This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A276044 #50 Aug 16 2025 22:23:43
%S A276044 1,3,5,7,17,13,85,31,37,65,1285,61,4369,193,185,143,65537,181,327685,
%T A276044 241,577,3281,5570645,403,1297,12289,1057,1037,286331153,779,
%U A276044 1431655765,899,9509,197633,5629,1333,137438953472,786433,42653,1763,2199023255552,2993,8796093022208,15361,3737,12648641
%N A276044 Least k such that phi(k) has exactly n divisors.
%C A276044 Least k such that A000005(A000010(k)) = n.
%C A276044 From _Jon E. Schoenfield_, Nov 13 2016: (Start)
%C A276044 For every n > 0, phi(2^n) = 2^(n-1) has exactly n divisors, so a(n) <= 2^n.
%C A276044 For every prime p, since phi(a(p)) has exactly p divisors, phi(a(p)) must be of the form q^(p-1), where q is a prime number. If q >= 3, we would have phi(a(p)) >= 3^(p-1), and since k > phi(k) for every k > 1, we would have a(p) >= 3^(p-1)+1, which would be contradicted by the upper bound a(p) <= 2^p (see above) unless 3^(p-1)+1 <= 2^p, which is true only for p = 2. Thus, for every prime p > 2, phi(a(p)) = 2^(p-1), so a(p) > 2^(p-1). In summary, we can state that, for every prime p > 2:
%C A276044 (1) a(p) is the least k such that phi(k) = 2^(p-1), and
%C A276044 (2) 2^(p-1) < a(p) <= 2^p.
%C A276044 After a(36)=1333, the next few known terms are a(38)=786433, a(39)=42653, a(40)=1763, and a(42)=2993; as shown above, known bounds on a(37) and a(41) are 2^36 < a(37) <= 2^37 and 2^40 < a(41) <= 2^41.
%C A276044 For prime p < 37, a(p) = A001317(p-1).
%C A276044 Observation: for prime p < 37, a(p) is the product of distinct Fermat primes 2^(2^j)+1 for j=0..4, i.e., 3, 5, 17, 257, and 65537 (see A019434), according to the locations of the 1-bits in p-1:
%C A276044 . p-1 in
%C A276044 p a(p) prime factorization of a(p) binary
%C A276044 == ========== =========================== ======
%C A276044 2 3 = 3 1
%C A276044 3 5 = 5 10
%C A276044 5 17 = 17 100
%C A276044 7 85 = 17 * 5 110
%C A276044 11 1285 = 257 * 5 1010
%C A276044 13 4369 = 257 * 17 1100
%C A276044 17 65537 = 65537 10000
%C A276044 19 327685 = 65537 * 5 10010
%C A276044 23 5570645 = 65537 * 17 * 5 10110
%C A276044 29 286331153 = 65537 * 257 * 17 11100
%C A276044 31 1431655765 = 65537 * 257 * 17 * 5 11110
%C A276044 .
%C A276044 This pattern does not continue to p=37, since 2^(2^5)+1 is not prime. (See also A038183 and the observation there from _Arkadiusz Wesolowski_.) (End)
%C A276044 As noted, for every prime p, phi(a(p))=2^(p-1), decompose a(p) = p_1^(e_1) *...* p_m^(e_m), then phi(a(p)) = p_1^(e_1-1)*(p_1 - 1) * ... * p_m^(e_m-1)*(p_m - 1). Thus a(p) is of the form 2^e * F_(a_1) *...* F_(a_l), where F_(a_i) = 2^(a_i) + 1 denote distinct Fermat primes. If e = 0, a_1 + ... + a_l = p - 1, while if e > 0, e + a_1 + ... + a_l = p. It can be deduced that a(p) = 2^p unless p-1 can be written as a_1 + ... a_l where 2^(a_i) + 1 are distinct Fermat primes. The only Fermat primes known have a_i in {1,2,4,8,16} and it is known that 2^a + 1 is composite for 16 < a < 2^33 (cf. A019434). It follows from the fact that 1 + 2 + 4 + 8 + 16 = 31 that a(p) = 2^p for primes p with 32 < p <= 2^33. - _Pjotr Buys_, Sep 18 2019
%H A276044 David A. Corneth, <a href="/A276044/b276044.txt">Table of n, a(n) for n = 1..512</a>
%F A276044 a(p) = 2^p for primes p with 32 < p <= 2^33. - _Pjotr Buys_, Sep 18 2019
%e A276044 a(5) = 17 because phi(17) = 16 has 5 positive divisors.
%t A276044 Table[k = 1; While[DivisorSigma[0, #] &@ EulerPhi@ k != n, k++]; k, {n, 28}] (* _Michael De Vlieger_, Aug 21 2016 *)
%o A276044 (PARI) a(n) = {my(k = 1); while(numdiv(eulerphi(k)) != n, k++); k; }
%Y A276044 Cf. A000005, A000010, A001317, A002181, A019434, A038183, A062821.
%K A276044 nonn
%O A276044 1,2
%A A276044 _Altug Alkan_, Aug 17 2016
%E A276044 a(31)-a(36) from _Michel Marcus_ and _Jon E. Schoenfield_, Nov 13 2016
%E A276044 a(37)-a(46) from _Pjotr Buys_, Sep 18 2019