A083162 a(n) is the smallest unused proper divisor or proper multiple of n such that a(n)/n != a(m)/m for all m < n.

2, 1, 9, 16, 25, 36, 49, 64, 3, 90, 110, 132, 156, 182, 210, 4, 255, 288, 323, 360, 399, 440, 483, 528, 5, 598, 648, 700, 754, 810, 868, 928, 990, 1054, 1120, 6, 1221, 1292, 1365, 1440, 1517, 1596, 1677, 1760, 1845, 1932, 2021, 2112, 7, 2250, 2346, 2444, 2544

Offset

1,1

Comments

Previous name: a(n) is either a multiple or a divisor of n but not equal to n, such that a(n)/n = a(m)/m implies m = n and n/a(n)= m/a(m) also implies n = m. Also a(m) = a(n) if and only if m = n.

Equivalently, a(n) is the smallest integer k other than n that is a divisor or multiple of n such that k/n != a(m)/m for all m < n.

Links

Table of n, a(n) for n=1..53.

Formula

a(a(n)) = n; a(n) = (b(k)/(k + 1)) if n = b(k) for some k and a(n) = n*(n - max{k: b(k) < n} + 1) otherwise, where b(k) is the k-th number at which a(n) < n. (Equivalently, b(k) is the unique x for which a(x)/x = (k+1).) - Carl B. Carlson (carlsonc(AT)stolaf.edu), Jan 09 2005

Example

a(3) = 9, a(3)/3 = 3 hence for no other m > 3, a(m) = 3m.
a(1000) = 1000*(1000-max{k: b(k) < 1000} + 1) = 1000*(1000-29+1) = 972000.

Crossrefs

A022342 gives the sequence analogous to b(n) if we replace the multiplications in the definition by additions.

Sequence in context: A199887 A058876 A214884 * A178075 A221756 A094633

Adjacent sequences: A083159 A083160 A083161 * A083163 A083164 A083165

Keyword

nonn

Author

Amarnath Murthy, Apr 25 2003

Extensions

More terms from Carl B. Carlson (carlsonc(AT)stolaf.edu), Jan 09 2005

Edited and more terms added, Jon E. Schoenfield, Feb 21 2021

Status

approved