

A181801


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


10



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, 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, 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
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OFFSET

1,2


COMMENTS

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.)
a(n) = a(n+12) if n is not a multiple of 12.


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Eric Weisstein's World of Mathematics, Highly composite number


FORMULA

a(n) = Sum_{dn} A322586(d).  Antti Karttunen, Dec 20 2018


EXAMPLE

6 is a multiple of 3 highly composite integers (1, 2 and 6); therefore a(6) = 3.
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.


PROG

(PARI)
v002182 = vector(128); v002182[1] = 1; \\ For memoization.
A002182(n) = { my(d, k); if(v002182[n], v002182[n], k = A002182(n1); d = numdiv(k); while(numdiv(k) <= d, k=k+1); v002182[n] = k; k); };
A261100(n) = { my(k=1); while(A002182(k)<=n, k=k+1); (k1); };
A322586(n) = if(1==n, 1, (A261100(n)A261100(n1)));
A181801(n) = sumdiv(n, d, A322586(d)); \\ Antti Karttunen, Dec 20 2018


CROSSREFS

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.
Cf. A002182, A181804, A181805, A322584.
Inverse MÃ¶bius transform of A322586.
Sequence in context: A328050 A176982 A079728 * A029244 A067513 A116372
Adjacent sequences: A181798 A181799 A181800 * A181802 A181803 A181804


KEYWORD

nonn


AUTHOR

Matthew Vandermast, Nov 27 2010


STATUS

approved



