[seqfan] Re: A possible list of new sequences

[Richard Mathar]

Followup to http://list.seqfan.eu/pipermail/seqfan/2010-May/004550.html :

vs> I just have sent to OEIS the following two sequences :
vs> 
vs> %I A177025
vs> %S A177025 1,1,1,2,1,1,2,2,1,2,1,1,3,2,1,2,1,1,3,2,1,2,2,1,2,3,1,2,1,1,2,2,2,4,1,
vs> %T A177025 1,2,2,1,2
vs> %N A177025 Number of k>=3 for which n is k-gonal number 
vs> %C A177025 Since n is always n-gonal number, then a(n)>=1. Conjecture. Every positive integer appears in the sequence. 
vs> %Y A177025 A000217 A000326 A000384 A000566 A000567 A176949 A176948 A176774 A176744 A176747 A176775 A175873 A176874 
vs> %K A177025 nonn
vs> %O A177025 3,4
vs> %A A177025 Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 01 2010

A177025 starts as
1,1,1,2,1,1,2,2,1,2,1,1,3,2,1,2,1,1,3,2,1,2,2,1,2,3,1,2,1,1,2,2,2,4,1,1,2,2,1,
2,1,1,4,2,1,2,2,1,3,2,1,2,3,1,2,2,1,2,1,1,2,3,2,4,1,1,2,3,1,2,1,1,3,2,1,3,1,1,
4,2,1,2,2,1,2,2,1,2,3,2,2,2,2,3,1,1,2,3,1,2,1,1,4,2,1,2,1,1,3,3,1,2,2,1,3,2,1,4

The first occurrence of a=2,3,4,.. is at n = 6, 15, 36, 225 (see A063778).

See Also A139601, A139600 and A086280.

vs> %I A177029
vs> %S A177029 6,9,10,12,16,18,22,24,25,27,30,33,34,35,39,40,42
vs> %N A177029 Numbers m for which A177025(m)=2 
vs> %C A177029 Numbers u for which A177025(u)=1 are listed by A090467. 
vs> %Y A177029 A177025 A090467 A177028 A000217 A000326 A000384 A000566 A000567 A176949 A176948 A176774 A176744 A176747 A176775 A175873 A176874 
vs> %K A177029 nonn
vs> %O A177029 1,1
vs> %A A177029 Vladimir Shevelev (shevelev(AT)bgu.ac.il), May 01 2010

On should mention that A177029 is a subsequence of A090466.
A177029 starts

6,9,10,12,16,18,22,24,25,27,30,33,34,35,39,40,42,46,48,49,52,54,57,58,60,63,65,
69,72,76,82,84,85,87,88,90,92,93,94,95,99,102,106,108,114,115,118,121,123,124,
125,129,130,132,133,138,142,147,150,155,159,160,162,166,168,169,172,174,177,
178,180,183,184,185,192,198,202,205,207,208,213,214,215,217,219,220,222,226,
228,234,237,240,243,244,245,247,249,250,252,253,258,259,260,262,264,265,267,
268,270,273,274,275,279,282,287,289,292,294,295,298,301,303,304,305,309,310,
312,318,327,328,334,335,339

Maple implementation of A177025:
# see A139601 A139600 A086270
kgonal := proc(n,k)
        option remember ;
        n/2*( (k-2)*n-k+4) ;
end proc:

A177025 := proc(n)
        local a,k,m ;
        a := 0 ;
        for k from 3 to n do
                for m from 0 do
                        km := kgonal(m,k) ;
                        if km = n then
                                a := a+1 ;
                        elif m > n then
                                break;
                        end if;
                end do:
        end do:
        a ;
end proc:
seq(A177025(n),n=3..120) ;