A053577 -id:A053577 - cp's OEIS frontend

A053578 Values of cototient function for A053577.

1, 1, 2, 1, 4, 1, 4, 1, 8, 1, 8, 8, 1, 1, 1, 16, 16, 1, 1, 16, 1, 1, 1, 1, 32, 1, 32, 1, 1, 32, 32, 1, 1, 1, 1, 1, 1, 64, 1, 1, 1, 1, 1, 64, 1, 64, 1, 64, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 128, 1, 1, 1, 1, 1, 128, 1, 1, 1, 1, 1, 128, 1, 128, 128, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Labos Elemer, Jan 18 2000

nonn

Except for 2^0 = 1, there are only finitely many values of k such that cototient(k) = 2^m for fixed m.

			For p prime, cototient(p) = 1. Smallest values for which cototient(x) = 2^w are A058764(w) = A007283(w-1) = 3*2^(w-1) = 6, 12, 24, 48, 96, 192, .., 49152 for w = 2, 3, 4, 5, 6, ..., 15. [Corrected by _M. F. Hasler_, Nov 10 2016]

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

Cf. A051953, A053577, A058764, A007283.

Select[Table[k - EulerPhi[k], {k, 1, 400}], # == 2^IntegerExponent[#, 2] &] (* Amiram Eldar, Jun 09 2024 *)

lista(kmax) = {my(c); for(k = 2, kmax, c = k - eulerphi(k); if(c >> valuation(c, 2) == 1, print1(c, ", ")));} \\ Amiram Eldar, Jun 09 2024

Edited and corrected by M. F. Hasler, Nov 10 2016

A053159 Numbers n such that n+cototient(n) is a power of 2.

Original entry on oeis.org

1, 3, 7, 10, 20, 31, 40, 80, 127, 160, 320, 322, 640, 644, 1280, 1288, 2560, 2576, 5120, 5152, 8191, 10240, 10304, 20480, 20608, 40960, 41216, 81920, 82432, 131071, 163840, 164864, 327680, 329728, 333634, 524287, 655360, 659456, 667268, 1310720, 1318912

Offset: 1

Labos Elemer, Feb 29 2000

nonn

See especially A053579 and also A053576, A053577.

			Mersenne primes are a proper subset of this sequence: A(M)=2M-M+1=M+1=2^p

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A000043, A000668, A001348.

print(1); for(n=3, 10^9, if(omega(2*n-eulerphi(n))==1, print(n))) /* Donovan Johnson, Apr 04 2013 */

a(n)+A051953(n) = 2*a(n)-A000010(n) = 2^w for some w.

More terms from Reiner Martin, Dec 24 2001

A053162 Nonprimes n such that n+cototient(n) is a power of 2.

Original entry on oeis.org

1, 10, 20, 40, 80, 160, 320, 322, 640, 644, 1280, 1288, 2560, 2576, 5120, 5152, 10240, 10304, 20480, 20608, 40960, 41216, 81920, 82432, 163840, 164864, 327680, 329728, 333634, 655360, 659456, 667268, 1310720, 1318912, 1334536, 1378114, 2621440

Offset: 1

Labos Elemer, Feb 29 2000

nonn

See especially A053579 and also A053576, A053577.

			Mersenne primes were deleted from set of numbers with similar property. An infinite subset here is m(r)=5*2^r, since Phi[m(r)]=2^(r+1) and a(m(r))=5*2^(r+1)-2^(r+1)=2^(r+3). A different subset includes m = 322,644,1288,.. = Set of {(2^s)*7*23} generating 2^(s+8)=2m-Phi(m) powers of 2.

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A000043, A000668, A001348, A053159.

for(n=1, 2621440, if(isprime(n)==0, if(omega((2*n-eulerphi(n))*2)==1, print1(n ", ")))) \\ Donovan Johnson, Jan 09 2014

a(n)+A051953(n) = 2*a(n)-A000010(n) = 2^w for some w and a(n).

More terms from Olaf Voß, Feb 25 2008

A053163 n+cototient(n) produces these powers of 2 in order of magnitude.

Original entry on oeis.org

1, 4, 8, 16, 32, 32, 64, 128, 128, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 131072, 262144, 262144, 524288, 524288, 524288, 524288, 1048576, 1048576, 1048576, 2097152

Offset: 1

Labos Elemer, Feb 29 2000

See especially A053579 and also A053576, A053577.

			1+Mersenne primes powers of 2 are here, 2^p for special primes. Also because of other (infinite) subsequences, all 2-powers from 2^6 occurs at least twice.

Donovan Johnson, Table of n, a(n) for n = 1..100
Index to divisibility sequences

Cf. A000043, A000668, A001348.

Join[{1}, Reap[For[n=3, n<10^7, n++, If[PrimeNu[k = 2*n - EulerPhi[n]] == 1, Print[k]; Sow[k]]]][[2, 1]]] (* Jean-François Alcover, Jun 30 2015, after Donovan Johnson *)

print(1); for(n=3, 10^9, k=2*n-eulerphi(n); if(omega(k)==1, print(k))) /* Donovan Johnson, Apr 04 2013 */

a(n) = 2^w = m+A051953(m) = 2*m-A000010(m) for some m.

More terms from Olaf Voß, Feb 25 2008

cp's OEIS Frontend

A053578 Values of cototient function for A053577.

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A053159 Numbers n such that n+cototient(n) is a power of 2.

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A053162 Nonprimes n such that n+cototient(n) is a power of 2.

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A053163 n+cototient(n) produces these powers of 2 in order of magnitude.

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