 37, 2, 37, 2, 101, 2, 19, 37, 1, 1, 5, 36, 13, 1, 19, 101, 13333, 1, 55, 19, 13, 19, 985, 1, 1057, 4, 401, 4, 73, 18, 7, 13, 9, 4, 275, 18, 2649, 401, 9, 13333, 169285, 4, 1813, 50, 4, 73, 3385, 12, 25, 73, 7, 788, 40371, 4, 3737, 1057, 12, 2, 37, 401, 4357, 2, 6537, 73, 5401, 9, 35, 7, 25, 7, 3737, 9, 48673, 2

Offset: 1

Zhi-Wei Sun, Apr 23 2020

nonn

Conjecture: a(n) exists for any n > 0. In other words, for any positive integer n, there is a positive integer m with phi(m*n) equal to a cube.
We note that there is no positive integer m <= 10^8 with phi(107*m) equal to a fourth power.
The author's conjecture in A259915 implies that for any positive integer n there is a positive integer m with phi(m*n) equal to a square.
See also A334339 for a similar conjecture.

			a(3) = 5 with phi(3*5) = 2^3.
a(7) = 37 with phi(7*37) = 6^3.
a(863) = 21176773 with phi(863*21176773) = 17293606056 = 2586^3.

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
P. Pollack and C. Pomerance, Square values of Euler's function, Bull. London Math. Soc. 46 (2014), 403-414. Alternative link.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. See also arXiv, arXiv:1211.1588 [math.NT], 2012-2017. (Cf. Conjecture 4.5.)

Cf. A000010, A000578, A039771, A259915, A259916, A280988, A334339.

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
phi[n_]:=phi[n]=EulerPhi[n];
tab={};Do[m=0;Label[aa];m=m+1;If[CQ[phi[m*n]],tab=Append[tab,m],Goto[aa]],{n,1,80}];tab

a(n) = my(m=1); while (!ispower(eulerphi(n*m), 3), m++); m; \\ Michel Marcus, Apr 23 2020

A334350 Least positive integer m relatively prime to n such that phi(mn) = phi(m)phi(n) is a fourth power, where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 1, 16, 15, 8, 703, 247, 5, 247, 489, 1255, 5, 109, 247, 4, 3, 1, 247, 73, 3, 109, 1255, 13315, 163, 753, 109, 73, 109, 1373, 163, 27331, 1, 625, 1, 81, 109, 57, 73, 1295, 1, 251, 109, 74663, 625, 949, 13315, 1557377, 1, 74663, 753, 16, 81, 175765, 73, 251, 81, 37, 1373, 243895, 1

Offset: 1

Zhi-Wei Sun, Apr 24 2020

nonn

Conjecture: For any positive integers k and m, there is a positive integer n relatively prime to m such that phi(m*n) = phi(m)*phi(n) is a k-th power.
This conjecture implies that a(n) exists for every n = 1,2,3,....
See also A334353 for a similar conjecture involving the sigma function (A000203).
a(n) = 1 if and only if n is in A078164. - Charles R Greathouse IV, Apr 24 2020

			a(3) = 16 with gcd(3,16) = 1 and phi(3*16) = phi(3)*phi(16) = 2*8 = 2^4.
a(167) = 370517977 with gcd(167, 370517977) = 1 and phi(167*370517977) = phi(167)*phi(370517977) = 166*370517976 = 61505984016 = 498^4.

Zhi-Wei Sun, Table of n, a(n) for n = 1..226
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. See also arXiv, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A000010, A000203, A000583, A039770, A039771, A259915, A259916, A334337, A280988, A334339, A334353, A078164.

QQ[n_]:=QQ[n]=IntegerQ[n^(1/4)];
phi[n_]:=phi[n]=EulerPhi[n];
tab={};Do[m=0;Label[aa];m=m+1;If[GCD[m,n]==1&&QQ[phi[m]*phi[n]],tab=Append[tab,m],Goto[aa]],{n,1,60}];tab

a(n) = my(m=1,e=eulerphi(n)); while (!((gcd(n, m) == 1) && ispower(e*eulerphi(m), 4)), m++); m; \\ Michel Marcus, Apr 25 2020

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A259915 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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A334337 Least positive integer m such that phi(m*n) is a cube, where phi is Euler's totient function (A000010).

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A334350 Least positive integer m relatively prime to n such that phi(mn) = phi(m)phi(n) is a fourth power, where phi is Euler's totient function (A000010).

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cp's OEIS Frontend

A259915 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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Mathematica

Perl

A334337 Least positive integer m such that phi(m*n) is a cube, where phi is Euler's totient function (A000010).

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Author

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Comments

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Crossrefs

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Mathematica

PARI

A334350 Least positive integer m relatively prime to n such that phi(m*n) = phi(m)*phi(n) is a fourth power, where phi is Euler's totient function (A000010).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A334350 Least positive integer m relatively prime to n such that phi(mn) = phi(m)phi(n) is a fourth power, where phi is Euler's totient function (A000010).