A039770 -id:A039770 - cp's OEIS frontend

A078169 Numbers k such that phi(k) is a perfect 9th power.

1, 2, 771, 1024, 1028, 1088, 1280, 1360, 1536, 1542, 1632, 1920, 2040, 327685, 524288, 524296, 526336, 557056, 559232, 655360, 655370, 657920, 696320, 699040, 786432, 786444, 789504, 835584, 838848, 983040, 986880, 1044480, 1048560

Offset: 1

Labos Elemer, Nov 27 2002

nonn

			phi of the sequence includes 1, 512, 262144,.. etc.; powers arise several times; a(3) = A053576(9) = 771; in sequence smoother ranges and quite large jumps arise when power of new numbers appear as phi-values.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A039770 (square), A039771 (cube), A078164 (4th), A078165 (5th), A078166 (6th), A078167 (7th), A078168 (8th), A078169 (9th, this sequence), A078170 (10th power), A001317, A053576, A045544, A000010.

k=9; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 10000000}]

is(n)=ispower(eulerphi(n),9) \\ Charles R Greathouse IV, Apr 24 2020

A078170 Numbers k such that phi(k) is a perfect tenth power.

Original entry on oeis.org

1, 2, 1285, 2048, 2056, 2176, 2560, 2570, 2720, 3072, 3084, 3264, 3840, 4080, 1114129, 2097152, 2097184, 2105344, 2228224, 2228258, 2236928, 2621440, 2621480, 2631680, 2785280, 2796160, 3145728, 3145776, 3158016, 3342336

Offset: 1

Labos Elemer, Nov 27 2002

nonn

			phi of the sequence includes 1, 1024, 1048576,.. etc.; powers emerge several times; a(3) = A053576(10) = 1285; in sequence smoother ranges and quite large jumps alternate when power of new numbers appear as phi-values.

Amiram Eldar, Table of n, a(n) for n = 1..2250

Cf. A039770 (square), A039771 (cube), A078164 (4th), A078165 (5th), A078166 (6th), A078167 (7th), A078168 (8th), A078169 (9th), A078170 (10th power, this sequence), A001317, A053576, A045544, A000010.

k=10; Do[s=EulerPhi[n]^(1/k); If[IntegerQ[s], Print[n]], {n, 1, 10000000}]

is(n)=ispower(eulerphi(n),10) \\ Charles R Greathouse IV, Apr 24 2020

A054754 Totient(n) and cototient(n) are squares.

Original entry on oeis.org

1, 2, 5, 8, 17, 32, 37, 101, 125, 128, 197, 257, 401, 468, 512, 577, 677, 1297, 1417, 1601, 1872, 2048, 2340, 2917, 3125, 3137, 3145, 4100, 4212, 4357, 4913, 5477, 7057, 7488, 8101, 8192, 8837, 9360, 12101, 13457, 14401, 14841, 15377, 15588, 15877

Offset: 1

Labos Elemer, Apr 25 2000

nonn

Subsequence of A039770, supersequence of A002496.
a(n) is an odd power of a prime q = w^2+1, like 4913 = 17^3, where A000010(a(31)) = phi(4913) = 4624 = 68^2 and A051953(4913) = 4913-4624 = 289 = 17^2.
a(n) is not an odd power of a prime of A002496, like a(14) = 468, where phi(468) = 144 and 468-phi(468) = 324 = 18^2.
Intersection of A039770 and A063752. - Altug Alkan, Aug 16 2017

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A000010, A002496, A039770, A051953, A063752.

Select[Range@ 16000, Function[n, AllTrue[{#, n - #} &@ EulerPhi@ n, IntegerQ@ Sqrt@ # &]]] (* Michael De Vlieger, Aug 16 2017 *)

isok(n) = issquare(eulerphi(n)) && issquare(n-eulerphi(n)); \\ Michel Marcus, Sep 09 2013

A000010(a(n))=x^2 and a(n)-A000010(a(n))=y^2.

A068560 Numbers k such that phi(k) = tau(k)^2.

Original entry on oeis.org

1, 5, 34, 63, 76, 128, 136, 170, 315, 364, 380, 444, 640, 680, 972, 1820, 1824, 1836, 2142, 2220, 4788, 4860, 6000, 8064, 8568, 8736, 9120, 9180, 10710, 23940, 40320, 42840, 43680

Offset: 1

Benoit Cloitre, Mar 25 2002

This sequence is finite because phi(k) >= sqrt(k) for all k >= 6, and for any e > 0, tau(k) < k^e for k large enough. Choosing e=1/4 gives tau(k)^2 < sqrt(k) <= phi(k). It remains unknown, however, if this sequence is full. - Nathaniel Johnston, Apr 28 2011

It can be shown that tau(k) <= 120 and the sequence is complete. - Max Alekseyev, May 30 2024

			a(2) = A107655(2) = 5.

Cf. A039770, A068559, A107655, A114063.

Do[If[EulerPhi[n] == DivisorSigma[0, n]^2, Print[n]], {n, 10^5}] (* Ryan Propper, Jun 09 2006 *)
Select[Range[10^5], EulerPhi[#] == DivisorSigma[0, #]^2 &] (* Alonso del Arte, Aug 25 2011 *)

More terms from Ryan Propper, Jun 09 2006

"full" keyword added by Max Alekseyev, May 30 2024

A247129 Semiprimes n such that phi(n) is a square.

Original entry on oeis.org

10, 34, 57, 74, 85, 185, 202, 219, 394, 451, 489, 505, 514, 629, 679, 802, 985, 1057, 1154, 1285, 1354, 1387, 1417, 1717, 2005, 2047, 2509, 2594, 2649, 2761, 2885, 3097, 3202, 3277, 3349, 3385, 3409, 3459, 3737, 4207, 4369, 4377, 4577

Offset: 1

Charles R Greathouse IV, Nov 19 2014

nonn

Freiberg & Pomerance show that this sequence is infinite and a(n) << n^2 log^2 n.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Tristan Freiberg, Carl Pomerance, A note on square totients, arXiv:1410.8109 [math.NT], 2014.

Cf. A039770, A221285, A002496.

is(n)=issquare(eulerphi(n))&&bigomega(n)==2

list(lim)=my(v=List()); forprime(p=2,sqrtint(lim\1), forprime(q=p+1, lim\p, if(issquare((p-1)*(q-1)), listput(v,p*q)))); Set(v)

A324747 Numbers k with exactly two distinct prime factors and such that phi(k) is a square, when: k = p^(2s) * q^(2t+1) with s >= 1, t >= 0, p <> q primes.

Original entry on oeis.org

12, 48, 63, 76, 108, 192, 292, 304, 432, 567, 652, 768, 873, 972, 1168, 1216, 1359, 1728, 2107, 2608, 3072, 3087, 3532, 3888, 4383, 4525, 4612, 4672, 4864, 5103, 5409, 5836, 6543, 6912, 7204, 7857, 8716, 8748, 10372, 10432, 12231, 12288

Offset: 1

Bernard Schott, Mar 13 2019

nonn

An integer belongs to this sequence iff p*(p-1)*(q-1) = m^2.
This is the second subsequence of A324745, the first one is A324746.
Some values of (k,p,q,m): (12,2,3,2), (63,3,7,6), (76,2,19,6), (292,2,73,12), (652,2,163,18), (873,3,97,24).
The primitive terms of this sequence are the products p^2 * q, with p<>q which satisfy p*(p-1)*(q-1) = m^2. The first few primitive terms are: 12, 63, 76, 292, 652, 873.. Then the integers (p^2 * q) * p^2 and (p^2 * q) * q^2 are new terms of the general sequence.

			63 = 3^2 * 7 and phi(63) = 3*2*6 = 6^2.
1728 = 2^6 * 3^3 and phi(1728) = (2^2 * 3^1 * 2)^2 = 24^2.

David A. Corneth, Table of n, a(n) for n = 1..9878 (terms <= 10^10)

Cf. A039770, A054755, A324745, A324746.

isok(k) = {if (issquare(eulerphi(k)), my(expo = factor(k)[,2]); if ((#expo == 2)&& (expo[1]%2) != (expo[2]%2), return (1)););} \\ Michel Marcus, Mar 18 2019

phi(p^2 * q) = p*(p-1)*(q-1) = m^2 for primitive terms.

phi(k) = (p^(s-1) * q^t * m)^2 with k as in the name of this sequence.

A272798 Carmichael numbers k such that Euler totient function of k (phi(k)) is a perfect square.

Original entry on oeis.org

1729, 63973, 75361, 172081, 278545, 340561, 658801, 997633, 1773289, 3224065, 5310721, 8719309, 8719921, 12945745, 13187665, 15888313, 17586361, 27402481, 29020321, 39353665, 40430401, 49333201, 67371265, 84417985, 120981601, 128697361, 129255841, 130032865, 151530401, 151813201, 158864833

Offset: 1

Altug Alkan, May 06 2016

nonn

Subsequence of A262406.
If n is a Carmichael number, then phi(n) = Product_{primes p dividing n} (p-1).
So the question is: What are the Carmichael numbers n such that Product_{primes p dividing n} (p-1) is a square?
The number of prime divisors of terms of this sequence are 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 5, 5, 5, 4, 4, 4, 4, 4, ...
1299963601 = 601*1201*1801 is the second term that has three prime divisors and it is a member of this sequence since 600*1200*1800 = 2^10*3^4*5^6 is a square.
This sequence is infinite. See links section for more details. - Altug Alkan, Jan 16 2017

			1729 is a term because A000010(1729) = 1729*(1-1/7)*(1-1/13)*(1-1/19) = 1296 = 36^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
William D. Banks, Carmichael Numbers with a Square Totient, Canad. Math. Bull. 52 (2009), 3-8.
Index entries for sequences related to Carmichael numbers.

Cf. A000010, A000290, A002997, A039770, A262406, A290793.

isA002997(n) = {my(f); bittest(n, 0) && !for(i=1, #f=factor(n)~, (f[2, i]==1 && n%(f[1, i]-1)==1)||return) && #f>1}
lista(nn) = for(n=1, nn, if(isA002997(n) && issquare(eulerphi(n)), print1(n, ", ")));

a(30) corrected by Amiram Eldar, Aug 11 2017

A280988 Least k such that phi(k*n) is a perfect square, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 4, 2, 1, 2, 9, 1, 7, 1, 41, 1, 21, 9, 4, 2, 1, 6, 3, 2, 3, 41, 89, 2, 5, 14, 4, 13, 113, 2, 143, 1, 25, 1, 9, 3, 1, 2, 7, 1, 11, 3, 49, 25, 7, 89, 1151, 1, 43, 5, 4, 7, 553, 2, 15, 9, 1, 113, 233, 1, 77, 122, 1, 2, 21, 25, 299, 2, 356, 9, 281, 6, 3, 1, 11, 1, 61, 6, 313

Offset: 1

Altug Alkan, Jan 12 2017

nonn

Pollack and Pomerance proved that if phi(a) = b^m, then m = 2 occurs only on a set of density 0.

			a(11) = 41 because phi(k*11) is not a perfect square for 0 < k < 41 and phi(41*11) = 20^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Paul Pollack and Carl Pomerance, Square values of Euler's function, Bulletin of the London Mathematical Society, Vol. 46, No. 2 (2014), pp. 403-414, alternative link.

Cf. A000010, A039770, A062732, A280986.

f:= proc(n) local k;
    for k from 1 do
      if issqr(numtheory:-phi(k*n)) then return k fi
   od
end proc:
map(f, [$1..100]); # Robert Israel, Jan 12 2017

a[n_] := Module[{k = 1}, While[!IntegerQ[Sqrt[EulerPhi[k*n]]], k++]; k]; Array[a, 80] (* Amiram Eldar, Jul 13 2019 *)

a(n) = {my(k = 1); while (!issquare(eulerphi(k*n)), k++); k; }

A306908 Numbers k with exactly three distinct prime factors and such that phi(k) is a square.

Original entry on oeis.org

60, 114, 126, 170, 204, 240, 273, 285, 315, 364, 370, 380, 438, 444, 456, 468, 504, 540, 680, 816, 825, 902, 960, 969, 978, 1010, 1026, 1071, 1095, 1100, 1134, 1212, 1258, 1292, 1358, 1456, 1460, 1480, 1500, 1520, 1729, 1746, 1752, 1776, 1824, 1836, 1872

Offset: 1

Bernard Schott, Mar 16 2019

nonn

This sequence is the intersection of A033992 and A039770.

The integers with only one prime factor and whose totient is a square are in A002496 and A054755, the integers with two prime factors and whose totient is a square are in A324745, A324746 and A324747.

			1st family: 273 = 3 * 7 * 13 and phi(273) = 12^2.
2nd family: 816 = 2^4 * 3 * 17 and phi(816) = 16^2.
3rd family: 6975 = 3^2 * 5^2 * 31 and phi(6975) = 60^2.

Robert Israel, Table of n, a(n) for n = 1..10000
Bernard Schott, Subfamilies and subsequences

Intersection of A033992 and A039770.

Cf. A002496, A054755 (only one prime factor), A324745, A324746, A324747 (two prime factors).

filter:= n -> issqr(numtheory:-phi(n)) and nops
(numtheory:-factorset(n))=3:
select(filter, [$1..2000]); # after Robert Israel in A324745

Select[Range[2000], And[PrimeNu@ # == 3, IntegerQ@ Sqrt@ EulerPhi@ #] &] (* Michael De Vlieger, Mar 31 2019 *)

isok(n) = (omega(n)==3) && issquare(eulerphi(n)); \\ Michel Marcus, Mar 19 2019

1st family: The primitive terms are p*q*r with p,q,r primes and phi(p*q*r) = (p-1)*(q-1)*(r-1) = m^2. These primitives generate the entire family formed by the numbers k = p^(2s+1) * q^(2t+1) * r^(2u+1) with s,t,u >= 0, and phi(k) = (p^s * q^t * r^u * m)^2.
2nd family: The primitive terms are p^2 * q * r with p,q,r primes and phi(p^2 * q * r) = p*(p-1)*(q-1)*(r-1) = m^2. These primitives generate the entire family formed by the numbers k = p^(2s) * q^(2t+1) * r^(2u+1) with s >= 1, t,u >= 0, and phi(k) = (p^(s-1) * q^t * r^u * m)^2.
3rd family: The primitive terms are p^2 * q^2 * r with p,q,r primes and phi(p^2 * q^2 * r) = p*q*(p-1)*(q-1)*(r-1) = m^2. These primitives generate the entire family formed by the numbers k = p^(2s) * q^(2t) * r^(2u+1) with s,t> = 1, u >= 0, and phi(k) = (p^(s-1) * q^(t-1) * r^u * m)^2.

A324745 Numbers k with exactly two distinct prime factors and such that phi(k) is a square.

Original entry on oeis.org

10, 12, 34, 40, 48, 57, 63, 74, 76, 85, 108, 136, 160, 185, 192, 202, 219, 250, 292, 296, 304, 394, 432, 451, 489, 505, 513, 514, 544, 567, 629, 640, 652, 679, 768, 802, 808, 873, 972, 985, 1000, 1057, 1154, 1168, 1184, 1216, 1285, 1354

Offset: 1

Bernard Schott, Mar 12 2019

nonn

This sequence is the intersection of A007774 and A039770.
The sequences A324746 and A324747 form a partition of this sequence.
See the file "Subfamilies and subsequences" (& II) in A039770 for more details, proofs with data, comments, formulas and examples.
The integers with only one prime factor and whose totient is a square are in A054755.

			1st family: 136 = 2^3 * 37 and phi(136) = 8^2.
2nd family: 652 = 2^2 * 163 and phi(652) = 18^2.

Intersection of A007774 and A039770.

Cf. A054755, A062732, A221285, A324746, A324747.

filter:= n -> issqr(numtheory:-phi(n)) and nops(numtheory:-factorset(n))=2:
select(filter, [$1..2000]); # Robert Israel, Mar 18 2019

Select[Range[1400], And[PrimeNu[#] == 2, IntegerQ@ Sqrt@ EulerPhi@ #] &] (* Michael De Vlieger, Mar 21 2019 *)

isok(n) = (omega(n)==2) && issquare(eulerphi(n)); \\ Michel Marcus, Mar 17 2019

1st family (A324746): The primitive terms are defined by p*q, p < q, with phi(p*q) = (p-1)*(q-1) = m^2. The general terms are defined by k = p^(2s+1) * q^(2t+1), s,t >= 0, with phi(k) = (p^s * q^t * m)^2.

2nd family (A324747): The primitive terms are defined by p^2 * q, p <> q, with phi(p^2 * q) = p*(p-1)*(q-1) = m^2. The general terms are defined by k = p^(2s ) * q^(2t+1), s >= 1, t >= 0, with phi(k) = (p^(s-1) * q^t * m)^2.

cp's OEIS Frontend

A078169 Numbers k such that phi(k) is a perfect 9th power.

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A078170 Numbers k such that phi(k) is a perfect tenth power.

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A054754 Totient(n) and cototient(n) are squares.

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A068560 Numbers k such that phi(k) = tau(k)^2.

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A247129 Semiprimes n such that phi(n) is a square.

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A324747 Numbers k with exactly two distinct prime factors and such that phi(k) is a square, when: k = p^(2s) * q^(2t+1) with s >= 1, t >= 0, p <> q primes.

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A272798 Carmichael numbers k such that Euler totient function of k (phi(k)) is a perfect square.

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A280988 Least k such that phi(k*n) is a perfect square, or 0 if no such k exists.

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