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Spacetime02 (talk | contribs) The article previously said the sum of divisors of the nth primorial was 2^n, which is untrue (p1# = 2, which has divisor sum 3). This edit corrects that to the number of divisors of the nth primorial. |
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:<math> \sigma_0(p_n\#) = 2^n </math>
since ''n'' prime factors allow a sequence of binary selection (<math>p_{i}</math> or 1) from ''n'' terms for each proper divisor formed. However, these are not in general the smallest numbers whose number of divisors is a [[power of two]]; instead, the smallest such number may be obtained by multiplying together the first ''n'' [[Fermi–Dirac prime]]s, prime powers whose exponent is a power of two.<ref>{{citation
| last = Ramanujan | first = S. | author-link = Srinivasa Ramanujan
| doi = 10.1112/plms/s2_14.1.347
| issue = 1
| journal = Proceedings of the London Mathematical Society
| pages = 347–409
| title = Highly Composite Numbers
| volume = s2-14
| year = 1915}}; see section 47, pp. 405–406, reproduced in ''Collected Papers of Srinivasa Ramanujan'', Cambridge Univ. Press, 2015, [https://books.google.com/books?id=h1G2CgAAQBAJ&pg=PA124 pp. 124–125]</ref>
Clearly, <math>1 < \sigma_0(n) < n</math> for all <math>n > 2</math>, and <math>\sigma_x(n) > n </math> for all <math>n > 1</math>, <math>x > 0</math> .
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