A039770 -id:A039770 - cp's OEIS frontend

A307245 First occurrence of n in A306722.

11, 1, 3, 10, 27, 60, 72, 120, 180, 270, 480, 252, 1155, 720, 792, 1260, 630, 1050, 4590, 1680, 1320, 7980, 3780, 4680, 5880, 5040, 5460, 4620, 9180, 10080, 10710, 6930, 9240, 7560, 21420, 20790, 27300, 52080, 15120, 13860, 48510, 23940, 62370, 46200, 16380, 30030

Offset: 0

Bernard Schott and Robert G. Wilson v, Mar 30 2019

nonn

Record values: 11, 27, 60, 72, 120, 180, 270, 480, 1155, 1260, 4590, 7980, 9180, 10080, 10710, 21420, 27300, 52080, 62370, 191520, 207480, 214200, 428400, ..., .

			a(0) = 11  because (2*11)^2 = 484 is the smaller integer that can't be written as (p-1)*(q-1) with p,q primes, p < q.
a(3) = 10 because (2*10)^2 = 400 is the smaller integer such that the Diophantine equation (p-1)*(q-1) = 400 has three solutions: (p,q) = (2,401) = (5,101) = (11,41); also, phi(2*401) = phi(5*101) = phi(11*41) = 20^2.

Robert G. Wilson v, Table of n, a(n) for n = 0..876

Cf. A039770, A247129, A306722.

f[n_] := Length@ Select[ Divisors[ 4n^2], # < 2n && PrimeQ[# +1] && PrimeQ[4n^2/# +1] &]; t[_] := 0; k = 1; While[k < 100000, a = f@k; If[t[a] == 0, t[a] = k]; k++]; t@# & /@ Range[0, 50]

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

A360944 Numbers m such that phi(m) is a triangular number, where phi is the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 7, 9, 11, 14, 18, 22, 29, 37, 57, 58, 63, 67, 74, 76, 79, 108, 114, 126, 134, 137, 143, 155, 158, 175, 183, 191, 211, 225, 231, 244, 248, 274, 277, 286, 308, 310, 329, 341, 350, 366, 372, 379, 382, 396, 417, 422, 423, 450, 453, 462, 554, 556, 604, 623, 631, 658, 682

Offset: 1

Bernard Schott, Feb 26 2023

nonn

Subsequence of primes is A055469 because in this case phi(k(k+1)/2+1) = k(k+1)/2.

Subsequence of triangular numbers is A287472.

			phi(57) = 36 = 8*9/2, a triangular number; so 57 is a term of the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000010, A000217, A055469, A287472.

Similar, but with phi(m) is: A039770 (square), A078164 (biquadrate), A096503 (repdigit), A117296 (palindrome), A236386 (oblong).

filter := m ->  issqr(1 + 8*numtheory:-phi(m)) : select(filter, [$(1 .. 700)]);

Select[Range[700], IntegerQ[Sqrt[8 * EulerPhi[#] + 1]] &] (* Amiram Eldar, Feb 27 2023 *)

isok(m) = ispolygonal(eulerphi(m), 3); \\ Michel Marcus, Feb 27 2023

from itertools import islice, count
from sympy.ntheory.primetest import is_square
from sympy import totient
def A360944_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:is_square((totient(n)<<3)+1), count(max(1,startvalue)))
A360944_list = list(islice(A360944_gen(),20)) # Chai Wah Wu, Feb 28 2023

A054756 Numbers k such that phi(k) and cototient(k) are squares but k is not in A054755.

Original entry on oeis.org

1, 468, 1417, 1872, 2340, 3145, 4100, 4212, 7488, 9360, 14841, 15588, 16400, 16848, 20329, 21060, 29952, 31417, 37440, 37908, 45097, 49833, 58500, 62352, 63529, 63945, 65600, 67392, 69700, 78625, 79092, 83569, 84169, 84240, 88929, 102500

Offset: 1

Labos Elemer, Apr 25 2000

nonn

			An even term is 2340 = 4*9*5*13 (phi = 576 = 24^2 and cototient = 1764 =  42^2).
An odd term is 14841 = 9*17*97 (phi = 9216 = 96^2, cototient = 5625 = 75^2).

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A000010, A002496, A005574, A039770, A051953.

Equals A054754 \setminus A054755. See also A063752.

Select[ Range[ 1, 200000 ], IntegerQ[ Sqrt[ eu[ # ] ] ]&& IntegerQ[ Sqrt[ co[ # ] ] ]&&!Equal[ lfi[ # ], 1 ]& ], where eu[ x_ ] =EulerPhi[ x ], co[ x_ ]=x-EulerPhi[ x ] and lfi[ x_ ]=Length[ FactorInteger[ x ] ]

phi(a(n)) = x^2, a(n) - phi(a(n)) = y^2, a(n) is not an odd power of prime from A002496.

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

A245199 Numbers n where phi(n) and tau(n) are perfect squares.

Original entry on oeis.org

1, 8, 10, 34, 57, 74, 85, 125, 185, 202, 219, 394, 451, 456, 489, 505, 514, 546, 570, 629, 640, 679, 680, 802, 985, 1000, 1026, 1057, 1154, 1285, 1354, 1365, 1387, 1417, 1480, 1717, 1752, 1938, 2005, 2016, 2047, 2176, 2190, 2340, 2457, 2509, 2565, 2594, 2649

Offset: 1

Reinhard Muehlfeld, Jul 13 2014

Numbers n such that A000005(n) and A000010(n) are perfect squares.

Intersection of A036436 and A039770. - Michel Marcus, Jul 15 2014

			8 is in the sequence because phi(8) = 4, tau(8) = 4, and 4 is a perfect square.
12 is not in the sequence because tau(12) = 6 is not a square.

Robert Israel, Table of n, a(n) for n = 1..8000

Cf. A000005, A000010, A036436, A039770.

filter:= proc(n) uses numtheory; issqr(phi(n)) and issqr(tau(n)) end proc:
select(filter, [$1..1000]); # Robert Israel, Jul 27 2014

fQ[n_] := IntegerQ@ Sqrt@ EulerPhi[n] && IntegerQ@ Sqrt@ DivisorSigma[0, n]; Select[ Range@ 3000, fQ] (* Robert G. Wilson v, Jul 21 2014 *)
Select[Range[3000],AllTrue[{Sqrt[EulerPhi[#]],Sqrt[DivisorSigma[0, #]]}, IntegerQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 01 2018 *)

isok(n) = issquare(numdiv(n)) && issquare(eulerphi(n)); \\ Michel Marcus, Jul 15 2014

from sympy import totient, divisor_count
from gmpy2 import is_square
[n for n in range(1,10**4) if is_square(int(divisor_count(n))) and is_square(int(totient(n)))] # Chai Wah Wu, Aug 04 2014

A280986 Least k > 0 such that (k*n)^2 is in A002202, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 6, 1, 2, 2, 8, 1, 2, 1, 2, 1, 4, 1, 4, 1, 2, 2, 2, 1, 2, 3, 4, 1, 6, 1, 10, 1, 2, 4, 2, 1, 4, 1, 4, 1, 8, 1, 2, 1, 2, 2, 4, 1, 12, 2, 2, 1, 2, 1, 2, 1, 4, 1, 4, 1, 2, 1, 2, 3, 4, 2, 6, 1, 2, 3, 8, 1, 2, 5, 2, 1, 6, 1, 4, 2, 2, 1, 4, 1, 8, 2, 2, 1

Offset: 1

Altug Alkan, Jan 12 2017

nonn

Pollack and Pomerance showed that almost all squares are missing from the range of Euler's totient function.

			a(11) = 4 because (k*11)^2 is not in A002202 for 0 < k < 4 and (4*11)^2 is in A002202.
a(95911) = 56 because (k*95911)^2 is not in A002202 for 0 < k < 56 and (56*95911)^2 is in A002202.

P. Pollack and C. Pomerance, Square values of Euler's function

Cf. A000010, A002202, A039770, A062732, A280988.

a(n) = {my(k = 1); while (!istotient((k*n)^2), k++); k; }

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.

A385822 Numbers k such that phi(k) is not a perfect square.

Original entry on oeis.org

3, 4, 6, 7, 9, 11, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 35, 36, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89

Offset: 1

Aidan Chen, Aug 11 2025

			Since phi(35) = 24 and there is no integer n such that n^2 = 24.

Cf. A000010. Complement of A039770.

Select[Range[100], !IntegerQ[Sqrt[EulerPhi[#]]] &] (* Amiram Eldar, Aug 18 2025 *)

isok(k) = !issquare(eulerphi(k)); \\ Michel Marcus, Aug 18 2025

from math import isqrt
from sympy import totient as phi
def ok(n): return isqrt(p:=phi(n))**2 != p
print([k for k in range(1, 110) if ok(k)]) # Michael S. Branicky, Aug 17 2025

cp's OEIS Frontend

A307245 First occurrence of n in A306722.

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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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A360944 Numbers m such that phi(m) is a triangular number, where phi is the Euler totient function (A000010).

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A054756 Numbers k such that phi(k) and cototient(k) are squares but k is not in A054755.

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

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A245199 Numbers n where phi(n) and tau(n) are perfect squares.

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A280986 Least k > 0 such that (k*n)^2 is in A002202, or 0 if no such k exists.

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

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).