A053576 -id:A053576 - cp's OEIS frontend

A082506 a(n) = gcd(2^n, n - phi(n)); largest power of 2 dividing cototient(n) = A051953(n).

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

Offset: 1

Labos Elemer, Apr 28 2003

nonn

a(n)=1 if and only if n is odd or n = 2. - Robert Israel, May 31 2018

			Different from A069177, analogous sequence with totient, instead of cototient.

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

Cf. A000010, A051953, A009195, A083250, A007283, A050339, A053576, A069177.

f:= n -> padic:-ordp(n - numtheory:-phi(n), 2):
map(f, [$1..100]); # Robert Israel, May 31 2018

A281627 a(n) is the smallest number k such that sigma(phi(k)) = A062402(k) is the n-th Mersenne prime (A000668(n)), or 0 if no such k exists.

Original entry on oeis.org

3, 5, 17, 85, 4369, 65537, 327685, 1431655765, 2305843009213693952, 618970019642690137449562112, 162259276829213363391578010288128, 170141183460469231731687303715884105728

Offset: 1

Jaroslav Krizek, Feb 11 2017

nonn

Conjecture 1: If A062402(n) = A000203(A000010(n)) = sigma(phi(n)) is a prime p for some n, then p is Mersenne prime (A000668); a(n) > 0 for all n.
Conjecture 2: a(n) = the smallest number k such that phi(k) has exactly A000043(n)-1 divisors; see A276044.
Conjecture 3: a(n) = the smallest number k such that phi(k) has exactly A000043(n)-1 prime factors (counted with multiplicity); see A275969.
a(n) <= A000668(n) for n = 1-18; conjecture: a(n) <= A000668(n) for all n.
Equals A002181 (index in A002202 of (intersection of A023194 and A002202)). - Michel Marcus, Feb 12 2017

Amiram Eldar, Table of n, a(n) for n = 1..13
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000010, A000043, A000203, A000668, A062402, A062514, A275969, A276044.
Cf. A002181, A002202, A023194.
Cf. A053576 (includes the first 13 known terms of this sequence).

A281627:=func; Set(Sort([A281627(n): n in [SumOfDivisors(EulerPhi(n)): n in[1..1000000] | IsPrime(SumOfDivisors(EulerPhi(n)))]]));

terms() = {v = readvec("b023194.txt"); for(i=1, #v, if (istotient(v[i], &n), print1(n/2, ", ")););} \\ Michel Marcus, Feb 12 2017

f(p) = {my(s = invsigma(p), t, minv = 0); for(i = 1 ,#s, t = invphi(s[i]); for(j = 1, #t, if(minv == 0, minv = t[j]); if(t[j] < minv, minv = t[j]))); minv;} \\ using Max Alekseyev's invphi.gp
list(pmax) = forprime(p = 1, pmax, if(isprime(2^p-1), print1(f(2^p-1), ", "))); \\ Amiram Eldar, Dec 23 2024

a(8) from Michel Marcus, Feb 12 2017

a(9)-a(12) from Amiram Eldar, Dec 23 2024

cp's OEIS Frontend

A082506 a(n) = gcd(2^n, n - phi(n)); largest power of 2 dividing cototient(n) = A051953(n).

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