# Smallest square-free k-gonal number with n prime factors

Jonathan Post jvospost3 at gmail.com
Thu Feb 1 18:15:05 CET 2007

```Refer, for the moment, just to the polygonal numbers, as a subset of
2-D figurate numbers, which are a subset of n-Dimensional figurate
numbers. I've elsewhere referred to my web page which tabulates
polygonal, centered polygonal, polhedral, pyramidal, polytopic, and
related numbers complete through 10^6 and extensive beyond that.
Table of Polytope Numbers, Sorted, Through 1,000,000
http://magicdragon.com/poly.html

A086271 gives the array of nth k-gonal number. OEIS has its main
diagonal, and the like.

I have elsewhere given the complement of the sequence of its elements,
sorted, namely those natural numbers which are not (nontrvially)
polygonal numbers.  These are the "uninteresting numbers" from a
polygonal viewpoint. Your question can be rephrased in terms of the
array of nth k-gonal number, its elements, and the difference set of
those elements.

I do not see in OEIS the partial sums of rows of A086271.  For row 1,
this is n.  For row 2, it is (n(n+1)/2)-3.  For row 3 it is 6, 15, 27,
42, 60, 81, 105, ...
= 3rd triangular number + 3rd square + 3rd pentagonal number + 3rd
hexagonal number + ... + 3rd k-gonal number  =
SUM[k=3..n] (1/2)k[(n-2)k-(k-4)].

I have recently submitted these sequences:

>%S A000001 1,3,4,8,6,10,11,19,17,9,22,2,7
>%N A000001 a(1)=1. a(n) = the smallest positive integer not occurring
>earlier in the sequence such that (sum{k=1 to n-1} a(k)) is congruent to
>a(n) (mod n).
>%C A000001 This sequence seems likely to be a permutation of the positive
>integers.
>%O A000001 1
>%K A000001 ,more,nonn,

>%S A000001 1,3,6,2,7,13,20,4,22,12,23,11
>%N A000001 a(1)=1. a(n) = the smallest positive integer not occurring
>earlier in the sequence such that a(n-1) is congruent to a(n) (mod n).
>%C A000001 This sequence seems likely to be a permutation of the positive
>integers.
>%O A000001 1
>%K A000001 ,more,nonn,

>%S A000001 1,3,2,7,6,13,10,11,5,9,20,25,15,29
>%N A000001 a(1)=1. a(n) = the smallest positive integer not occurring
>earlier in the sequence such that the nth prime is congruent to a(n) (mod n).
>%C A000001 This sequence seems likely to be a permutation of the positive
>integers.
>%Y A000001 A004648
>%O A000001 1
>%K A000001 ,more,nonn,

>%S A000001 1,2,3,5,8,7,15,6,12,18,19,13,32,17
>%N A000001 a(1)=1. a(2)=2. a(n) = the smallest positive integer not
>occurring earlier in the sequence such that a(n-2)+a(n-1) is congruent to
>a(n) (mod n).
>%O A000001 1
>%K A000001 ,more,nonn,

My question is, can any of these sequences be PROVED by someone (besides
me, of course) to really be permutations of the positive integers?

It seems like there would be an easy proof of permutation-ality for at
least one or two of these sequences.

Thanks,
Leroy Quet

```