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A062854 First differences of A027424.

%I A062854 #32 Aug 21 2024 21:14:02
%S A062854 1,2,3,3,5,4,7,5,6,6,11,6,13,8,9,8,17,9,19,10,12,12,23,10,16,14,15,13,
%T A062854 29,12,31,15,18,18,20,13,37,20,21,16,41,17,43,20,21,24,47,17,31,22,27,
%U A062854 23,53,22,31,22,30,30,59,19,61,32,28,26,36,26,67,30,36,26,71,23,73,38
%N A062854 First differences of A027424.
%C A062854 For prime p, a(p) = p. - _Ralf Stephan_, Jun 02 2005
%C A062854 a(n) is the number of times n appears in A033677. - _Franklin T. Adams-Watters_, Nov 18 2005
%C A062854 Conjecture: a(n) > n/log(n) for n > 2. - _Thomas Ordowski_, Jan 28 2017
%C A062854 a(n) is the number of integers 1<=i<=n such that all divisors of i*n are either <=i or >=n. - _Chai Wah Wu_, Oct 13 2023
%H A062854 Chai Wah Wu, <a href="/A062854/b062854.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%e A062854 a(4)=3 because there are 9 unique products in the 4 X 4 multiplication table (1 2 3 4 6 8 9 12 16), which is 3 more than the 6 unique products in the 3 X 3 multiplication table (1 2 3 4 6 9).
%p A062854 A062854 := proc(n)
%p A062854     A027424(n)-A027424(n-1) ;
%p A062854 end proc:
%p A062854 seq(A062854(n),n=1..40) ; # _R. J. Mathar_, Oct 02 2020
%t A062854 Prepend[Differences@ #, First@ #] &@ Module[{ u = {}}, Table[Length[u = Union[u, n Range@ n]], {n, 100}]] (* _Michael De Vlieger_, Jan 30 2017 *)
%o A062854 (PARI) b(n) = #setbinop((x, y)->x*y, vector(n, i, i); );
%o A062854 a(n) = b(n) - b(n-1); \\ _Michel Marcus_, Jan 28 2017
%o A062854 (Python)
%o A062854 from itertools import takewhile
%o A062854 from sympy import divisors
%o A062854 def A062854(n): return sum(1 for i in range(1,n+1) if all(d<=i for d in takewhile(lambda d:d<n,divisors(n*i)))) # _Chai Wah Wu_, Oct 13 2023
%Y A062854 Cf. A027421, A062854, A062855, A062856, A062857, A062859.
%K A062854 nonn
%O A062854 1,2
%A A062854 Ron Lalonde (ronronronlalonde(AT)hotmail.com), Jun 25 2001
%E A062854 More terms from _Ralf Stephan_, Jun 02 2005