A075460 Odd primitive numbers such that n! divided by product of factorials of all proper divisors of n is not an integer.

1575, 2835, 3465, 4095, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 8085, 9135, 9765, 11655, 12705, 12915, 13545, 14805, 15015, 15435, 16695, 18585, 19215, 19635, 21105, 21945, 22275, 22365, 22995, 23205, 24885, 25245, 25935, 26145, 26565

Offset: 1

Author

Robert G. Wilson v, Sep 16 2002

Keywords

• base
• nonn

Comments

If a number is in the sequence, then all of its multiples would also meet the criterion, but are not included. This is meant by the word "primitive" in the definition.

Examples

			1575 = 3^2*5^2*7 is in the sequence, because the product of the factorials of its proper divisors { 1, 3, 5, 7, 9, 15, 21, 25, 35, 45, 63, 75, 105, 175, 225, 315, 525 } does not divide 1575!. (For example, the former's 2-adic valuation equals 1588 while the latter's 2-adic valuation equals only 1569.) This is the smallest odd number with this property. - _M. F. Hasler_, Dec 30 2016
		

Crossrefs

Cf. A075071. The first primitive n's with this property (most of which are even) are in A075422.

Programs

Mathematica

f[n_] := n!/Apply[Times, Drop[Divisors[n], -1]! ]; a = {}; Do[b = f[n]; If[ !IntegerQ[b], If[ Select[n/a, IntegerQ] == {}, Print[n]; a = Append[a, n]]], {n, 1, 28213, 2}]; a

Extensions

Edited by M. F. Hasler, Dec 30 2016