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A264961 Numbers that are products of two triangular numbers in more than one way. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

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

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

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

Sequence in context: A195528 A144291 A068143 * A188630 A167310 A083674

Adjacent sequences:  A264958 A264959 A264960 * A264962 A264963 A264964

KEYWORD

nonn

AUTHOR

R. J. Mathar, Nov 29 2015

STATUS

approved

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Last modified December 4 08:16 EST 2021. Contains 349479 sequences. (Running on oeis4.)