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A276044
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Least k such that phi(k) has exactly n divisors.
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3
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1, 3, 5, 7, 17, 13, 85, 31, 37, 65, 1285, 61, 4369, 193, 185, 143, 65537, 181, 327685, 241, 577, 3281, 5570645, 403, 1297, 12289, 1057, 1037, 286331153, 779, 1431655765, 899, 9509, 197633, 5629, 1333, 137438953472, 786433, 42653, 1763, 2199023255552, 2993, 8796093022208, 15361, 3737, 12648641
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OFFSET
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1,2
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COMMENTS
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For every n > 0, phi(2^n) = 2^(n-1) has exactly n divisors, so a(n) <= 2^n.
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:
(1) a(p) is the least k such that phi(k) = 2^(p-1), and
(2) 2^(p-1) < a(p) <= 2^p.
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.
For prime p < 37, a(p) = A001317(p-1).
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:
. p-1 in
p a(p) prime factorization of a(p) binary
== ========== =========================== ======
2 3 = 3 1
3 5 = 5 10
5 17 = 17 100
7 85 = 17 * 5 110
11 1285 = 257 * 5 1010
13 4369 = 257 * 17 1100
17 65537 = 65537 10000
19 327685 = 65537 * 5 10010
23 5570645 = 65537 * 17 * 5 10110
29 286331153 = 65537 * 257 * 17 11100
31 1431655765 = 65537 * 257 * 17 * 5 11110
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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.)
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
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LINKS
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FORMULA
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a(p) = 2^p for primes p with 32 < p <= 2^33. - Pjotr Buys, Sep 18 2019
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EXAMPLE
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a(5) = 17 because phi(17) = 16 has 5 positive divisors.
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MATHEMATICA
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Table[k = 1; While[DivisorSigma[0, #] &@ EulerPhi@ k != n, k++]; k, {n, 28}] (* Michael De Vlieger, Aug 21 2016 *)
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PROG
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(PARI) a(n) = {my(k = 1); while(numdiv(eulerphi(k)) != n, k++); k; }
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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