

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.


16



1, 2, 4, 6, 12, 24, 36, 48, 60, 72, 120, 180, 240, 360, 420, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 30240, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 360360, 498960, 554400, 665280, 720720
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OFFSET

1,2


COMMENTS

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.
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
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 occurrence of a(n) and M(n) = max{F(i),i <= n}. Then the following table indicates this connection.
n 1,2,3,4, 5, 6, 7, 8, 9,10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21
a(n)1,2,4,6,12,24,36,48,60,72,120,180,240,360,420,720,840,1260,1680,2520,5040
F(n)1,2,4,3, 4, 8,18,16, 5, 9, 8, 18, 16, 9, 7, 16, 15, 21, 16, 9, 16
M(n)1,2,4,4, 4, 8,18,18,18,18, 18, 18, 18, 18, 18, 18, 18, 21, 21, 21, 21
 Peter Luschny, Dec 29 2010


LINKS

Table of n, a(n) for n=1..42.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 844.
J. Britton, Perfect Number Analyzer
Wikipedia, Table of divisors.
Peter Luschny, More terms, OEIS wiki.


EXAMPLE

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.


PROG

(PARI) \\ Computes the first 50 terms of A094348
A094348() = {local(n, i, R, A, len, count, change, high, lim);
lim = 7208000; R = vector(500); A = vector(50); A[1]=1;
A[2]=2; A[3]=4; A[4]=6; A[5]=12; count=5; high=0; n=12;
while(n < lim, d=divisors(n); len=length(d); change=0;
for(i=1, min(len, high), if(R[i]>d[i], R[i]=d[i]; change=1));
if(len>high, for(i=high+1, len, R[i]=d[i]); high=len);
if(change, count++; A[count] = n); n += 12; );
write("A094348.txt", vector(count, i, A[i])); }
\\ Peter Luschny, Dec 29 2010


CROSSREFS

Cf. A096179, A003418, A002182, A002201, A072938, A106037, A002110.
Sequence in context: A141420 A141551 A181804 * A242298 A002182 A340840
Adjacent sequences: A094345 A094346 A094347 * A094349 A094350 A094351


KEYWORD

nonn


AUTHOR

Matthew Vandermast, Jun 18 2004, Oct 12 2008


EXTENSIONS

More terms from David Wasserman, Jun 28 2007
Title edited by Matthew Vandermast, Nov 20 2010


STATUS

approved



