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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.

%I A080360 #28 Feb 16 2025 08:32:48
%S A080360 10,16,28,40,46,58,66,70,96,100,106,126,148,150,166,178,180,226,228,
%T A080360 232,238,240,262,268,280,306,310,346,348,366,372,400,408,418,430,432,
%U A080360 438,460,486,490,502,568,570,586,592,598,600,606,640,642,646,652,658,676
%N 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.
%D A080360 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.
%H A080360 Amiram Eldar, <a href="/A080360/b080360.txt">Table of n, a(n) for n = 1..10000</a>
%H A080360 S. Ramanujan, <a href="http://ramanujan.sirinudi.org/Volumes/published/ram24.html">A proof of Bertrand's postulate</a>, J. Indian Math. Soc., 11 (1919), 181-182.
%H A080360 V. Shevelev, <a href="http://arxiv.org/abs/0909.0715">Ramanujan and Labos primes, their generalizations and classifications of primes</a>, arXiv:0909.0715 [math.NT], 2009-2011.
%H A080360 J. Sondow, <a href="https://mathworld.wolfram.com/RamanujanPrime.html">Ramanujan Prime in MathWorld</a>
%H A080360 J. Sondow and E. W. Weisstein, <a href="https://mathworld.wolfram.com/BertrandsPostulate.html">Bertrand's Postulate in MathWorld</a>
%H A080360 J. Sondow, <a href="http://arxiv.org/abs/0907.5232">Ramanujan primes and Bertrand's postulate</a>, Amer. Math. Monthly, 116 (2009), 630-635; arXiv:0907.5232 [math.NT], 2009-2010.
%H A080360 Wikipedia, <a href="http://en.wikipedia.org/wiki/Ramanujan_prime">Ramanujan prime</a>
%F A080360 a(n) = Max{x; Pi[x]-Pi[x/2]=n} = Max{x; A056171(x)=n} = Max{x; A056169(n!)=n}; where Pi()=A000720().
%F A080360 a(n) = A104272(n+1) - 1. [_Jonathan Sondow_, Aug 11 2008]
%e A080360 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.
%t A080360 nn = 60; R = Table[0, {nn}]; s = 0;
%t A080360 Do[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; If[s < nn, R[[s+1]] = k], {k, Prime[3*nn]}];
%t A080360 Rest[R] (* _Jean-François Alcover_, Dec 02 2018, after _T. D. Noe_ in A104272 *)
%Y A080360 Cf. A056171, A056169, A000720, A000142, A080359.
%Y A080360 Cf. A104272 (Ramanujan primes).
%K A080360 nonn
%O A080360 1,1
%A A080360 _Labos Elemer_, Feb 21 2003
%E A080360 Definition corrected by _Jonathan Sondow_, Aug 10 2008