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A273379 Ramanujan's largely composite numbers n (A067128) which are not divisible by all the primes < p, where p is the greatest prime divisor of n.

%I A273379 #86 Jan 24 2024 10:12:53
%S A273379 3,10,20,84,168,336,504,660,672,3960,4680,32760,42840,43680,65520,
%T A273379 98280,131040,163800,196560,262080,327600,393120,471240,491400,655200,
%U A273379 942480,982800,1053360,1413720,1884960,1965600,2106720,2827440,5654880,6320160,13693680,14137200,20540520,27387360,28274400
%N A273379 Ramanujan's largely composite numbers n (A067128) which are not divisible by all the primes < p, where p is the greatest prime divisor of n.
%C A273379 No term is a highly composite number (A002182), since as P. Erdős noted, every highly composite number is divisible by every prime less than p; on the other hand, the sequence is a strict subsequence of A244353.
%C A273379 Let p=prime(k), k=k(n), be the greatest prime divisor of a(n). _David A. Corneth_ noted that a(11)=4680 is the first term which is not divisible by both prime(k-2) and prime(k-1).
%C A273379   What is the next such term?
%C A273379 Is there a case when prime(k-1) divides a(n), but prime(k-2) does not? - _Vladimir Shevelev_, May 21 2016
%C A273379 From _David A. Corneth_, May 22 2016: (Start)
%C A273379 It seems that most terms have the property mentioned in the second comment. 105753195046699200 is the first term whose three largest distinct prime divisors are consecutive primes. No other terms below that number have that property.
%C A273379 There are many terms where prime(k-1) divides a(n), but prime(k-2) does not. They are 107442720, 537213600, 1074427200, 1889727840, 9448639200, 18897278400, 37794556800, ... (End)
%C A273379 Intersection of A067128 and A080259. - _Michel Marcus_, May 22 2016; edited by _Michael De Vlieger_, Jan 23 2024
%C A273379 From _Vladimir Shevelev_, May 23 2016: (Start)
%C A273379 To explain what happens, note that, if p = p_k = prime(k), k = k(n), is the greatest prime divisor of a(n), then b(n)=a(n)*prime(k+r)/prime(k) is also in the sequence, if between a(n) and b(n) there is no highly composite number(HCN). Indeed, d(a(n))=d(b(n)) and in the interval [a(n),b(n)] there is no number x with a greater d(x).
%C A273379 For example, between the term 3960=2^3*3^2*5*11 and 4680=3960*13/11 there is no HCN, so 4680 is also a term; however, between 3960 and 3960*17/11=6120 there is an HCN, so 6120 is not a term.
%C A273379 If n is a large HCN, then its greatest prime divisor is O(log n)[Erdős]. So we can also say that prime(k)=O(log n), since, by Erdős's theorem, if n_1 is the next HCN, then n < n_1 < n + n*(log n)^-c, where c > 0 is constant. So if
%C A273379       p_(k+r)/p_k < 1 + (log n)^-c (1)
%C A273379 then there is a real possibility that there are numbers a(m)*p_(k+i)/p_k, i=1,...,r, which are all terms, where a(m) is between n and the next HCN. Note that (1) means that (1+o(1))*(k+r)*log(k+r)/(k*log k) < 1 + (log n)^-c, where k=O(log n/loglog n). Since, for r < k, log(k+r) = log k + log(1+r/k) ~ log k  +r/k + O((r/k)^2), then log(k+r)/log k = 1 + O(r/(k*log k). Thus (1) means r < k/(log n)^c = O(log n)^(1-c)/loglog n. Thus if c < 1, then r could be arbitrary large for sufficiently large n.
%C A273379   Moreover, the numerical results of _David A. Corneth_ (cf. A273415) allow us to conjecture that indeed 0 < c < 1. (End)
%H A273379 Amiram Eldar, <a href="/A273379/b273379.txt">Table of n, a(n) for n = 1..7692</a>
%H A273379 P. Erdős, <a href="https://www.renyi.hu/~p_erdos/1944-04.pdf">On Highly composite numbers</a>, J. London Math. Soc. 19 (1944), 130--133 MR7,145d; Zentralblatt 61,79.
%H A273379 Vladimir Shevelev, <a href="http://arxiv.org/abs/1605.08884">On Erdős constant</a>, arXiv:1605.08884 [math.NT], 2016.
%t A273379 r = 0; fQ[x_] := Nor[IntegerQ @Log2[x], And[EvenQ[x], Union@ Differences@ PrimePi[FactorInteger[x][[All, 1]]] == {1}]]; Reap[Do[If[# >= r, r = #; If[fQ[i], Sow[i]]] &[DivisorSigma[0, i] ], {i, 2^20}] ][[-1, 1]] (* _Michael De Vlieger_, Jan 23 2024 *)
%Y A273379 Cf. A002182, A067128, A244343.
%K A273379 nonn
%O A273379 1,1
%A A273379 _Vladimir Shevelev_ and _Peter J. C. Moses_, May 21 2016
%E A273379 Inserted a missed term a(11)=4680 by _David A. Corneth_, May 21 2016
%E A273379 Inserted missing a(13), a(28), a(32) and a(35) by _David A. Corneth_, May 22 2016