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

1,1

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).

Prog

(Magma) A281627:=func<n|exists(r){k:k in[1..1000000] | SumOfDivisors(EulerPhi(k)) eq n}select r else 0>; 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

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).

Sequence in context: A365518 A227335 A351718 * A102846 A100003 A283331

Adjacent sequences: A281624 A281625 A281626 * A281628 A281629 A281630

Keyword

nonn

Author

Jaroslav Krizek, Feb 11 2017

Extensions

a(8) from Michel Marcus, Feb 12 2017

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

Status

approved