cp's OEIS Frontend

A181801 Number of divisors of n that are highly composite (A002182).

%I A181801 #16 Feb 16 2025 08:33:13
%S A181801 1,2,1,3,1,3,1,3,1,2,1,5,1,2,1,3,1,3,1,3,1,2,1,6,1,2,1,3,1,3,1,3,1,2,
%T A181801 1,6,1,2,1,3,1,3,1,3,1,2,1,7,1,2,1,3,1,3,1,3,1,2,1,6,1,2,1,3,1,3,1,3,
%U A181801 1,2,1,7,1,2,1,3,1,3,1,3,1,2,1,5,1,2,1,3,1,3,1,3,1,2,1,7,1,2,1,3,1,3,1,3,1
%N A181801 Number of divisors of n that are highly composite (A002182).
%C A181801 A divisor d of integer n is highly composite iff more multiples of (n/d) divide n than divide any smaller positive integer. This is because the number of divisors of n that are multiples of (n/d) equals the number of divisors of d, or A000005(d). (Also see example.)
%C A181801 a(n) = a(n+12) if n is not a multiple of 12.
%H A181801 Antti Karttunen, <a href="/A181801/b181801.txt">Table of n, a(n) for n = 1..20000</a>
%H A181801 Antti Karttunen, <a href="/A181801/a181801.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%H A181801 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HighlyCompositeNumber.html">Highly composite number</a>.
%F A181801 a(n) = Sum_{d|n} A322586(d). - _Antti Karttunen_, Dec 20 2018
%F A181801 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A352418 = 2.132872... . - _Amiram Eldar_, Jan 01 2024
%e A181801 6 is a multiple of 3 highly composite integers (1, 2 and 6); therefore a(6) = 3.
%e A181801 As the first comment implies, there are also a(6) = 3 values of m such that 6 sets a record for number of divisors that are multiples of m. These values of m are 1, 3 and 6. All four of 6's divisors are multiples of 1; two (namely, 3 and 6) are multiples of 3; and one (namely, 6) is a multiple of 6. Each of these totals exceeds the corresponding total for any positive integer smaller than 6.
%o A181801 (PARI)
%o A181801 v002182 = vector(128); v002182[1] = 1; \\ For memoization.
%o A181801 A002182(n) = { my(d,k); if(v002182[n],v002182[n], k = A002182(n-1); d = numdiv(k); while(numdiv(k) <= d, k=k+1); v002182[n] = k; k); };
%o A181801 A261100(n) = { my(k=1); while(A002182(k)<=n,k=k+1); (k-1); };
%o A181801 A322586(n) = if(1==n,1,(A261100(n)-A261100(n-1)));
%o A181801 A181801(n) = sumdiv(n,d,A322586(d)); \\ _Antti Karttunen_, Dec 20 2018
%Y A181801 Row n of A181802 gives highly composite divisors of n. Row n of A181803 gives values of m such that n sets a record for the number of its divisors that are multiples of m. Numbers that set records for a(n) are in A181806.
%Y A181801 Cf. A002182, A181804, A181805, A322584, A352418.
%Y A181801 Inverse Möbius transform of A322586.
%K A181801 nonn
%O A181801 1,2
%A A181801 _Matthew Vandermast_, Nov 27 2010