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

1,1

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

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]

Example

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.

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 *)

Crossrefs

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

Cf. A104272 (Ramanujan primes).

Sequence in context: A155151 A104788 A249720 * A026320 A144206 A335675

Adjacent sequences: A080357 A080358 A080359 * A080361 A080362 A080363

Keyword

nonn

Author

Labos Elemer, Feb 21 2003

Extensions

Definition corrected by Jonathan Sondow, Aug 10 2008

Status

approved