A039771 -id:A039771 - cp's OEIS frontend

A290793 Carmichael numbers k such that Euler totient function of k (phi(k)) is a cube.

63973, 18162001, 26921089, 133205761, 225745345, 490503601, 496050841, 698548201, 1031750401, 1100674561, 1384157161, 2178944461, 3805181281, 11351100241, 12648201841, 26498875681, 26542598401, 28553256865, 28645206001, 37590868801, 39866123377, 40527674881

Offset: 1

Amiram Eldar, Aug 10 2017

nonn

Banks proved that for each positive integer N there are an infinite number of Carmichael numbers whose Euler totient function value is an N-th power. Therefore this sequence is infinite.

The terms were calculated using Pinch's tables of Carmichael numbers (see link below).

			phi(63973) = 36^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier)
William D. Banks, Carmichael Numbers with a Square Totient, Canadian Mathematical Bulletin, Vol. 52, No. 1 (2009), pp. 3-8.
Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
R. G. E. Pinch, Tables relating to Carmichael numbers.
Index entries for sequences related to Carmichael numbers.

Intersection of A002997 (Carmichael numbers) and A039771.

Cf. A000010, A000578, A272798.

With[{s = Import["b002997.txt", "Data"][[All, -1]]}, Select[s, IntegerQ@ Power[EulerPhi@ #, 1/3] &]] (* Michael De Vlieger, Aug 14 2017, using b-file at A002997 *)

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

A237123 Number of ways to write n = i + j + k with 0 < i < j < k such that phi(i), phi(j) and phi(k) are all cubes, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 2, 1, 0, 0, 2, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1

Offset: 1

Zhi-Wei Sun, Feb 03 2014

nonn

Conjecture: For each k = 2, 3, ... there is a positive integer s(k) such that any integer n >= s(k) can be written as i_1 + i_2 + ... + i_k with 0 < i_1 < i_2 < ... < i_k such that all those phi(i_1), phi(i_2), ..., phi(i_k) are k-th powers. In particular, we may take s(2) = 70640, s(3) = 935 and s(4) = 3273.

			a(18) = 1 since 18 = 1 + 2 + 15 with phi(1) = 1^3, phi(2) = 1^3 and phi(15) = 2^3.
a(101) = 1 since 101 = 1 + 15 + 85 with phi(1) = 1^3, phi(15) = 2^3 and phi(85) = 4^3.
a(1613) = 1 since 1613 = 192 + 333 + 1088 with phi(192) = 4^3, phi(333) = 6^3 and phi(1088) = 8^3.

Zhi-Wei Sun, Table of n, a(n) for n = 1..7000

Cf. A000010, A000578, A039771, A233386, A236998.

CQ[n_]:=IntegerQ[EulerPhi[n]^(1/3)]
a[n_]:=Sum[If[CQ[i]&&CQ[j]&&CQ[n-i-j],1,0],{i,1,n/3-1},{j,i+1,(n-1-i)/2}]
Table[a[n],{n,1,70}]

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

A114573 Numbers k such that phi(k) is a perfect 11th power.

Original entry on oeis.org

1, 2, 3855, 4096, 4112, 4352, 5120, 5140, 5440, 6144, 6168, 6528, 7680, 7710, 8160, 5570645, 8388608, 8388736, 8421376, 8912896, 8913032, 8947712, 10485760, 10485920, 10526720, 11141120, 11141290, 11184640, 12582912, 12583104

Offset: 1

Stefan Steinerberger, Feb 17 2006

nonn

Given the fact that phi(n) > sqrt(n) for all n except n=2 and n=6 we can see that every 11th power does appear as value only a finite number of times. What bounds on the density of this sequence can be proved?

			phi(4096) = 2048 = 2^11.

Amiram Eldar, Table of n, a(n) for n = 1..3625

Cf. A039770 (square), A039771 (cube), A078164 (4th), A078165 (5th), A078166 (6th), A078167 (7th), A078168 (8th), A078169 (9th), A078170 (10th power), A000010.

For[n = 1, n < 100000, n++, If[EulerPhi[n]^(1/11) == Floor[EulerPhi[n]^(1/11)], Print[n]]]

More terms from Stefan Steinerberger, May 16 2007

A113939 Numbers k such that sigma(k) and phi(k) are both perfect cubes.

Original entry on oeis.org

1, 300118, 2250885, 5294800, 76336260, 141483384, 242375035, 453280968, 456345036, 461356896, 478240392, 537375930, 1139523375, 1180718238, 1227111595, 1599634410, 5043835125, 5229921340, 5610392513, 6087775939, 6376494509, 11038341720, 11044670824

Offset: 1

Giovanni Resta, Jan 31 2006

nonn

phi(k) = A000010(k) is the Euler totient function, while sigma(k) = A000203(k) is the sum of divisors of k.

Intersection of A020477 and A039771. - Michel Marcus, Jan 04 2014

			sigma(2250885) = 168^3 and phi(2250885) = 96^3.

Cf. A000010, A000203.

for(n=1, 10^9, if(ispower(eulerphi(n),3), if(ispower(sigma(n),3), print1(n ", ")))) \\ Donovan Johnson, Jan 04 2014

a(7)-a(20) from Donovan Johnson, Feb 06 2010

a(21)-a(23) from Donovan Johnson, Jan 04 2014

A216452 The fourth roots of the fourth powers arising in A078164.

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 10, 8, 8, 8, 10, 8, 10, 8, 8, 8, 10, 10, 10, 12, 12, 12, 12, 10, 12

Offset: 1

V. Raman, Sep 07 2012

nonn

Cf. A039770, A039771, A039772, A062732, A078164, A166955.

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A290793 Carmichael numbers k such that Euler totient function of k (phi(k)) is a cube.

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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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A237123 Number of ways to write n = i + j + k with 0 < i < j < k such that phi(i), phi(j) and phi(k) are all cubes, where phi(.) is Euler's totient function.

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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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A114573 Numbers k such that phi(k) is a perfect 11th power.

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A113939 Numbers k such that sigma(k) and phi(k) are both perfect cubes.

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PARI

Extensions

A216452 The fourth roots of the fourth powers arising in A078164.

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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).