A137849 - OEIS

%I #8 Oct 31 2013 12:17:44

%S 1,2,2,4,2,6,2,7,5,7,2,12,2,9,8,14,2,17,2,17,10,10,2,24,9,12,12,23,2,

%T 28,2,25,13,14,12,35,2,15,14,34,2,36,2,30,23,17,2,48,14,33,18,34,2,46,

%U 18,45,19,19,2,60,2,21,30,49,20,54,2,41,22,47,2,71,2,24,36,45,21,63,2,67

%N Number of integers m from 1 through n inclusive such that d_i(n)<=d_i(m) for 1<=i<=Min(d(n),d(m)) where d_i(n) denotes the i-th smallest divisor of n and d(n) denotes the number of divisors of n (A000005).

%C In other words, number of integers m in {1,...,n} such that the i-th divisor of m is >= the i-th divisor of n, for i=1,...,min(A000005(m),A000005(n)).

%H Robert G. Wilson v, <a href="/A137849/b137849.txt">Table of n, a(n) for n = 1..10000</a>.

%F a(n) = 2 iff n is prime.

%e a(10) = 7 because there are 7 integers, 1, 2, 3, 5, 7, 9 and 10, whose divisors meet the criterion for n = 10 (4 does not meet this criterion in that 4's 3rd smallest divisor is 4 and 10's third smallest divisor is 5; similarly 6 and 8 do not meet the criterion).

%t f[n_] := Block[{c = 1, d = Divisors@ n, k = DivisorSigma[0, n], m = 1}, While[m != n, If[ Min[ PadRight[ Divisors@ m, k, n] - d] > -1, c++ ]; m++ ]; c]; Array[f, 80] (* _Robert G. Wilson v_ *)

%o (PARI) A137849(n)={ local( d=divisors(n), d2 ); sum( m=1,n, d2=divisors(m); vecmin( vector(min(#d,#d2),i,d2[i]-d[i]))>=0 )} \\ - M. F. Hasler, May 01 2008

%Y Cf. A094783 (numbers where a(n) = n), Records: A140067.

%K nonn

%O 1,2

%A _J. Lowell_, Apr 29 2008

%E Edited and extended by _M. F. Hasler_ and _Ray Chandler_, May 01 2008