A051533 -id:A051533 - cp's OEIS frontend

A265138 Numbers that are the sum of two distinct nonzero triangular numbers in exactly four ways.

471, 531, 601, 616, 786, 871, 906, 991, 1056, 1126, 1156, 1186, 1281, 1296, 1341, 1446, 1486, 1551, 1576, 1602, 1641, 1656, 1771, 1806, 1836, 1896, 1906, 1921, 2044, 2061, 2146, 2226, 2265, 2281, 2316, 2401, 2421, 2451, 2486, 2551, 2646, 2716, 2746, 2781

Offset: 1

Arkadiusz Wesolowski, Dec 02 2015

nonn

Cf. A000217, A051533, A260647, A265140 (exactly one way), A262749 (more than one way), A265134 (exactly two ways), A265135 (more than two ways), A265136 (exactly three ways), A265137 (more than three ways).

r = 2781; lst = Table[0, {r}]; lim = Floor[Sqrt[8*r - 7]]; Do[num = (i^2 + i)/2 + (j^2 + j)/2; If[num <= r, lst[[num]]++], {i, lim}, {j, i - 1}]; Flatten@Position[lst, 4]
Module[{nn=80,trnos},trnos=Accumulate[Range[nn]];Select[PositionIndex[ Sort[ Counts[Total/@Subsets[trnos,{2}]]]][4],#<=Last[trnos]&]] (* The program uses the PositionIndex and Counts functions from Mathematica version 10 *) (* Harvey P. Dale, Dec 25 2015 *)

A265140 Numbers that are the sum of two distinct nonzero triangular numbers in exactly one way.

Original entry on oeis.org

4, 7, 9, 11, 13, 18, 21, 22, 24, 25, 27, 29, 34, 36, 37, 38, 39, 42, 43, 48, 49, 55, 56, 57, 58, 60, 61, 64, 65, 66, 67, 69, 70, 72, 73, 79, 83, 84, 87, 88, 91, 92, 93, 97, 99, 100, 101, 102, 108, 112, 114, 115, 119, 120, 127, 130, 135, 136, 137, 139, 142, 144

Offset: 1

Arkadiusz Wesolowski, Dec 02 2015

nonn

Cf. A000217, A051533, A260647, A262749 (more than one way), A265134 (exactly two ways), A265135 (more than two ways), A265136 (exactly three ways), A265137 (more than three ways), A265138 (exactly four ways).

r = 144; lst = Table[0, {r}]; lim = Floor[Sqrt[8*r - 7]]; Do[num = (i^2 + i)/2 + (j^2 + j)/2; If[num <= r, lst[[num]]++], {i, lim}, {j, i - 1}]; Flatten@Position[lst, 1]

A342326 a(n) is the smallest nonnegative integer that can be written as a sum of two distinct nonzero triangular numbers in exactly n ways or -1 if no such integer exists.

Original entry on oeis.org

0, 4, 16, 81, 471, 2031, 1381, 11781, 6906, 17956, 34531, 123256, 40056, 305256, 863281, 448906, 200281, 1957231, 520731, 10563906, 1001406, 11222656, 7631406, 3454506, 1482081, 75865156, 7172606106, 8852431, 25035156, 334020781, 13018281, 38531031, 7410406, 7014160156

Offset: 0

Robert G. Root, Mar 08 2021

nonn

Conjecture: This sequence has a positive a(n) for every positive integer n, and each sequence in the infinite indexed family, of which this sequence offers the initial terms, is infinite, as well.
From David A. Corneth, Mar 08 2021: (Start)
a(40) = 37052031, a(45) = 221310781, a(48) = 60765331, a(39) <= 2782318906, a(42) <= 325457031, a(47) <= 927577056, a(50) <= 2200089531, a(54) <= 327539956, a(56) <= 926300781, a(60) <= 481676406, a(63) <= 4598740656, a(64) <= 303826656, a(71) <= 4579579956, a(72) <= 789949306, a(80) <= 1519133281, a(96) <= 3220562556. Terms for n <= 96 not listed here and terms for which only upper bounds are known are >= 3*10^8.
Is a(n) == 6 (mod 25) for n >= 5? It holds for all terms known to date.
The triangular numbers mod 25 are periodic with period 25. Constructing all 25*25 = 625 sums of two distinct triangular numbers mod 25 gives 65 cases for 6 (mod 25). The second largest occurs 40 times. (End)
a(47) = 550240551, a(59) = 7629645156, a(67) = 6418012656, a(81) = 9498658731, a(90) = 8188498906. All upper bounds listed in the above comments for n other than 47 are the exact values of a(n). For all n for which no value is listed here or above, a(n) > 10^10 (or a(n) = -1). - Jon E. Schoenfield, Mar 09 2021
From Martin Ehrenstein, Mar 09 2021: (Start)
a(44) = 15646972656. For n<=51, all terms not mentioned here or above, a(n) >= 6.5*10^10 (or a(n) = -1).
a(47) == 1 (mod 25) and a(95) = 47652012541 == 16 (mod 25). Thus the answer to Corneth's question is 'No'. (End)

			a(1) =  4 = 1 +  3;
a(2) = 16 = 1 + 15 =  6 + 10;
a(3) = 81 = 3 + 78 = 15 + 66 = 36 + 45.

Martin Ehrenstein, Table of n, a(n) for n = 0..37

Cf. A000217, A260647, A051533, A262749, A265140, A265134, A265136, A265138, A307597.

r = 125000; (* generates the first 12 terms of the sequence *)
lst = Table[0, {r}];
lim = Floor[Sqrt[2r]];
Do[ num = (i^2 + i)/2 + (j^2 + j)/2;
If[num <= r, lst[[num]]++], {i, lim}, {j,  i - 1}];
First /@ (Flatten@Position[lst, #] & /@ Range[Max[lst]])

upto(n) = {my(v = vector(n)); res = vector(10); for(i = 1, (sqrtint(8*n + 1)-1)\2, bi = binomial(i + 1, 2); for(j = i+1, (sqrtint(8*(n - bi))-1)\2, v[bi + binomial(j+1, 2)]++ ) ); for(i = 1, #v, if(v[i] > 0, if(v[i] > #res, res = concat(res, vector(v[i] - #res)); ); if(res[v[i]] == 0, res[v[i]] = i ) ) ); concat(0, res) } \\ David A. Corneth, Mar 08 2021

a(n) = min { m >= 0 : A307597(m) = n }. - Alois P. Heinz, Mar 08 2021

a(13)-a(18) from Alois P. Heinz, Mar 08 2021
a(19)-a(25) from David A. Corneth, Mar 08 2021
a(26)-a(33) from Jon E. Schoenfield, Mar 09 2021 (some terms first found by David A. Corneth)

A051611 Numbers that are not the sum of 2 nonzero triangular numbers.

Original entry on oeis.org

1, 3, 5, 8, 10, 14, 15, 17, 19, 23, 26, 28, 32, 33, 35, 40, 41, 44, 45, 47, 50, 52, 53, 54, 59, 62, 63, 68, 71, 74, 75, 77, 78, 80, 82, 85, 86, 89, 95, 96, 98, 103, 104, 105, 107, 109, 113, 116, 117, 118, 122, 124, 125, 128, 129, 131, 134, 138, 140, 143, 145, 147

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

A053603(a(n)) = 0. - Reinhard Zumkeller, Jun 28 2013

T. D. Noe, Table of n, a(n) for n=1..1000

Integers not in the sequence A051533. Cf. A002097, A000217, A007294, A051611, A053603.

a051611 n = a051611_list !! (n-1)
a051611_list = filter ((== 0) . a053603) [1..]
-- Reinhard Zumkeller, Jun 28 2013

notSumQ[n_] := Reduce[ 0 < x <= y && n == x*(x + 1)/2 + y*(y + 1)/2, {x, y}, Integers] === False; Select[ Range[150], notSumQ] (* Jean-François Alcover, Jun 27 2012 *)
With[{trnos=Accumulate[Range[100]]},Complement[Range[150],Total/@ Tuples[ trnos,2]]] (* Harvey P. Dale, Jun 01 2016 *)

A089982 Triangular numbers that can be expressed as the sum of 2 positive triangular numbers.

Original entry on oeis.org

6, 21, 36, 55, 66, 91, 120, 136, 171, 210, 231, 276, 351, 378, 406, 496, 561, 666, 703, 741, 820, 861, 946, 990, 1035, 1081, 1176, 1225, 1326, 1378, 1431, 1485, 1540, 1596, 1653, 1711, 1770, 1891, 1953, 2016, 2080, 2211, 2278, 2346, 2556, 2701, 2775, 2850

Offset: 1

Jon Perry, Jan 13 2004

Intersection of triangular numbers with sumset of triangular numbers. Triangular number analog of what for squares is {A057100(n)^2} = {A009000(n)^2}. {A000217} INTERSECT {A000217 + A000217}. - Jonathan Vos Post, Mar 09 2007

A subsequence of A051533. - Wolfdieter Lang, Jan 11 2017

			Generally, A000217(A000217(n)) = A000217(A000217(n)-1) + A000217(n) and so is automatically included. These are 6=T(3), 21=T(6), 55=T(10), etc. Other solutions occur when a partial sum from x to y is triangular, e.g., 15 + 16 + 17 + 18 = 66 = T(11), so T(14) + T(11) = T(18). This particular example arises since 10+4k is triangular (at k=14, 10 + 4k = 66), and we therefore have a solution.
All other solutions occur when 3+2k, 6+3k, 10+4k, etc. -- in general, T(j) + j*k -- is triangular.

Michael S. Branicky, Table of n, a(n) for n = 1..11915

Cf. A000217, A012132, A051533, A057100.

trn[i_]:=Module[{trnos=Accumulate[Range[i]],t2s},t2s=Union[Total/@ Tuples[ trnos,2]];Intersection[trnos,t2s]] (* Harvey P. Dale, Nov 08 2011 *)
Select[Range[75], ! PrimeQ[#^2 + (# + 1)^2] &] /. Integer_ -> (Integer^2 + Integer)/2 (* Arkadiusz Wesolowski, Dec 03 2015 *)

t(i) = i*(i+1)/2;
{ v=vector(100,i,t(i)); y=vector(100); c=0; for (i=1,30, for (j=i,30, x=t(i)+t(j); f=0; for (k=1,100,if (x==v[k],f=1;break)); if (f==1,y[c++ ]=x))); select(x->(x>0), vecsort(y,,8)) } \\ slightly edited by Michel Marcus, Apr 15 2021

lista(nn) = {for (n=1, nn, my(t = n*(n+1)/2); for (k=1, n-1, if (ispolygonal(t - k*(k+1)/2, 3), print1(t, ", "); break;)););} \\ Michel Marcus, Apr 15 2021

from itertools import count, takewhile
def aupto(lim):
    t = list(takewhile(lambda x: x<=lim, (i*(i+1)//2 for i in count(1))))
    s = set(a+b for i, a in enumerate(t) for b in t[i:])
    return sorted(s & set(t))
print(aupto(3000)) # Michael S. Branicky, Jun 21 2021

Triangular number m is in this sequence iff A000161(4*m+1)>1 or, alternatively, A083025(4*m+1)>1. - Max Alekseyev, Oct 24 2008

a(n) = A000217(A012132(n)). - Ivan N. Ianakiev, Jan 17 2013

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net) and David Wasserman, Sep 23 2005

A117048 Prime numbers that are expressible as the sum of two positive triangular numbers.

Original entry on oeis.org

2, 7, 11, 13, 29, 31, 37, 43, 61, 67, 73, 79, 83, 97, 101, 127, 137, 139, 151, 157, 163, 181, 191, 193, 199, 211, 227, 241, 263, 277, 281, 307, 331, 353, 367, 373, 379, 389, 409, 421, 433, 443, 461, 463, 487, 499, 541, 571, 577, 587, 601, 619, 631, 659, 661

Offset: 1

Andrew S. Plewe, Apr 15 2006

If the triangular number 0 is allowed, only one additional prime occurs: 3. In that case, the sequence becomes A117112.

A subsequence of A051533. - Wolfdieter Lang, Jan 11 2017

			2 = 1 + 1
7 = 1 + 6
11 = 1 + 10
13 = 10 + 3, etc.

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

Cf. A000040, A000217, A002243, A002244, A020756, A051533, A053614, A060773, A002636.

tri = Table[n (n + 1)/2, {n, 40}]; Select[Union[Flatten[Outer[Plus, tri, tri]]], # <= tri[[-1]]+1 && PrimeQ[#] &] (* T. D. Noe, Apr 07 2011 *)

is(n)=for(k=sqrtint(4*n+1)\2+1,(sqrtint(8*n+1)-1)\2, if(ispolygonal(n-k*(k+1)/2,3), return(n>3 && isprime(n)))); n==2 \\ Charles R Greathouse IV, Nov 07 2014

A265135 Numbers that are the sum of two distinct nonzero triangular numbers in more than two ways.

Original entry on oeis.org

81, 106, 181, 211, 256, 276, 331, 361, 381, 406, 456, 471, 531, 556, 601, 606, 616, 631, 666, 681, 706, 718, 731, 781, 786, 856, 861, 871, 906, 931, 939, 956, 981, 991, 1051, 1056, 1126, 1131, 1156, 1186, 1206, 1231, 1281, 1296, 1341, 1381, 1446, 1456

Offset: 1

Arkadiusz Wesolowski, Dec 02 2015

nonn

Cf. A000217, A051533, A260647, A265140 (exactly one way), A262749 (more than one way), A265134 (exactly two ways), A265136 (exactly three ways), A265137 (more than three ways), A265138 (exactly four ways).

r = 1456; lst = Table[0, {r}]; lim = Floor[Sqrt[8*r - 7]]; Do[num = (i^2 + i)/2 + (j^2 + j)/2; If[num <= r, lst[[num]]++], {i, lim}, {j, i - 1}]; Flatten@Position[lst, n_ /; n > 2]
Module[{nn=60,trnos},trnos=Accumulate[Range[nn]];Select[Sort[Flatten[ Table[ PositionIndex[Counts[Total/@Subsets[trnos,{2}]]][i],{i,3,nn}]]], #<= Last[trnos]&]] (* The program uses the PositionIndex and Counts functions from Mathematica version 10 *) (* Harvey P. Dale, Dec 26 2015 *)

A265137 Numbers that are the sum of two distinct nonzero triangular numbers in more than three ways.

Original entry on oeis.org

471, 531, 601, 616, 786, 871, 906, 991, 1056, 1126, 1156, 1186, 1281, 1296, 1341, 1381, 1446, 1486, 1551, 1576, 1602, 1641, 1656, 1771, 1806, 1836, 1896, 1906, 1921, 2031, 2044, 2061, 2146, 2226, 2265, 2281, 2316, 2356, 2401, 2421, 2451, 2486, 2551, 2646

Offset: 1

Arkadiusz Wesolowski, Dec 02 2015

nonn

If there exists any 3 X 3 magic square composed of triangular numbers, then its magic sum is a(n) + A000217(m) for some m and n.

Cf. A000217, A051533, A260647, A265140 (exactly one way), A262749 (more than one way), A265134 (exactly two ways), A265135 (more than two ways), A265136 (exactly three ways), A265138 (exactly four ways).

r = 2646; lst = Table[0, {r}]; lim = Floor[Sqrt[8*r - 7]]; Do[num = (i^2 + i)/2 + (j^2 + j)/2; If[num <= r, lst[[num]]++], {i, lim}, {j, i - 1}]; Flatten@Position[lst, n_ /; n > 3]
Module[{nn=80,trnos},trnos=Accumulate[Range[nn]];Select[Sort[ Flatten[ Table[ PositionIndex[Counts[Total/@Subsets[trnos,{2}]]][i],{i,4,nn}]]], #<= Last[trnos]&]] (* The program uses the PositionIndex and Counts functions from Mathematica version 10 *) (* Harvey P. Dale, Dec 26 2015 *)

A332987 Sums of two nonzero pentagonal numbers.

Original entry on oeis.org

2, 6, 10, 13, 17, 23, 24, 27, 34, 36, 40, 44, 47, 52, 56, 57, 63, 70, 71, 73, 75, 82, 86, 92, 93, 97, 102, 104, 105, 114, 118, 121, 122, 127, 129, 139, 140, 143, 146, 150, 152, 157, 162, 167, 168, 177, 180, 181, 184, 187, 188, 196, 198, 209, 211, 215, 222, 227

Offset: 1

Olivier Gérard, Mar 05 2020

nonn

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A000326 (pentagonal numbers).

Analogs are A000404 (square numbers), A051533 (triangular numbers), A286636 (centered square numbers), A287960 (centered triangular numbers), A288631 (square pyramidal numbers).

Module[{nn=15,pn},pn=PolygonalNumber[5,Range[nn]];Select[Union[ Total/@ Tuples[ pn,2]],#<=Last[pn]&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 15 2021 *)

A061208 Numbers which can be expressed as sum of distinct triangular numbers (A000217).

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Amarnath Murthy, Apr 21 2001

nonn

These numbers were called "almost-triangular" numbers during the Peru's Selection Test for the XII IberoAmerican Olympiad (1998). All numbers >= 34 are almost-triangular: see link. [Bernard Schott, Feb 04 2013]

			25 = 1 + 3 + 6 + 15

R. E. Woodrow, The Olympiad Corner, No. 198, Crux Mathematicorum, v25-n4(2002), 207-208, exercise 2.

Cf. A000217, A007294, A051611, A051533. Complement of A053614.

gf := product(1+x^(j*(j+1)/2), j=1..100): s := series(gf, x, 200): for i from 1 to 200 do if coeff(s, x, i) > 0 then printf(`%d,`,i) fi:od:

Corrected and extended by James Sellers, Apr 24 2001

cp's OEIS Frontend

A265138 Numbers that are the sum of two distinct nonzero triangular numbers in exactly four ways.

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A265140 Numbers that are the sum of two distinct nonzero triangular numbers in exactly one way.

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A342326 a(n) is the smallest nonnegative integer that can be written as a sum of two distinct nonzero triangular numbers in exactly n ways or -1 if no such integer exists.

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A051611 Numbers that are not the sum of 2 nonzero triangular numbers.

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A089982 Triangular numbers that can be expressed as the sum of 2 positive triangular numbers.

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A117048 Prime numbers that are expressible as the sum of two positive triangular numbers.

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A265135 Numbers that are the sum of two distinct nonzero triangular numbers in more than two ways.

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A265137 Numbers that are the sum of two distinct nonzero triangular numbers in more than three ways.

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A332987 Sums of two nonzero pentagonal numbers.