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

Author

Amarnath Murthy, Apr 25 2003

Keywords

• nonn

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.

Examples

			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.

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

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