Smarandacheials

edited by J. Dezert

ONERA, France

Let n>k≥1 be two integers.  Then the Smarandacheial is defined as:

!n!k =  (n-k·i)

0<|n-k·i|≤n

icN

For examples:

1) In the case k=1:

conv

!n!1≡ !n! =  (n-i) = n(n-1)(n-2)…(2)(1)(-1)(-2)…(-n+2)(-n+1)(-n) = (-1)n(n!)2.

0<|n-i|≤n

i=0, 1, 2, … .

Thus !5! = 5(5-1)(5-2)(5-3)(5-4)(5-6)(5-7)(5-8)(5-9)(5-10)=5·4·3·2·1·(-1)·(-2)·(-3)·(-4)·(-5) = -14400.

The sequence is: 4, -36, 576, -14400, 518400, -25401600, 1625702400, -131681894400, 13168189440000,

-1593350922240000, 229442532802560000, -38775788043632640000, 7600054456551997440000,

-1710012252724199424000000, … .

2) In the case k=2:

a) If n is odd, then

!n!2 =  (n-2i) = n(n-2)(n-4)…(3)(1)(-1)(-3)…(-n+4)(-n+2)(-n) = (-1)(n+1)/2(n!!)2.

0<|n-2i|≤n

i=0, 1, 2, … .

a) If n is even, then

!n!2 =  (n-2i) = n(n-2)(n-4)…(4)(2)(-2)(-4)…(-n+4)(-n+2)(-n) = (-1)n/2(n!!)2.

0<|n-2i|≤n

i=0, 1, 2, … .

Thus: !3!2 = 3(3-2)(3-4)(3-6) = 9  and !4!2 = 4(4-2)(4-6)(4-8) = 64.

The sequence is: 9, 64, -225, -2304, 11025, 147456, -893025, -14745600, 108056025, 2123366400, … .

3) In the case k=3:

!n!3 =  (n-3i) = n(n-3)(n-6)… .

0<|n-3i|≤n

i=0, 1, 2, … .

Thus !7!3 = 7(7-3)(7-6)(7-9)(7-12) = 7(4)(1)(-2)(-5) = 280.

The sequence is: -8, 40, 324, 280, -2240, -26244, -22400, 246400, 3779136, 3203200, -44844800, … .

4) In the case k=4:

!n!4 =  (n-4i) = n(n-4)(n-8)… .

0<|n-4i|≤n

i=0, 1, 2, … .

Thus !9!4 = 9(9-4)(9-8)(9-12)(9-16) = 9(5)(1)(-3)(-7) = 945.

The sequence is: -15, 144, 105, 1024, 945, -14400, -10395, -147456, -135135, 2822400, 2027025, … .

5) In the case k=5:

!n!5 =  (n-5i) = n(n-5)(n-10)… .

0<|n-5i|≤n

i=0, 1, 2, … .

Thus !11!5 = 11(11-5)(11-10)(11-15)(11-20) = 11(6)(1)(-4)(-9) = 2376.

The sequence is: -24, -42, 336, 216, 2500, 2376, 4032, -52416, -33264, -562500, -532224,

-891072, 16039296, … .

More general:

Let n>k≥1 be two integers and m≥1 another integer.  Then the generalized Smarandacheial is defined as:

!n!mk =  (n-k·i)

0<|n-k·i|[m

i cN

For examples:

!7!32 = 7(7-2)(7-4)(7-6)(7-8)(7-10) = 7(5)(3)(1)(-1)(-3) = 315.

!7!92 = 7(7-2)(7-4)(7-6)(7-8)(7-10)(7-12)(7-14)(7-16) = 7(5)(3)(1)(-1)(-3)(-5)(-7)(-9)

= -99225.

References:

J. Dezert, editor, “Smarandacheials”, Mathematics Magazine, Aurora, Canada, No. 4/2004; http://www.mathematicsmagazine.com/corresp/J_Dezert/JDezert.htm, and

www.gallup.unm.edu/~smarandache/Smarandacheials.htm.

F. Smarandache, “Back and Forth Factorials”, Arizona State Univ., Special Collections, 1972.