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A181801
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Number of divisors of n that are highly composite (A002182).
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10
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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
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A352418 = 2.132872... . - Amiram Eldar, Jan 01 2024
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EXAMPLE
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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.
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PROG
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(PARI)
v002182 = vector(128); v002182[1] = 1; \\ For memoization.
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); };
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CROSSREFS
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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.
Inverse Möbius transform of A322586.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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