A232099 Numbers n such that {largest m such that 1, 2, ..., m divide n} is different from {largest m such that m! divides n^2}.

840, 2520, 4200, 5880, 7560, 9240, 10920, 12600, 14280, 15960, 17640, 19320, 21000, 22680, 24360, 26040, 27720, 29400, 31080, 32760, 34440, 36120, 37800, 39480, 41160, 42840, 44520, 46200, 47880, 49560, 51240, 52920, 54600, 55440, 56280, 57960, 59640, 61320, 63000

Offset: 1

Author

Antti Karttunen, Nov 18 2013

Keywords

• nonn

Comments

Numbers n such that A055874(n) differs from A232098(n). (By the definition of the sequence).

This sequence is a subset of A055926. Please see there for a proof. From that follows that A055881(a(n))+1 is always composite (in range n=1..100000, only values 6, 8, 9 and 10 occur).

Also, incidentally, for the first five terms, n=1..5, a(n) = 70*A055926(n), then a(6)=77*A055926(6), and the next time the ratio A232099(n)/A055926(n) is integral is at n=21, where a(n) = 82*A055926(21), at n=41 (a(41) = 79*A055926(41) = 79*840 = 66360), at n=136, a(136) = 80*A055926(136) = 80*2772 = 221760 and at n=1489, where a(1489) = 80*A055926(1489) = 80 * 30492 = 2439360. The ratio seems to converge towards some value a little less than 80. Please see the plot generated by Plot2 in the links section.

Examples

			840 (= 3*5*7*8) is in the sequence as all natural numbers up to 8 divide 840, but the largest factorial that divides its square, 705600, is 7! (840^2 = 140 * 5040), and 7 differs from 8.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• OEIS Server, Ratio A232099(n)/A055926(n) plotted with Plot 2
• OEIS Server, Ratio A232099(n)/A232743(n) plotted with Plot 2
• Wikipedia, Wilson's theorem (Please see especially the section "Composite modulus")

Crossrefs

Subset of A055926.

Cf. A055881, A055874, A232098, A232100.

Formula

For all n, a(n) = A055926(A232100(n)). [Follows from the definition of A232100, but cannot as such be used to compute the sequence. Use the given Scheme-program instead.]