A020756 -id:A020756 - cp's OEIS frontend

A287960 Numbers that are the sum of two centered triangular numbers (A005448).

2, 5, 8, 11, 14, 20, 23, 29, 32, 35, 38, 41, 47, 50, 56, 62, 65, 68, 74, 77, 83, 86, 89, 92, 95, 104, 110, 113, 116, 119, 128, 131, 137, 140, 146, 149, 155, 167, 170, 173, 176, 182, 185, 194, 197, 200, 203, 209, 212, 218, 221, 230, 236, 239, 245, 251, 254, 263, 266, 272, 275, 278, 281, 284, 293, 299

Offset: 1

Ilya Gutkovskiy, Jun 03 2017

nonn

Eric Weisstein's World of Mathematics, Centered Triangular Number
Index entries for sequences related to centered polygonal numbers

Cf. A005448, A020756, A051533, A102795, A286636.

nmax = 300; f[x_] := Sum[x^(3 k (k - 1)/2 + 1), {k, 1, 20}]^2; Exponent[#, x] & /@ List @@ Normal[Series[f[x], {x, 0, nmax}]]

8*a(n) = 10+3*A097269(n). - R. J. Mathar, Jul 26 2017

A357504 Numbers that are the sum of two distinct triangular numbers.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 11, 13, 15, 16, 18, 21, 22, 24, 25, 27, 28, 29, 31, 34, 36, 37, 38, 39, 42, 43, 45, 46, 48, 49, 51, 55, 56, 57, 58, 60, 61, 64, 65, 66, 67, 69, 70, 72, 73, 76, 78, 79, 81, 83, 84, 87, 88, 91, 92, 93, 94, 97, 99, 100, 101, 102, 105, 106, 108

Offset: 1

Stefano Spezia, Oct 01 2022

This sequence differs from A020756 in excluding the terms that are twice a triangular number and that cannot be expressed as a sum of two distinct triangular numbers: 0, 2, 12, 20, 30, 90, 110, 132, ... = 2*A357529.

Cf. A000217 (subsequence, excluding 0), A020756 (supersequence), A339952, A357505 (complement).

Cf. A357529.

TriangularQ[n_]:=IntegerQ[(Sqrt[1+8n]-1)/2]; A000217[n_]:=n(n+1)/2; a={}; kn=0; For[k=0, k<=110, k++, For[h=0, A000217[h]A000217[h]] && k>kn, AppendTo[a, k]; kn=k]]]; a (* Stefano Spezia, Nov 06 2022 *)

a(n) = (A339952(n) - 1)/4.

A357529 Triangular numbers k such that 2*k cannot be expressed as a sum of two distinct triangular numbers.

Original entry on oeis.org

0, 1, 6, 10, 15, 45, 55, 66, 91, 120, 136, 231, 276, 300, 406, 435, 496, 561, 595, 630, 741, 780, 820, 861, 1081, 1225, 1326, 1431, 1830, 2016, 2080, 2145, 2211, 2415, 2485, 2701, 2850, 3240, 3321, 3486, 3655, 3916, 4005, 4465, 4560, 4950, 5050, 5356, 5460, 5565

Offset: 1

Stefano Spezia, Oct 02 2022

Subset of even terms of A357505, divided by 2. - Michel Marcus, Nov 05 2022

Cf. A000217 (supersequence), A002378.
Half of the complement of A357504 in A020756.
Half of the complement of A020757 in A357505.
Subsequence of A008851.

TriangularQ[n_]:=IntegerQ[(Sqrt[1+8n]-1)/2]; A000217[n_]:=n(n+1)/2; a={}; For[k=0, k<=105, k++, ok=1; For[h=0, h<2k, h++, If[TriangularQ[2*A000217[k] - A000217[h]] && k!=h, ok=0]]; If[ok==1, AppendTo[a,k(k+1)/2]]]; a (* Stefano Spezia, Nov 05 2022 *)

A111908 Numbers that are not the sum of a prime and a nonzero triangular number.

Original entry on oeis.org

1, 2, 7, 36, 61, 105, 171, 210, 211, 216, 325, 351, 406, 528, 561, 630, 741, 780, 990, 1081, 1176, 1275, 1596, 1711, 1830, 1953, 2016, 2145, 2346, 2628, 2775, 3003, 3081, 3240, 3321, 3655, 3741, 3916, 4278, 4371, 4465, 4560, 4851, 5253, 5460, 5565, 5886

Offset: 1

Stefan Steinerberger, Nov 25 2005

nonn

Can anybody prove or disprove a(n) = O(n^c) for some constant c?

Jonathan Vos Post has observed that every term in A076768 also occurs in this sequence.

			7 = 1+6 = 2+5 = 3+4; 7 is in the sequence because there is no sum where the first element is a prime and the second one a triangular number.

Donovan Johnson, Table of n, a(n) for n = 1..10000

Cf. A064233, A000217, A076768, A020756, A072386, A014089, A014090, A000404, A001481, A002654, A100570, A101181.

lim=6000;plim=PrimePi[lim];tlim=Ceiling[Sqrt[2lim]];Complement[Range[lim],Union[Flatten[Table[Prime[i]+PolygonalNumber[j],{i,plim},{j,tlim}]]]] (* James C. McMahon, Jun 04 2024 *)

a(47) and offset corrected by Donovan Johnson, Feb 09 2013

A113796 Numbers k such that k = T(x) + T(y) where T(m) is the m-th triangular number and k is concatenate(x, y) in base 10.

Original entry on oeis.org

190, 191, 19900, 19901, 90415, 585910, 1201545, 1414910, 1501726, 1909415, 1999000, 1999001, 2442196, 7003676, 7693846, 14745226, 28296970, 30307171, 42009156, 47748526, 61549231, 63249300, 78049756, 82749850, 84559880, 115449880, 117259850, 121959756

Offset: 1

Giovanni Resta, Jan 21 2006

Contains (2*10^i - 1)*10^i and (2*10^i - 1)*10^i + 1 for all i >= 1. - Michael S. Branicky, Jan 22 2022

			90415 = T(90) + T(415).

David A. Corneth, Table of n, a(n) for n = 1..56 (first 41 terms from Michael S. Branicky)

Cf. A000217.

Subsequence of A020756.

lst = {}; t[n_] := n(n + 1)/2; Do[p=10; While[n > p, If[t[Mod[n, p]] + t[Floor[n/p]] == n, AppendTo[lst, n]]; p*= 10], {n, 10^6}]; lst

def T(n): return n*(n+1)//2
def ok(n):
    if n < 10: return False
    s = str(n)
    splits = ((int(s[:i]), int(s[i:])) for i in range(1, len(s)))
    return any(n == T(x) + T(y) for x, y in splits)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jan 22 2022

a(26)-a(28) from Michael S. Branicky, Jan 22 2022

A185980 Nontriangular numbers that are the sum of two (positive) triangular numbers in more than one way.

Original entry on oeis.org

16, 31, 42, 46, 51, 56, 72, 76, 81, 94, 106, 111, 121, 123, 126, 133, 141, 146, 156, 157, 172, 174, 181, 186, 191, 196, 198, 211, 216, 225, 226, 237, 241, 246, 256, 259, 268, 281, 286, 289, 291, 297, 301, 306, 310, 315, 321, 326, 328, 331, 336, 342, 346, 354

Offset: 1

Wolfdieter Lang, Feb 15 2011

A000217(0)=0, but it will not appear as a summand.

A185979 uses only positive triangular summands, and includes triagonal (or body diagonal) numbers like 231, 276, 406, 666, ...

			16 = 15 + 1 = 10 + 6.
81 = 45 + 36 = 66 + 15 = 78 + 3.

Cf. A000217, A020756 (sums of two triangular numbers), A051533 (sums of two positive triangular numbers), A064816, A185978, A185979.

See the one given for A185978, taking into account multiplicities.

A191766 Integers that are a sum of two triangular numbers and also the sum of two square numbers (including zeros).

Original entry on oeis.org

0, 1, 2, 4, 9, 10, 13, 16, 18, 20, 25, 29, 34, 36, 37, 45, 49, 58, 61, 64, 65, 72, 73, 81, 90, 97, 100, 101, 106, 121, 130, 136, 137, 144, 146, 148, 153, 157, 160, 164, 169, 181, 193, 196, 200, 202, 205, 208, 218, 225, 226, 232, 234, 241, 244, 245

Offset: 1

Ant King, Jun 22 2011

nonn

This sequence is infinite as, for example, all integers of the form m^8+m^4-2*m^2*n^2+12*m^6*n^2+n^4+38*m^4*n^4+12*m^2*n^6+n^8 are included.

The sequence includes all squares, since n^2 = T(n-1) + T(n), where T(n) = A000217(n) is the n-th triangular number. - Franklin T. Adams-Watters, Jun 24 2011

			9 is the sum of two triangular numbers: 6 + 3, and also two squares: 9 + 0. Hence 9 is in the sequence.

P. A. Piza, Problems for Solution: 4425 The American Mathematical Monthly, Vol. 58, No. 2, (February 1951), p. 113.
P. A. Piza, G. W. Walker, and C. M. Sandwick, Sr., 4425, The American Mathematical Monthly, Vol. 59, No. 6, (June - July 1952), pp. 417-419.

Cf. A000217, A000290, A191765, intersection of A001481 and A020756.

data=Length[Reduce[a^2+b^2==1/2 c (c+1)+1/2 d(d+1) == # && a>=0 && b>=0 && c>=0 && d>=0,{a,b,c,d},Integers]] &/@Range[0,250];Prepend[DeleteCases[Table[If[data[[k]]>0,k-1,0],{k,1,Length[data]}],0],0]
With[ {n = 250}, Pick[ Range[ 0, n], {} != FindInstance[ a*a + b*b == # && c (c + 1) + d (d + 1) == 2 # && a >= 0 && b >= 0 && c >= 0 && d >= 0, {a, b, c, d}, Integers] & /@ Range[ 0, n]]] (* Michael Somos, Jun 24 2011 *)

A119961 Semiprimes that are the sum of two triangular numbers.

Original entry on oeis.org

4, 6, 9, 10, 15, 21, 22, 25, 34, 38, 39, 46, 49, 51, 55, 57, 58, 65, 69, 87, 91, 93, 94, 106, 111, 115, 119, 121, 123, 133, 141, 142, 146, 159, 169, 177, 183, 202, 205, 213, 214, 218, 219, 226, 235, 237, 249, 253, 254, 259, 262, 265, 267, 274, 289, 291, 295

Offset: 1

Jonathan Vos Post, Aug 02 2006

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

Cf. A001358, A020756, A117048.

With[{nn=60},Take[Union[Select[Total/@Tuples[Accumulate[Range[0,nn]],2],PrimeOmega[ #] ==2&]],nn]] (* Harvey P. Dale, Nov 04 2020 *)

A020756 intersection A001358.

Missing a(2) and a(19)-a(53) from Giovanni Resta, Jun 13 2016

A274794 Numbers n such that n^3 is the sum of two triangular numbers in exactly one way.

Original entry on oeis.org

0, 1, 3, 4, 7, 9, 10, 19, 24, 25, 34, 37, 39, 42, 49, 54, 55, 72, 73, 78, 85, 87, 93, 94, 102, 108, 109, 118, 138, 142, 147, 157, 160, 165, 168, 175, 192, 195, 202, 210, 214, 220, 228, 232, 243, 247, 249, 250, 252, 253, 258, 267, 273, 274, 279, 289, 297, 312, 333

Offset: 1

Altug Alkan, Jul 07 2016

nonn

A115104 is a subsequence. Terms such that 4*n^3 + 1 is not prime are 24, 337, 457, 750, 840, 1015, ...

			3 is a term because 3^3 = 27 = 6 + 21.

Cf. A000217, A000578, A020756, A052343, A115104, A119345.

Select[Range@ 333, Length[PowersRepresentations[4 #^3 + 1, 2, 2]] == 1 &] (* after Ant King at A052343, or *)
nn = 20; t = (#^2 + #)/2 & /@ Range[0, nn^3]; Select[Range[0, nn], Function[n, Count[Transpose@ {#, n^3 - #} &@ Range[0, Floor[n^3/2]], k_ /; Times @@ Boole@ Map[MemberQ[t, #] &, k] == 1] == 1]] (* Michael De Vlieger, Jul 07 2016 *)

a052343(n) = sum(i=0, (sqrtint(4*n + 1) - 1)\2, issquare(n - i - i^2));
lista(nn) = for(n=0, nn, if(a052343(n^3) == 1, print1(n, ", ")));

cp's OEIS Frontend

A287960 Numbers that are the sum of two centered triangular numbers (A005448).

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A357504 Numbers that are the sum of two distinct triangular numbers.

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A357529 Triangular numbers k such that 2*k cannot be expressed as a sum of two distinct triangular numbers.

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A111908 Numbers that are not the sum of a prime and a nonzero triangular number.

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A113796 Numbers k such that k = T(x) + T(y) where T(m) is the m-th triangular number and k is concatenate(x, y) in base 10.

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A185980 Nontriangular numbers that are the sum of two (positive) triangular numbers in more than one way.

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A191766 Integers that are a sum of two triangular numbers and also the sum of two square numbers (including zeros).

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A119961 Semiprimes that are the sum of two triangular numbers.

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Extensions

A274794 Numbers n such that n^3 is the sum of two triangular numbers in exactly one way.

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