A085780 -id:A085780 - cp's OEIS frontend

A265010 Numbers which are the product of two tetrahedral numbers.

0, 1, 4, 10, 16, 20, 35, 40, 56, 80, 84, 100, 120, 140, 165, 200, 220, 224, 286, 336, 350, 364, 400, 455, 480, 560, 660, 680, 700, 816, 840, 880, 969, 1120, 1140, 1144, 1200, 1225, 1330, 1456, 1540, 1650, 1680, 1771, 1820, 1960, 2024, 2200, 2240

Offset: 1

R. J. Mathar, Nov 30 2015

nonn

This is for the tetrahedral numbers A000292 what A085780 is for the triangular numbers.

The subsequence of numbers with more than one factorization starts 0, 560 (= 65*10 = 1*560), 19600 (= 560*15 = 19600*1), 28560 (=816 *35 = 7140*4), 43680, 292600, 416640, ...

			Contains 480=4*120, 560=1*560, 660=4*165, 680=1*680, 700=20*35, ....

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A085780. Subsequences: A000292, A001249, A027790, A105939, A104474, A104677.

# reuses code of A000292
isA265010 := proc(n)
    if n = 0 then
        return true;
    end if;
    for d in numtheory[divisors](n) do
        if isA000292(d) and isA000292(n/d) then
            return true;
        end if;
    end do:
    false;
end proc:
for n from 0 to 4000 do
    if isA265010(n) then
        printf("%d, ",n);
    end if;
end do:

lim = 2240; t = Table[Binomial[n + 2, 3], {n, 0, 10^3}]; f[n_] := Select[{#, n/#} & /@ Select[Divisors[n], # <= Sqrt@ n && MemberQ[t, #] &], MemberQ[t, Last@ #] &]; Select[Range@ lim, Length@ f@ # > 0 &] (* Michael De Vlieger, Nov 30 2015 *)

{n: n = A000292(i)*A000292(j) for some i,j>=0}.

A334129 Numbers that can be written as a product of one or more consecutive triangular numbers.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 18, 21, 28, 36, 45, 55, 60, 66, 78, 91, 105, 120, 136, 150, 153, 171, 180, 190, 210, 231, 253, 276, 300, 315, 325, 351, 378, 406, 435, 465, 496, 528, 561, 588, 595, 630, 666, 703, 741, 780, 820, 861, 900, 903, 946, 990, 1008, 1035

Offset: 1

Ilya Gutkovskiy, Apr 14 2020

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triangular Number
Index to sequences related to polygonal numbers

Cf. A000217, A006472, A085780, A085782, A334130.

lmt = 1050; t = PolygonalNumber[3, #] & /@ Range[0, Sqrt[ 2lmt]]; f[n_] := Select[ Times @@@ Partition[t, n +1, 1], # < lmt &]; lst = {}; k = 0; While[f@k != {}, lst = Join[lst, f@k]; k++]; Union@lst (* Robert G. Wilson v, Apr 16 2020 *)

list(lim)=if(lim<1, return(if(lim<0,[],[0]))); my(v=List([0,1]),t=1,m=2); lim\=1; while(t<=lim, listput(v,t); t=m*m++/2); for(e=1,m, for(i=3,m-e, t=factorback(Vec(v[i..i+e])); if(t>lim, break); listput(v,t))); Set(v) \\ Charles R Greathouse IV, Apr 16 2020

A334130 Numbers that can be written as a product of distinct triangular numbers.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 18, 21, 28, 30, 36, 45, 55, 60, 63, 66, 78, 84, 90, 91, 105, 108, 120, 126, 135, 136, 150, 153, 165, 168, 171, 180, 190, 198, 210, 216, 231, 234, 253, 270, 273, 276, 280, 300, 315, 325, 330, 351, 360, 378, 396, 406, 408, 420, 435, 450

Offset: 1

Ilya Gutkovskiy, Apr 14 2020

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triangular Number
Index to sequences related to polygonal numbers

Cf. A000217, A006472, A054731, A085780, A085782, A334129.

N:= 1000: # for all terms <= N
S:= {0,1}:
for i from 2 do
  t:= i*(i+1)/2;
  if t > N then break fi;
  S:= S union select(`<=`,map(`*`,S,t),N)
od:
sort(convert(S,list)); # Robert Israel, Apr 21 2020

A356748 Numbers k such that k and k+1 are both products of 2 triangular numbers.

Original entry on oeis.org

0, 9, 90, 135, 945, 1710, 1890, 4959, 5670, 8910, 10584, 11025, 11934, 13860, 19305, 21735, 26334, 32130, 36855, 44550, 49140, 65340, 107415, 138600, 172080, 239085, 305370, 351540, 366795, 459360, 849555, 873180, 933660, 1100385, 1413720, 1516410, 1904175, 2297295

Offset: 1

Amiram Eldar, Aug 25 2022

nonn

Numbers k such that k and k+1 are both terms of A085780.
Are all the terms divisible by 9?
Yes, because the product of two triangular numbers == 0, 1, 3 or 6 (mod 9). - Robert Israel, Apr 05 2023

			9 is a term since 9 = 3*3 and 10 = 1*10 are both products of 2 triangular numbers.

Robert Israel, Table of n, a(n) for n = 1..138

Cf. A085780.

N:= 10^9: # for terms <= N
S:= {0}:
for x from 1 do
  s:= x*(x+1)/2;
  if s^2 > N then break fi;
  for y from x do
    t:= y*(y+1)/2;
    if s*t > N then break fi;
    S:= S union {s*t};
od od:
L:= sort(convert(S,list)):
DL:= L[2..-1]-L[1..-2]:
J:= select(t -> DL[t]=1, [$1..nops(DL)]):
L[J]; # Robert Israel, Apr 05 2023

t = Table[n*(n + 1)/2, {n, 0, 3000}]; s = Select[Union[Flatten[Outer[Times, t, t]]], # <= t[[-1]] &]; i = Position[Differences[s], 1] // Flatten; s[[i]]
Take[Select[Partition[Union[Times@@@Tuples[Accumulate[Range[0,2500]],2]],2,1],#[[2]] - #[[1]]==1&][[All,1]],40] (* Harvey P. Dale, Oct 23 2022 *)

from itertools import count, islice
from sympy import divisors, integer_nthroot
def A356748_gen(startvalue=0): # generator of terms >= startvalue
    if startvalue <= 0:
        yield 0
    flag = False
    for n in count(max(startvalue,1)):
        for d in divisors(m:=n<<2):
            if d**2 > m:
                flag = False
                break
            if integer_nthroot((d<<2)+1,2)[1] and integer_nthroot((m//d<<2)+1,2)[1]:
                if flag: yield n-1
                flag = True
                break
        else:
            flag = False
A356748_list = list(islice(A356748_gen(),10)) # Chai Wah Wu, Aug 28 2022

A210446 Largest integer which is both the product of two integers summing to n+1 and the product of two integers summing to n-1.

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 0, 0, 16, 18, 0, 30, 0, 36, 48, 0, 0, 70, 0, 90, 96, 90, 0, 126, 144, 126, 160, 180, 0, 210, 0, 0, 240, 216, 288, 300, 0, 270, 336, 378, 0, 420, 0, 450, 480, 396, 0, 510, 576, 594, 576, 630, 0, 700, 720, 756, 720, 630, 0, 858, 0, 720, 960, 0

Offset: 1

Enric Reverter i Bigas, Jan 20 2013

nonn

a(n) is also the difference between ((n+1)/2)^2 and Q, where Q is the smallest square which exceeds n by a square q (or by 0 if n itself is a square): ((n+1) / 2)^2  - a(n) = Q; Q - n = q; (Q, q squares of an integer if n is odd).
If n is an odd nonprime > 1, a(n)/16 is the product of two triangular numbers (see A085780).
If n is 1, a prime or a power of 2, a(n) = 0.

			a(15) = 48 because 6*8 = 12*4 = 48 and 6 + 8 = 15 - 1; 12 + 4 = 15 + 1.
a(45) = 480 because 20*24 = 16*30 = 480 and 20 + 24 = 45 - 1; 16 + 30 = 45 + 1.
(Also 448 = 28*16 = 14*32, but 480 is larger.)

Cf. A085780.

a[n_] := Module[{x,y,p}, Max[p /. List@ToRules@Reduce[p == x*(n-1-x) == y*(n+1-y), {x, y, p}, Integers]]]; Table[a[n], {n, 100}] (* Giovanni Resta, Jan 22 2013 *)

a(n) = {my(x=vector(n\2), y=vector(n\2)); for(k=1, n\2, x[k]=k*(n-1-k); y[k]=k*(n+1-k)); v=setintersect(x, y); if(#v>0, v[#v], 0); } \\ Jinyuan Wang, Oct 13 2019

a(n) = (f1^2 - 1)*(f2^2 - 1)/4 (with f1 and f2 the nearest integers such that f1*f2 = n).

A346919 Numbers that are both palindromes (A002113) and terms of A072389.

Original entry on oeis.org

0, 4, 252, 2112, 2772, 6336, 21012, 27072, 42924, 48384, 48984, 63036, 252252, 297792, 407704, 2327232, 2572752, 2747472, 2774772, 2958592, 4457544, 4811184, 6378736, 6396936, 25777752, 27633672, 29344392, 63099036, 63399336, 404080404, 409757904, 441525144

Offset: 1

Dumitru Damian, Aug 07 2021

Cf. A002113, A072389, A085780.

p[n_] :=  n*(n + 1); m = 1000; Select[Union[ Times @@@ Tuples[p /@ Range[0, m], {2}]], # <= 2*p[m] && PalindromeQ[#] &] (* Amiram Eldar, Aug 13 2021 *)

# the sequence numbers up to a limit
def a346919_upto(lim, alst=set([0])):
    for i in range(1, int(lim**0.25)):
        for j in range(i, int((lim/(i*(i+1)))**0.5)+1):
            if all([(k:=i*(i+1)*j*(j+1))<=lim, str(k)==str(k)[::-1]]): alst.add(k)
    return sorted(alst)
print(a346919_upto(10**9)) # Dumitru Damian, Apr 05 2023

Intersection of A072389 and A002113.

Intersection of 4*A085780 and A002113.

More terms from Jinyuan Wang, Aug 07 2021

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A265010 Numbers which are the product of two tetrahedral numbers.

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A334129 Numbers that can be written as a product of one or more consecutive triangular numbers.

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A334130 Numbers that can be written as a product of distinct triangular numbers.

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A356748 Numbers k such that k and k+1 are both products of 2 triangular numbers.

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A210446 Largest integer which is both the product of two integers summing to n+1 and the product of two integers summing to n-1.

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A346919 Numbers that are both palindromes (A002113) and terms of A072389.

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