

A045619


Numbers that are the products of 2 or more consecutive integers.


14



0, 2, 6, 12, 20, 24, 30, 42, 56, 60, 72, 90, 110, 120, 132, 156, 182, 210, 240, 272, 306, 336, 342, 360, 380, 420, 462, 504, 506, 552, 600, 650, 702, 720, 756, 812, 840, 870, 930, 990, 992, 1056, 1122, 1190, 1260, 1320, 1332, 1406, 1482, 1560, 1640, 1680
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OFFSET

1,2


COMMENTS

Erdős and Selfridge proved that, apart from the first term, these are never perfect powers (A001597).  T. D. Noe, Oct 13 2002
Numbers of the form x!/y! with y+1 < x.  Reinhard Zumkeller, Feb 20 2008
a(n)=A000142(A137911(n))/A000142(A137912(n)1) for n>1.  Reinhard Zumkeller, Feb 27 2008


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000
P. Erdős and J. L. Selfridge, The product of consecutive integers is never a power, Illinois Jour. Math. 19 (1975), 292301.


FORMULA

Since the oblong numbers (A002378) have relative density of 100%, we have a(n) ~ (n1) n ~ n^2.  Daniel Forgues, Mar 26 2012
a(n) = n^2  2n^(5/3) + O(n^(4/3)).  Charles R Greathouse IV, Aug 27 2013


MATHEMATICA

maxNum = 1700; lst = {}; For[i = 1, i <= Sqrt[maxNum], i++, j = i + 1; prod = i*j; While[prod < maxNum, AppendTo[lst, prod]; j++; prod *= j]]; lst = Union[lst]


PROG

(PARI) is(n)=my(s=1, F=2, t); while(n%F==0, t=round(n^(1/(s+1))s/2); if(prod(i=0, s, t+i)==n, return(1)); s++; F*=s+1); 0 \\ Charles R Greathouse IV, Aug 27 2013


CROSSREFS

Cf. A001597, A000142, A137895, A053625, A093449, A064224, A084720, A137899, A137900, A120436.
Sequence in context: A102711 A235375 A141406 * A028690 A270878 A120344
Adjacent sequences: A045616 A045617 A045618 * A045620 A045621 A045622


KEYWORD

easy,nonn,nice


AUTHOR

Erich Friedman


EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Jul 20 2000
More terms from Reinhard Zumkeller, Feb 27 2008


STATUS

approved



