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.

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

Offset: 1

Author

Jaroslav Krizek, Feb 11 2017

Keywords

• nonn

Comments

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

Links

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

Crossrefs

Cf. A000010, A000043, A000203, A000668, A062402, A062514, A275969, A276044.

Cf. A002181, A002202, A023194.

Cf. A053576 (includes the first 13 known terms of this sequence).

Programs

Magma

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

PARI

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

PARI

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

Extensions

a(8) from Michel Marcus, Feb 12 2017

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