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 A094348 Numbers n such that, for some numbers (j,k), j<=k, n is the smallest positive multiple of j (or more) of the first k positive integers. 15

%I

%S 1,2,4,6,12,24,36,48,60,72,120,180,240,360,420,720,840,1260,1680,2520,

%T 5040,7560,10080,15120,20160,25200,27720,30240,45360,50400,55440,

%U 83160,110880,166320,221760,277200,332640,360360,498960,554400,665280,720720

%N Numbers n such that, for some numbers (j,k), j<=k, n is the smallest positive multiple of j (or more) of the first k positive integers.

%C Includes all highly composite numbers (A002182) and least common multiples of 1 through n (A003418). It would be interesting to know: 1) whether or not all deeply composite numbers (A095848, which includes all members of A003418) also belong to this sequence; 2) if 72 is the only member of this sequence not also belonging to A002182 or A095848.

%C 465585120 is the first member of A095848 that is not a member of this sequence. The first members that belong to neither A002182 nor A095848 are 72, 30240, 64864800 and 1470268800. - _David Wasserman_, Jun 28 2007

%C This sequence is also A096179 with duplicates deleted and sorted. Let F(n) be the number of the row of A096179 which has the first occurence of a(n) and M(n) = max{F(i),i <= n}. Then the following table indicates this connection.

%C n |1,2,3,4, 5, 6, 7, 8, 9,10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21

%C a(n)|1,2,4,6,12,24,36,48,60,72,120,180,240,360,420,720,840,1260,1680,2520,5040

%C F(n)|1,2,4,3, 4, 8,18,16, 5, 9, 8, 18, 16, 9, 7, 16, 15, 21, 16, 9, 16

%C M(n)|1,2,4,4, 4, 8,18,18,18,18, 18, 18, 18, 18, 18, 18, 18, 21, 21, 21, 21

%C - _Peter Luschny_, Dec 29 2010

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 844.

%H J. Britton, <a href="http://britton.disted.camosun.bc.ca/perfect/jbperfect.htm">Perfect Number Analyzer</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Table_of_divisors">Table of divisors</a>.

%H Peter Luschny, <a href="/wiki/User:Peter_Luschny/A057641A094348">More terms</a>, OEIS wiki.

%e 72 is a multiple of seven of the first nine positive integers (namely, 1, 2, 3, 4, 6, 8 and 9). It is the smallest positive integer for which this is true.

%o \\ (PARI) Computes the first 50 terms of A094348

%o A094348() = {local(n,i,R,A,len,count,change,high,lim);

%o lim = 7208000; R = vector(500); A = vector(50); A[1]=1;

%o A[2]=2; A[3]=4; A[4]=6; A[5]=12; count=5; high=0; n=12;

%o while(n < lim, d=divisors(n); len=length(d); change=0;

%o for(i=1,min(len,high),if(R[i]>d[i],R[i]=d[i];change=1));

%o if(len>high,for(i=high+1,len,R[i]=d[i]); high=len);

%o if(change, count++; A[count] = n); n += 12; );

%o write("A094348.txt", vector(count, i, A[i])); }

%o - _Peter Luschny_, Dec 29 2010

%Y Cf. A096179, A003418, A002182, A002201, A072938, A106037, A002110.

%K nonn

%O 1,2

%A _Matthew Vandermast_, Jun 18 2004, Oct 12 2008

%E More terms from _David Wasserman_, Jun 28 2007

%E Title edited by _Matthew Vandermast_, Nov 20 2010

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