A072389 -id:A072389 - cp's OEIS frontend

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

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

A085780 Numbers that are a product of 2 triangular numbers.

Original entry on oeis.org

0, 1, 3, 6, 9, 10, 15, 18, 21, 28, 30, 36, 45, 55, 60, 63, 66, 78, 84, 90, 91, 100, 105, 108, 120, 126, 135, 136, 150, 153, 165, 168, 171, 190, 198, 210, 216, 225, 231, 234, 253, 270, 273, 276, 280, 300, 315, 325, 330, 351, 360, 378, 396, 406, 408, 420, 435, 441

Offset: 1

Jon Perry, Jul 23 2003

nonn

Is there a fast algorithm for detecting these numbers? - Charles R Greathouse IV, Jan 26 2013

The number of rectangles with positive width 1<=w<=i and positive height 1<=h<=j contained in an i*j rectangle is t(i)*t(j), where t(k)=A000217(k), see A096948. - Dimitri Boscainos, Aug 27 2015

			18 = 3*6 = t(2)*t(3) is a product of two triangular numbers and therefore in the sequence.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000217, A085782, A068143, A000537 (subsequence), A006011 (subsequence), A033487 (subsequence), A188630 (subsequence).

Cf. A072389 (this times 4).

isA085780 := proc(n)
     local d;
     for d in numtheory[divisors](n) do
        if d^2 > n then
            return false;
        end if;
        if isA000217(d) then
            if isA000217(n/d) then
                return true;
            end if;
        end if;
    end do:
    return false;
end proc:
for n from 1 to 1000 do
    if isA085780(n) then
        printf("%d,",n) ;
    end if ;
end do: # R. J. Mathar, Nov 29 2015

t1 = Table[n (n+1)/2, {n, 0, 100}];Select[Union[Flatten[Outer[Times, t1, t1]]], # <= t1[[-1]] &] (* T. D. Noe, Jun 04 2012 *)

A003056(n)=(sqrtint(8*n+1)-1)\2
list(lim)=my(v=List([0]),t); for(a=1, A003056(lim\1), t=a*(a+1)/2; for(b=a, A003056(lim\t), listput(v,t*b*(b+1)/2))); vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jan 26 2013

from itertools import count, islice
from sympy import divisors, integer_nthroot
def A085780_gen(startvalue=0): # generator of terms
    if startvalue <= 0:
        yield 0
    for n in count(max(startvalue,1)):
        for d in divisors(m:=n<<2):
            if d**2 > m:
                break
            if integer_nthroot((d<<2)+1,2)[1] and integer_nthroot((m//d<<2)+1,2)[1]:
                yield n
                break
A085780_list = list(islice(A085780_gen(),10)) # Chai Wah Wu, Aug 28 2022

Conjecture: There are about sqrt(x)*log(x) terms up to x. - Charles R Greathouse IV, Jul 11 2024

More terms from Max Alekseyev and Jon E. Schoenfield, Sep 04 2009

A053990 Numbers of the form x(x + 1)y*(y + 1) ("bipronics") where x and y are distinct.

Original entry on oeis.org

0, 12, 24, 40, 60, 72, 84, 112, 120, 144, 180, 220, 240, 252, 264, 312, 336, 360, 364, 420, 432, 480, 504, 540, 544, 600, 612, 660, 672, 684, 760, 792, 840, 864, 924, 936, 1012, 1080, 1092, 1104, 1120, 1200, 1260, 1300, 1320, 1404, 1440, 1512, 1584, 1624

Offset: 1

Stuart M. Ellerstein (ellerstein(AT)aol.com), Apr 04 2000

			Taking x=1, y=2 gives 2 * 6 =12

Cf. A054731, A054734, A072389.

bipr[{a_,b_}]:=a(a+1)b(b+1); Take[Union[bipr/@Subsets[Range[0,40], {2}]],50] (* Harvey P. Dale, Jun 01 2012 *)

More terms from James Sellers, Apr 22 2000

A386303 Positive integers k such that the set {d+k/d : d|k} contains four consecutive integers.

Original entry on oeis.org

15120, 712800, 3341520, 10533600, 23284800, 85503600, 147026880, 171097920, 302702400, 477338400, 2058376320, 2633510880, 4204418400, 7342876800, 9673606800, 13035884400, 13734761040, 14895223200, 22388788800, 22647794400, 26108082000, 34183749600, 62246804400, 89169141600

Offset: 1

Giedrius Alkauskas, Jul 18 2025

nonn

a(n) is divisible by 720.
Subsequence of A072389 (with two consecutive instead of four).
Integers k with five consecutive integers in the set {d+k/d : d|k} seem not to exist.
As terms must be of the form k * (k + 1) * m * (m + 1) and divisible by 720 we can restrict the search based on g = gcd(k * (k + 1), 720) which is at least 2. We must have (720 / g) | m * (m + 1). - David A. Corneth, Jul 19 2025
If q is the number of divisors of a(n) then the first of these four divisors is generally d[q/2 + 1] at least for nonsquares. For three consecutive integers (cf. A386302) there is the exception 180180. - David A. Corneth, Jul 20 2025

			a(1)=15120=M is a term of this sequence since 105, 108, 112, 120 are divisors of M, and 120+M/120=246, 112+M/112=247, 108+M/108=248, 105+M/105=249. It is the first term since no smaller such positive integer exists.

Giedrius Alkauskas, Consecutive integers in the set S_n={d+n/d: d|n}
David A. Corneth, 360 x 360 pixels image where white pixels with coordinate (k, m) have 720 | (k * (k + 1) * m * (m + 1))
David A. Corneth, PARI program

Cf. A002496, A027862, A072389, A335572, A386302.

M:=2*10^10:
Ki:={}:
Vi:=floor(sqrt(2*M)):
Ski:=floor((19*M)^(1/4)/2):
for F from 1 to Vi-4 do
  for y from 1 to min(floor((Vi-F)/2),Ski) do
     G:=F+2*y+1:
     if issqr(2*F^2-G^2+2) and issqr(3*F^2-2*G^2+6) then
       x:=(F+G-1)/2:
       n:=x*(x+1)*y*(y+1):
       Ki:=Ki union {n}:
     end if:
  end do:
end do:
Ki;

More terms from David A. Corneth, Jul 19 2025

A049207 Array T(m,n) of products of pronic numbers m(m+1) * n(n+1) read by antidiagonals ("bipronics").

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 0, 12, 12, 0, 0, 24, 36, 24, 0, 0, 40, 72, 72, 40, 0, 0, 60, 120, 144, 120, 60, 0, 0, 84, 180, 240, 240, 180, 84, 0, 0, 112, 252, 360, 400, 360, 252, 112, 0, 0, 144, 336, 504, 600, 600, 504, 336, 144, 0, 0, 180, 432, 672, 840, 900, 840, 672, 432, 180, 0, 0

Offset: 0

Stuart M. Ellerstein (ellerstein(AT)aol.com), Jul 20 2002

			Array begins
0 0 0 0 0 0 ...
0 4 12 24 40 ...
0 12 36 72 120 ...
0 24 72 144 240 ...

Cf. A002378, A072389, A053990.

T := (m,n)->m*(m+1)*n*(n+1): seq(seq(T(q-p,p),p=0..q),q=0..12);

Corrected and extended by Emeric Deutsch, Mar 04 2004

A218391 Let k be the n-th odd composite, then a(n) is the smallest wx such that w + x = (k-1)/2, y + z = (k+1)/2, and wx = y*z.

Original entry on oeis.org

4, 12, 24, 36, 40, 60, 72, 84, 112, 144, 144, 180, 180, 220, 252, 264, 312, 360, 364, 432, 420, 504, 480, 540, 544, 612, 684, 792, 760, 864, 900, 840, 936, 924, 1080, 1012, 1104, 1260, 1260, 1200, 1300, 1440, 1404, 1584, 1512, 1764, 1624, 1836, 1740, 1860

Offset: 1

Bill McEachen, Oct 27 2012

nonn

If a number w + x + y + z with w, x, y, z > 0 has w*x = y*z then it is composite.

Appears to be a subset of A072389. - Bill McEachen, Feb 14 2025

			15=7+8 (partition is x,x+1)
col 1 sum(to products)
1*6=6
2*5=10
3*4=12
col 2 sum(to products)
1*7=7
2*6=12
3*5=15
4*4=16
There is an overlapping product, and the lowest is 12.
This indicates the original N of 15 is composite.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
John F. Richardson, A Property of Odd Composites, Math Forums, 2012.

Cf. A071904.

do(n)=my(X=vector(n\4,i,i*(n\2-i)),Y=vector((n+1)\4,i,i*(n\2-i+1)),i=1,j=1);while(X[i]!=Y[j],if(X[i]Charles R Greathouse IV, Oct 28 2012

A352796 Numbers m such that {d + m/d : d | m } does not contain consecutive integers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74

Offset: 1

José María Grau Ribas, Jun 08 2022

nonn

Conjecture: Complement of A072389.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A072389, A335572.

S[n_]:=Divisors[n]+n/Divisors[n]//Union; Test[n_]:= {aux=S[n];Union[ {False},Table[aux[[i+1]]-aux[[i]] ==1,{i,Length[aux]-1}]]}[[1]]   //Last; Select[Range[1000],Test[#]&]

isok(m) = my(list=List()); fordiv(m, d, listput(list, d+m/d)); my(w = Set(vector(#list-1, k, list[k+1]-list[k]))); #select(x->(x==1), w) == 0; \\ Michel Marcus, Jun 09 2022

from sympy import divisors
def ok(n):
    s = sorted(set(d + n//d for d in divisors(n)))
    return 1 not in set(s[i+1]-s[i] for i in range(len(s)-1))
print([k for k in range(1, 75) if ok(k)]) # Michael S. Branicky, Jul 10 2022

A386302 Positive integers k such that the set {d+k/d : d|k} contains three consecutive integers.

Original entry on oeis.org

144, 180, 1260, 1440, 2520, 5040, 5544, 7200, 14040, 15120, 25200, 31680, 33660, 37800, 46800, 59400, 62244, 65520, 70560, 83160, 107100, 110880, 115920, 166320, 169344, 176400, 180180, 183600, 190944, 221760, 277200, 287280, 297540

Offset: 1

Giedrius Alkauskas, Jul 17 2025

nonn

Terms are divisible by 36.

Subsequence of A072389 (with two consecutive rather than three).

			a(1)=144, since 144/12+12=24, 144/9+9=25, 144/8+8=26, and no smaller integer with such property exists.

Giedrius Alkauskas, Consecutive integers in the set S_n={d+n/d: d|n}

Cf. A002496, A027862, A072389, A335572, A386303.

M:=300000:
Ki:={}:
Vi:=floor(sqrt(2*M)):
Ski:=floor((19*M)^(1/4)/2):
for F from 1 to Vi-4 do
  for y from 1 to min(floor((Vi-F)/2),Ski) do
     G:=F+2*y+1:
     if issqr(2*F^2-G^2+2) then
       x:=(F+G-1)/2;
       n:=x*(x+1)*y*(y+1):
       Ki:=Ki union {n}:
     end if:
  end do:
end do:
Ki;

isok(m, nb=3) = nb--; my(v = Set(apply(x->x+m/x, divisors(m)))); if (#v >= nb, select(x->(x==nb), vector(#v-nb, k, v[k+nb]-v[k]))); \\ Michel Marcus, Jul 18 2025

A274187 Least number that is the product of n consecutive positive numbers and the product of 2 oblong numbers.

Original entry on oeis.org

4, 12, 24, 24, 120, 5040, 5040, 362880, 362880, 3628800, 39916800, 6227020800, 6227020800, 3379030566912000

Offset: 1

Gionata Neri, Jun 12 2016

			a(3) = 24 = 2*3*4 = 2*12.
a(6) = 5040 = 2*3*4*5*6*7 = 12*420 = 56*90.

Cf. A045619, A072389.

N:= 10^10: # to get all terms <= N
A072389:= {seq(seq(n*(n+1)*m*(m+1),m=n..floor((sqrt(1+4*N/(n*(n+1))-1)/2))),n=1..floor((sqrt(1+2*N)-1)/2))}:
for n from 1 do
  x:= n!;
  for m from 1 while x <= N and not member(x, A072389) do
    x:= x*(n+m)/m
  od;
  if x > N then break fi;
  A[n]:= x;
od:
seq(A[i],i=1..n-1); # Robert Israel, Jun 16 2016

a(14) from Robert Israel, Jun 16 2016

cp's OEIS Frontend

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

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Mathematica

SageMath

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A085780 Numbers that are a product of 2 triangular numbers.

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Maple

Mathematica

PARI

Python

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A053990 Numbers of the form x*(x + 1)*y*(y + 1) ("bipronics") where x and y are distinct.

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Programs

Mathematica

Extensions

A386303 Positive integers k such that the set {d+k/d : d|k} contains four consecutive integers.

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Author

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Comments

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Programs

Maple

Extensions

A049207 Array T(m,n) of products of pronic numbers m(m+1) * n(n+1) read by antidiagonals ("bipronics").

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Examples

Crossrefs

Programs

Maple

Extensions

A218391 Let k be the n-th odd composite, then a(n) is the smallest w*x such that w + x = (k-1)/2, y + z = (k+1)/2, and w*x = y*z.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A352796 Numbers m such that {d + m/d : d | m } does not contain consecutive integers.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

A386302 Positive integers k such that the set {d+k/d : d|k} contains three consecutive integers.

Views

Author

Keywords

A053990 Numbers of the form x(x + 1)y*(y + 1) ("bipronics") where x and y are distinct.

A218391 Let k be the n-th odd composite, then a(n) is the smallest wx such that w + x = (k-1)/2, y + z = (k+1)/2, and wx = y*z.