A264961 Numbers that are products of two triangular numbers in more than one way.

36, 45, 210, 315, 360, 630, 780, 990, 1260, 1386, 1540, 1800, 2850, 2970, 3510, 3570, 3780, 4095, 4788, 4851, 6300, 7920, 8415, 8550, 8778, 9450, 11700, 11781, 14850, 15400, 15561, 16380, 17640, 17955, 18018, 18648, 19110, 20790, 21420, 21450, 21528, 25116, 25200, 26565, 26775, 26796, 27720, 28980

Offset: 1

Author

R. J. Mathar, Nov 29 2015

Keywords

• nonn

Comments

One of the factors in the product may be 1 = A000217(1). We count the ways of writing n = A000217(i)*A000217(j) with i <= j, unordered factorizations.

Examples

			36 = 1*36 = 6*6. 45 = 1*45 = 3*15. 210 = 1*210 = 10*21. 315 = 3*105 = 15*21. 360 = 3*120 = 10*36. 630 = 1*630 = 3*210 = 6*105. 3780= 6*360 = 10 * 378 = 36*105.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10602

Crossrefs

Subsequence of A085780. A188630 and A110904 are subsequences of this.

Programs

Maple

A264961ct := proc(n)
    local ct,d ;
    ct := 0 ;
    for d in numtheory[divisors](n) do
        if d^2 > n then
            return ct;
        end if;
        if isA000217(d) then
            if isA000217(n/d) then
                ct := ct+1 ;
            end if;
        end if;
    end do:
    return ct;
end proc:
for n from 1 to 30000 do
    if A264961ct(n) > 1 then
        printf("%d,",n) ;
    end if;
end do:

Mathematica

lim = 10000; t = Accumulate[Range@lim]; f[n_] := Select[{#, n/#} & /@ Select[Divisors@ n, # <= Sqrt@ n && MemberQ[t, #] &], MemberQ[t, Last@ #] &]; Select[Range@ lim, Length@ f@ # == 2 &] (* Michael De Vlieger, Nov 29 2015 *)

Python

from _future_ import division
mmax = 10**3
tmax, A264961_dict = mmax*(mmax+1)//2, {}
ti = 0
for i in range(1,mmax+1):
    ti += i
    p = ti*i*(i-1)//2
    for j in range(i,mmax+1):
        p += ti*j
        if p <= tmax:
            A264961_dict[p] = 2 if p in A264961_dict else 1
        else:
            break
A264961_list = sorted([i for i in A264961_dict if A264961_dict[i] > 1]) # Chai Wah Wu, Nov 29 2015