Acta Universitatis Danubius. Œconomica, Vol 11, No 4 (2015)
Mathematical and Quantative Methods
A Study of Integers Using Software Tools – III
Cătălin Angelo Ioan^{1},^{ }Alin Cristian Ioan^{2}
Abstract: The paper deals with a generalization of polite numbers that is of those numbers that are sums of consecutive integers to the numbers called almost polite numbers of order p,m which can be written in m ways as sum of two or more consecutive of same powers p of natural numbers.
Keywords: polite numbers; divisibility
1 Introduction
Let note for any nN^{*}, pN^{*}, S_{n,p}= and S_{k,n,p}= =S_{n,p}S_{k1,p}, k= .
It is well known that:
and also:
=
It is easly to see that the first 10 sums are:
S_{n,1}= =
S_{n,2}= =
S_{n,3}= =
S_{n,4}= =
S_{n,5}= =
S_{n,6}= =
S_{n,7}= =
S_{n,8}= =
S_{n,9}= =
S_{n,10}= =
All over in this paper, the software presented was written in Wolfram Mathematica 9.0.
2 Almost Polite Numbers of Order p,m
A natural number N greather than 2 will be called almost polite number of order p if it can be written as sum of two or more consecutive of a same power p of natural numbers.
If N is odd it is natural that for N=2k+1 we have N=k+(k+1) therefore each odd natural number is polite of order 1.
A natural number N greather than 2 will be called almost polite number of order p,m if it can be written in m ways as a sum of two or more consecutive of same powers p of natural numbers.
Therefore N is a polite number of order p,m if:
N= = =...= with p_{1}p_{2}...p_{m}
The software for determining the almost polite numbers of order p,m (m3) limited to 30000 and powers less than or equal with 10 is:
Clear["Global`*"];
limit=30000;
pmax=10;
nrorimax=3;
nrk=Table[i,{i,nrorimax}];
nrn=Table[i,{i,nrorimax}];
nrp=Table[i,{i,nrorimax}];
S[0]=n;
(*The calculus of sums of powers from 1 to n*)
For[p=1,ppmax,p++,
suma=0;
For[j=1,jp,j++,suma=suma+Binomial[p+1,j+1]*S[pj]];
S[p]=Factor[((n+1)^(p+1)1suma)/(p+1)]]
(*The calculus of sums of powers from k to n*)
For[p=1,ppmax,p++,sumpower[n_,p]=S[p]];
For[p=1,ppmax,p++,sumpowerkn[n_,k_,p]=Factor[Simplify[sumpower[n,p]
sumpower[k1,p]]]]
(*The analysis*)
For[number=2,numberlimit,number=number+1,nrori=0;
For[p=1,ppmax,p++,
For[n=2,nnumber^(1/p),n++,
For[k=1,kn1,k++,
If[sumpowerkn[n,k,p]==number,nrori=nrori+1;nrk[[nrori]]=k;nrn[[nrori]]=n;
nrp[[nrori]]=p];
If[nrori2&& nrp[[nrori]]==nrp[[nrori1]],nrori=nrori1]]]];
If[nrorinrorimax,For[k=1,knrori,k++,Print[number,"=\[Sum](power=",nrp[[k]],") from ",nrk[[k]]," to ",nrn[[k]]]];Print[""]]]
We found (till 30000):

91=1+2+...+13=1^{2}+...+6^{2}=3^{3}+4^{3}

559=9+10+...+34=7^{2}+...+12^{2}=6^{3}+7^{3}

855=4+5+...+41=11^{2}+...+15^{2}=7^{3}+8^{3}

6985=9+10+...+118=20^{2}+...+30^{2}=9^{3}+...+13^{3}

19721=200+201+...+281=14^{2}+...+39^{2}=4^{6}+5^{6}

24979=12489+12490=62^{2}+...+67^{2}=5^{4}+...+10^{4}

29240=29+30+...+243=35^{2}+...+50^{2}=2^{3}+...+18^{3}
3. References
Adler, A. & Coury, J.E. (1995). The Theory of Numbers. London, UK: Jones and Bartlett Publishers International.
Baker, A. (1984). A Concise Introduction to the Theory of Numbers. Cambridge: Cambridge University Press.
Coman, M. (2013). Mathematical Encyclopedia of Integer Classes. Educational Publishers.
Guy, R.K. (1994). Unsolved Problems in Number Theory. Second Edition. New York: Springer Verlag.
Hardy, G.H. & Wright, E.M. (1975). Introduction to the Theory of Numbers. Fourth Edition. Oxford: Oxford University Press.
Krantz, S.G. (2001). Dictionary of Algebra, Arithmetic and Trigonometry. CRC Press.
Niven, I., Zuckerman, H.S. & Montgomery, H.L. (1991). An Introduction to the Theory of Numbers. Fifth Edition. New York: John Wiley & Sons, Inc.
Sierpinski, W. (1995). Elementary theory of numbers. Second Edition. Elsevier.
Wai, Y.P. (2008). Sums of Consecutive Integers. arXiv:math/0701149v1 [math.HO].
1 Associate Professor, PhD, Danubius University of Galati, Faculty of Economic Sciences, Romania, Address: 3 Galati Blvd, Galati, Romania, Tel.: +40372 361 102, Fax: +40372 361 290, Corresponding author: catalin_angelo_ioan@univdanubius.ro.
2 Nicolae Oncescu College, Braila, Address: 13 ŞOS. Brăilei, City: Ianca, Brăila County, Tel.: +40239668 494, Email: alincristianioan@yahoo.com.
AUDŒ, Vol. 11, no. 4, pp. 4144
Refbacks
 There are currently no refbacks.
This work is licensed under a Creative Commons Attribution 4.0 International License.