A080360 a(n) is the largest positive integer x such that the number of unitary-prime-divisors of x! equals n. Same as the largest positive integer x such that the number of primes in (x/2,x] equals n.

10, 16, 28, 40, 46, 58, 66, 70, 96, 100, 106, 126, 148, 150, 166, 178, 180, 226, 228, 232, 238, 240, 262, 268, 280, 306, 310, 346, 348, 366, 372, 400, 408, 418, 430, 432, 438, 460, 486, 490, 502, 568, 570, 586, 592, 598, 600, 606, 640, 642, 646, 652, 658, 676

Offset: 1

Author

Labos Elemer, Feb 21 2003

Keywords

• nonn

Examples

			n=5: in 46! five unitary-prime-divisors[UPD] appear: {29,31,37,41,43}. In larger factorials number of UPD is not more equal 5. Thus a(5)=46.
		

References

• S. Ramanujan, Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, S. Aiyar, P. Venkatesvara and B. M. Wilson), Amer. Math. Soc., Providence, 2000, pp. 208-209.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.
• V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009-2011.
• J. Sondow, Ramanujan Prime in MathWorld
• J. Sondow and E. W. Weisstein, Bertrand's Postulate in MathWorld
• J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, 116 (2009), 630-635; arXiv:0907.5232 [math.NT], 2009-2010.
• Wikipedia, Ramanujan prime

Crossrefs

Cf. A056171, A056169, A000720, A000142, A080359.

Cf. A104272 (Ramanujan primes).

Programs

Mathematica

nn = 60; R = Table[0, {nn}]; s = 0;
Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, R[[s+1]] = k], {k, Prime[3*nn]}];
Rest[R] (* Jean-François Alcover, Dec 02 2018, after T. D. Noe in A104272 *)

Formula

a(n) = Max{x; Pi[x]-Pi[x/2]=n} = Max{x; A056171(x)=n} = Max{x; A056169(n!)=n}; where Pi()=A000720().

a(n) = A104272(n+1) - 1. [Jonathan Sondow, Aug 11 2008]

Extensions

Definition corrected by Jonathan Sondow, Aug 10 2008