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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.

%I A281627 #47 Jul 26 2025 19:58:56
%S A281627 3,5,17,85,4369,65537,327685,1431655765,2305843009213693952,
%T A281627 618970019642690137449562112,162259276829213363391578010288128,
%U A281627 170141183460469231731687303715884105728
%N 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.
%C A281627 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.
%C A281627 Conjecture 2: a(n) = the smallest number k such that phi(k) has exactly A000043(n)-1 divisors; see A276044.
%C A281627 Conjecture 3: a(n) = the smallest number k such that phi(k) has exactly A000043(n)-1 prime factors (counted with multiplicity); see A275969.
%C A281627 a(n) <= A000668(n) for n = 1-18; conjecture: a(n) <= A000668(n) for all n.
%C A281627 Equals A002181 (index in A002202 of (intersection of A023194 and A002202)). - _Michel Marcus_, Feb 12 2017
%H A281627 Amiram Eldar, <a href="/A281627/b281627.txt">Table of n, a(n) for n = 1..13</a>
%H A281627 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%o A281627 (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)))]]));
%o A281627 (PARI) terms() = {v = readvec("b023194.txt"); for(i=1, #v, if (istotient(v[i], &n), print1(n/2, ", ")););} \\ _Michel Marcus_, Feb 12 2017
%o A281627 (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
%o A281627 list(pmax) = forprime(p = 1, pmax, if(isprime(2^p-1), print1(f(2^p-1), ", "))); \\ _Amiram Eldar_, Dec 23 2024
%Y A281627 Cf. A000010, A000043, A000203, A000668, A062402, A062514, A275969, A276044.
%Y A281627 Cf. A002181, A002202, A023194.
%Y A281627 Cf. A053576 (includes the first 13 known terms of this sequence).
%K A281627 nonn
%O A281627 1,1
%A A281627 _Jaroslav Krizek_, Feb 11 2017
%E A281627 a(8) from _Michel Marcus_, Feb 12 2017
%E A281627 a(9)-a(12) from _Amiram Eldar_, Dec 23 2024