pyramidal numbers

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pyramidal numbers

[¦pir·ə‚mid·əl ′nəm·bərz]
(mathematics)
The numbers 1, 4, 10, 20, 35, …, which are the number of dots in successive pyramidal arrays and are given by (1/6) n (n + 1) (n + 2), where n = 1, 2, 3, ….
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In this paper we consider a figurate sequence of Triangular numbers of dimension 3, also called as Pyramidal numbers.
We can obtain the first k terms of Pyramidal numbers in Maple as; > t := n->(1/6)*n*(n+1)*(n+2): > first :+ k -> seq (t(n), n=1 ...
For constructing Smarandache sequence of Pyramidal numbers we use the operation of concatenation on the terms of the above sequence.This operation is defined as ;
We define Smarandache consecutive sequence {[scs.sub.n]} for Pyramidal numbers recursively as; [scs.sub.1] = [u.sub.1], [scs.sub.n] = conc([scs.sub.n-1], [u.sub.n])
Using MapleWe have obtained first 20 terms of Smarandache consecutive sequence of Pyramidal numbers; >conc :=(n,m)-> n*10^length(m)+m: > scs_n := (u,n)-> if n = 1 then u(1)else conc(scs_n(u,n-1),u(n))fi: > scs := (u,n)-> seq (scs_n(u,i),i=1 ...
We get the first 20 terms of reversed smarandache sequence of Pyramidal numbers as; > rss(t,20); 1, 41, 1041, 201041, 35201041, 5635201041, 845635201041, 120845635201041, 165120845635201041, 220165120845635201041, 286220165120845635201041, 364286220165120845635201041, 455364286220165120845635201041, 560455364286220165120845635201041, 680560455364286220165120845635201041, 816680560455364286220165120845635201041, 969816680560455364286220165120845635201041, 1140969816680560455364286220165120845635201041, 13301140969816680560455364286220165120845635201041, 154013301140969816680560455364286220165120845635201041
There exist no prime in the first 500 terms of Smarandache consecutive sequence of Pyramidal numbers; > st:= time():select(isprime,Lscs500); [] > printf("%a minutes",round((time()-st)/60)); 9 minutes
There is only one prime in the first 500 terms of reversed Smarandache sequence of Pyramidal numbers; > st:= time(): > select(isprime,Lrss500); [41] > printf("%a minutes",round((time()-st)/60)); 119 minutes