A323918 -id:A323918 - cp's OEIS frontend

A063752 Numbers k such that cototient(k) is a square.

1, 2, 3, 5, 6, 7, 8, 11, 13, 17, 19, 21, 23, 24, 27, 28, 29, 31, 32, 37, 41, 43, 47, 53, 54, 59, 61, 67, 68, 69, 71, 73, 79, 83, 89, 96, 97, 101, 103, 107, 109, 112, 113, 124, 125, 127, 128, 131, 133, 137, 139, 141, 149, 151, 157, 163, 167, 173, 179, 181, 189, 191

Offset: 1

Jason Earls, Aug 11 2001

nonn

Some different families and subsequences of integers belong to this sequence, see the file "Subfamilies and subsequences" for more details, with data, comments, proofs, formulas and examples. - Bernard Schott, Mar 05 2019

Harry J. Smith, Table of n, a(n) for n = 1..1000
Thomas E. Moore, Problem 1204, Crux Mathematicorum, page 93, Vol. 14, Mar. 88.
Bernard Schott, Subfamilies and subsequences

Cf. A000010, A051953.

Subsequences: A000396, A246551, A323916, A323917, A323918, A306670.

[n: n in [1..200] | IsSquare(n - EulerPhi(n))]; // Vincenzo Librandi, Jan 11 2019

Select[Range[200], IntegerQ[Sqrt[# - EulerPhi[#]]]&] (* Jean-François Alcover, Nov 06 2016 *)

j=[]; for(n=1,400,x=n-eulerphi(n); if(issquare(x),j=concat(j,n))); j

{ n=0; for (m=1, 10^9, if (issquare(m - eulerphi(m)), write("b063752.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Aug 29 2009

a(n) seems to be asymptotic to c * n * log(n) with c = 1.7... (all primes are in the sequence since cototient(p) = 1). - Benoit Cloitre, Sep 08 2002

A306670 Numbers k with exactly three distinct prime factors and such that cototient(k) is a square.

Original entry on oeis.org

345, 465, 468, 1332, 1545, 1833, 1872, 2628, 2737, 2769, 3105, 3145, 3585, 3657, 3945, 4081, 4100, 4185, 4212, 4345, 5328, 6465, 6516, 6785, 6945, 7105, 7488, 8428, 8569, 8625, 8961, 10257, 10512, 10785, 10833, 10945, 11625, 11988, 12132, 12865

Offset: 1

Bernard Schott, Mar 04 2019

nonn

The integers with only one prime factor and whose cototient is a square are in A246551. The integers with two prime factors and whose cototient is a square are in A323916, and the subsequences A323917 and A323918.

There are exactly three different families of integers which realize a partition of this sequence. See the file "Subfamilies and subsequences" (& III) in A063752 for more details, proofs with data, comments, formulas and examples.

			1st family: 2769 = 3 * 13 * 71 and cototient(2769) = 33^2.
2nd family: 14841 = 3^2 * 17 * 97 and cototient(14841) = 75^2.
3rd family: 1872 = 2^4 * 3^2 * 13 and cototient(1872) = 36^2.

Subsequence of A063752.

Cf. A246551 (only one prime factor), A323916, A323917, A323918 (two prime factors), A000396 (even perfect numbers).

1st family: The primitive terms are p*q*r with p,q,r primes and cototient(p*q*r) = 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 cototient(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 cototient(p^2 * q * r) = p * (p*q*r-(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 cototient(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 cototient(p^2 * q^2 * r) = p * q * (p*q*r-(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 cototient(k) = (p^(s-1) * q^(t-1) * r^u * M)^2.

A323916 Numbers k with exactly two distinct prime divisors and such that cototient(k) is a square.

Original entry on oeis.org

6, 21, 24, 28, 54, 68, 69, 96, 112, 124, 133, 141, 189, 216, 237, 272, 284, 301, 384, 388, 448, 481, 486, 496, 501, 508, 589, 621, 657, 669, 781, 796, 864, 964, 1025, 1029, 1077, 1088, 1136, 1141, 1269, 1317, 1348, 1357, 1372, 1417, 1536, 1537, 1552, 1701, 1792, 1796

Offset: 1

Bernard Schott, Feb 09 2019

nonn

The integers with only one prime factor and whose cototient is a square are in A246551.
This sequence is the intersection of A007774 and A063752.
There are exactly two different families of integers which realize a partition of this sequence (A323917 and A323918); there is also another family with the even perfect numbers of A000396 which is a subsequence of this sequence.
See the file "Subfamilies of terms" (& II) in A063752 for more details, proofs with data, comments, formulas and examples.

			1st family: 189 = 3^3 * 7 and cototient(189) = 9^2;
2nd family: 272 = 2^4 * 17 and cototient(272) = 12^2;
3rd family: 8128 = 2^6 * 127 and cototient(8128) = 64^2.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Subsequence of A063752.
Cf. A051953.
Cf. A000396, A323917, A323918

Select[Range[1800], 2 == Length@ FactorInteger@ # == 2 && IntegerQ@ Sqrt[# - EulerPhi@ #] &] (* Giovanni Resta, Feb 27 2019 *)
Select[Range[2000],PrimeNu[#]==2&&IntegerQ[Sqrt[#-EulerPhi[#]]]&] (* Harvey P. Dale, Jan 06 2022 *)

isok(n) = (omega(n)==2) && issquare(n - eulerphi(n)); \\ Michel Marcus, Feb 10 2019

[n for n in (1..2500) if len([1 for d in divisors(n) if is_prime(d)])==2 and is_square(n - euler_phi(n))] # G. C. Greubel, Mar 01 2019

1st family (A323917): if k = p^(2s+1) * q^(2t+1) with s,t >=0, p, q primes and p + q - 1 = M^2, then cototient(k) = (p^r * q^s * M)^2. The primitive terms are p*q with cototient(p*q) = p+q-1 = M^2
2nd family (A323918): if k = p^(2s) * q^(2t+1) with s>=1, t>=0, p, q primes, p < q and such that p*(p+q-1)= M^2, then cototient(k) = (p^(s-1) * q^t * M)^2. The primitive terms are p^2 *q with cototient(p^2 * q) = p * (p+q-1) = M^2
3rd family (A000396): the even perfect Numbers, if 2^p - 1 is a Mersenne prime, then cototient(2^(p-1) * (2^p - 1)) = (2^(p-1))^2.

A323917 Numbers k with exactly two distinct prime divisors and such that cototient(k) is square, where k = p^(2s+1) * q^(2t+1) with s,t >=0, p, q primes and p + q - 1 = M^2.

Original entry on oeis.org

6, 21, 24, 54, 69, 96, 133, 141, 189, 216, 237, 301, 384, 481, 486, 501, 589, 621, 669, 781, 864, 1029, 1077, 1141, 1269, 1317, 1357, 1417, 1536, 1537, 1701, 1944, 1957, 1981, 2041, 2133, 2181, 2517, 2869, 3261, 3397, 3456, 3601, 3661, 3669, 4101, 4309, 4333, 4374, 4509

Offset: 1

Bernard Schott, Feb 09 2019

nonn

This is the first subsequence of A323916, the second one is A323918 and A323916 = {this sequence} Union A323918 with empty intersection.
Some values of (k,p,q,M): (6,2,3,2), (21,3,7,3), (69,3,23,5), (133,7,19,5), (141,3,47,9), (301,7,43,7), (481,13,37,7).
The primitive terms of this sequence are the products p * q, with p,q which satisfy p+q-1 = M^2, the first ones are: 6, 21, 69, 133, 141, 237. Then the integers (p*q) * p^2 and (p*q) * q^2 are new terms of the general sequence.
There is only one even perfect number in this sequence: 6. The other ones are in A323918.
See the file "Subsequences and Subfamilies of terms" (&2.1) in A063752 for more details, proofs with data, comments, formulas and examples.

			Perfect number 6 = 2 * 3 and cototient(6) = 2^2.
781 = 11 * 71 and cototient(781) = 11 + 71 - 1 = 9^2.
864 = 2^5 * 3^3 and cototient(864)= (2^2 * 3^1 * 2)^2 = 24^2.

Cf. A051953, A063752, A246551, A323916, A323918.

isok(n) = (omega(n)==2) && issquare(n - eulerphi(n)) && ((factor(n)[1,2] % 2) == (factor(n)[2,2] % 2)); \\ Michel Marcus, Feb 10 2019

cototient(p*q) = p + q - 1 = M^2 for primitive terms.

cototient(k) = (p^s * q^t * M)^2 with k as in the name of this sequence.

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A063752 Numbers k such that cototient(k) is a square.

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A306670 Numbers k with exactly three distinct prime factors and such that cototient(k) is a square.

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A323916 Numbers k with exactly two distinct prime divisors and such that cototient(k) is a square.

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A323917 Numbers k with exactly two distinct prime divisors and such that cototient(k) is square, where k = p^(2s+1) * q^(2t+1) with s,t >=0, p, q primes and p + q - 1 = M^2.

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