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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.
0
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.
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
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