A083162 - OEIS

%I #12 Feb 27 2021 21:35:55

%S 2,1,9,16,25,36,49,64,3,90,110,132,156,182,210,4,255,288,323,360,399,

%T 440,483,528,5,598,648,700,754,810,868,928,990,1054,1120,6,1221,1292,

%U 1365,1440,1517,1596,1677,1760,1845,1932,2021,2112,7,2250,2346,2444,2544

%N 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.

%C 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.

%C 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.

%F 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

%e a(3) = 9, a(3)/3 = 3 hence for no other m > 3, a(m) = 3m.

%e a(1000) = 1000*(1000-max{k: b(k) < 1000} + 1) = 1000*(1000-29+1) = 972000.

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

%K nonn

%O 1,1

%A _Amarnath Murthy_, Apr 25 2003

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

%E Edited and more terms added, _Jon E. Schoenfield_, Feb 21 2021