A245646 -id:A245646 - cp's OEIS frontend

A245647 The middle member 'b' of the Pythagorean triples (a,b,c) ordered by increasing c, where the triples consist of a triangular number, a square number and a pentagonal number.

4, 12, 105, 2625

Offset: 1

Ivan N. Ianakiev, Jul 28 2014

Next term comes from a triple with c > 10^5.

			a(1) = 4 as the first such Pythagorean triple is (3,4,5). The next three triples are (9,12,15), (100,105,145), (900,2625,2775).

Cf. A000217, A000290, A000326, A245646, A245648.

n=10^3;ppt={};list={};pos=1;t[x_]:=(IntegerPart[Sqrt[2*x]])*(IntegerPart[Sqrt[2*x]]+1)/2;ls[x_]:=Length[Sqrt[x]];lis[x_]:=Length[IntegerPart[Sqrt[x]]];lp[x_]:=Length[(Sqrt[24*x+1]+1)/6];lip[x_]:=Length[IntegerPart[(Sqrt[24*x+1]+1)/6]];Do[y=x+1;z=y+1;While[z<=n,While[z^2

A245648 The largest member 'c' of the Pythagorean triples (a,b,c) ordered by increasing c, where the triples consist of a triangular number, a square number and a pentagonal number.

Original entry on oeis.org

5, 15, 145, 2775

Offset: 1

Ivan N. Ianakiev, Jul 28 2014

Next term comes from a triple with c > 10^5.
From Michel Marcus, Apr 08 2021: (Start)
The 4 known triples that satisfy the requisite are [3,4,5], [9,12,15], [100, 105, 145], [900, 2625, 2775].
Let po(n) be A176774(n), the least polygonality of a number.
  po([3,4,5]) = [3,4,5]; <-----
  po([9,12,15]) = [4,5,3];
  po([100,105,145]) = [4,3,5]; <-----
  po([900,2625,2775]) = [4,5,3].
So for the 2 highlighted triples, we have a-gonal^2 + b-gonal^2 = c-gonal^2. Are there other Pythagorean triples with the same property?
Let nb(n) be A177025(n) is the number of ways to represent n as a polygonal number.
  nb([3,4,5]) = [1,1,1]; <-----
  nb([9,12,15]) = [4,5,3];
  nb([100,105,145]) = [4,3,5];
  nb([900,2625,2775]) = [4,5,3].
So for the highlighted triple, we get [1,1,1]. Are there other Pythagorean triples with the same property? (End)
Regarding the first question by Michel Marcus, if such triple [x,y,z] exists, then z > 10^4. Regarding his second question, if such triple exists, then z > 10^7. - Ivan N. Ianakiev, Dec 16 2021
a(5) > 10^11, if it exists. - Giovanni Resta, Apr 15 2021

			a(1) = 5 as the first such Pythagorean triple is (3,4,5). The next three triples are (9,12,15), (100,105,145), (900,2625,2775).

Cf. A000217, A000290, A000326, A245646, A245647.

Cf. A176774, A177025.

n=10^3;ppt={};list={};pos=1;t[x_]:=(IntegerPart[Sqrt[2*x]])*(IntegerPart[Sqrt[2*x]]+1)/2;ls[x_]:=Length[Sqrt[x]];lis[x_]:=Length[IntegerPart[Sqrt[x]]];lp[x_]:=Length[(Sqrt[24*x+1]+1)/6];lip[x_]:=Length[IntegerPart[(Sqrt[24*x+1]+1)/6]];Do[y=x+1;z=y+1;While[z<=n,While[z^2

A342491 a(n) = f(x)+f(y)+f(z), where (x,y,h) is the n-th Pythagorean triple listed in (A046083, A046084, A009000), and f(m)=A176774(m) is the smallest polygonality of m.

Original entry on oeis.org

12, 14, 23, 12, 28, 29, 27, 20, 38, 52, 27, 22, 11, 47, 20, 49, 53, 16, 69, 81, 17, 47, 59, 59, 34, 41, 93, 32, 76, 33, 34, 121, 76, 93, 88, 33, 37, 39, 101, 102, 83, 27, 90, 52, 73, 183, 75, 37, 45, 130, 105, 15, 155, 83, 120, 54, 106, 133, 129, 15, 123, 42, 225

Offset: 1

Michel Marcus, Mar 14 2021

nonn

Inspired by (A245646, A245647, A245648), for which a(n) = 12.

Examples of lower terms: 11 for (21, 28, 35), 10 for (64, 120, 136) and 9 for (8778, 10296, 13530).

			a(1) = 12 because (3, 4, 5) are (3-, 4-, 5-) gonal numbers, and 3+4+5=12.

Michel Marcus, Table of n, a(n) for n = 1..12471 (hypotenuses <= 10000).

Cf. A009000, A046083, A046084, A176774.
Cf. A213188 (see 2nd comment).
Cf. A245646, A245647, A245648.

tp(n) = my(k=3); while( !ispolygonal(n,k), k++); k; \\ A176774
f(v) = vecsum(apply(tp, v));
list(lim) = {my(v=List(), m2, s2, h2, h); for(middle=4, lim-1, m2=middle^2; for(small=1, middle, s2=small^2; if(issquare(h2=m2+s2, &h), if(h>lim, break); listput(v, [h, middle, small]);););); v = vecsort(Vec(v)); apply(f, v);} \\ adapted from A009000

a(n) = f(A046083(n)) + f(A046084(n)) + f(A009000(n)) where f is A176774.

A342858 a(n) is the least integer h such that there exists a Pythagorean triple (x, y, h) that satisfies f(x)+f(y)+f(h)=n where f(m)=A176774(m) is the smallest polygonality of m; a(n) = 0 if no such h exists.

Original entry on oeis.org

13530, 136, 35, 5, 4510, 10, 100, 45, 51, 1404

Offset: 9

Michel Marcus, Mar 26 2021

a(19) > 10^9 if it exists.

It appears that the triples whose sum is 10 (as in the 2nd example below) have legs n^6 = A001014(n), (n^8 - n^4)/2 = A218131(n+1)/2 and (n^8 + n^4)/2 = A071231(n) for n >= 2; they consist of 2 triangular numbers and 1 square number. - Michel Marcus, Apr 12 2021

			a(9)  = 13530 with A176774([8778, 10296, 13530]) = [3,3,3].
a(10) = 136   with A176774([64, 120, 136])       = [4,3,3].
a(11) = 35    with A176774([21, 28, 35])         = [3,3,5].
a(12) = 5     with A176774([3, 4, 5])            = [3,4,5].
a(13) = 4510  with A176774([2926, 3432, 4510])   = [3,5,5].
a(14) = 10    with A176774([6, 8, 10])           = [3,8,3].
a(15) = 100   with A176774([28, 96, 100])        = [3,8,4].
a(16) = 45    with A176774([27, 36, 45])         = [10,3,3].
a(17) = 51    with A176774([45, 24, 51])         = [3,9,5].
a(18) = 1404  with A176774([540, 1296, 1404])    = [7,4,7].

Michel Marcus, Table of n, a(n) for n = 9..10000, with 0 for a(19)

Cf. A009000, A046083, A046084, A176774.
Cf. A213188 (see 2nd comment).
Cf. A245646, A245647, A245648, A342491.
Cf. A001014, A071231, A218131.

tp(n) = if (n<3, [n], my(v=List()); fordiv(2*n, k, if(k<2, next); if(k==n, break); my(s=(2*n/k-4+2*k)/(k-1)); if(denominator(s)==1, listput(v, s))); v = Vec(v); v[#v]); \\ A176774
vsum(v) = vecsum(apply(tp, v));
lista(limp, lim) = {my(vr = vector(limp)); for(u = 2, sqrtint(lim), for(v = 1, u, if (u*u+v*v > lim, break); if ((gcd(u,v) == 1) && (0 != (u-v)%2), for (i = 1, lim, if (i*(u*u+v*v) > lim, break); my(w = [i*(u*u - v*v), i*2*u*v, i*(u*u+v*v)]); my(h = i*(u*u+v*v)); my(sw = vsum(w)); if (sw <= limp, if (vr[sw] == 0, vr[sw] = h, if (h < vr[sw], vr[sw] = h))););););); vector(#vr - 8, k, vr[k+8]);}
lista(80, 15000) \\ Michel Marcus, Apr 16 2021

A342469 Positive numbers that are triangular, square or pentagonal.

Original entry on oeis.org

1, 3, 4, 5, 6, 9, 10, 12, 15, 16, 21, 22, 25, 28, 35, 36, 45, 49, 51, 55, 64, 66, 70, 78, 81, 91, 92, 100, 105, 117, 120, 121, 136, 144, 145, 153, 169, 171, 176, 190, 196, 210, 225, 231, 247, 253, 256, 276, 287, 289, 300, 324, 325, 330, 351, 361, 376, 378, 400

Offset: 1

Michel Marcus, Mar 13 2021

nonn

Cf. A000290, A000217, A000326, A005214.

Cf. A245646, A245647, A245648.

isok(m) = ispolygonal(m,3) || ispolygonal(m,4) || ispolygonal(m,5);

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A245647 The middle member 'b' of the Pythagorean triples (a,b,c) ordered by increasing c, where the triples consist of a triangular number, a square number and a pentagonal number.

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A245648 The largest member 'c' of the Pythagorean triples (a,b,c) ordered by increasing c, where the triples consist of a triangular number, a square number and a pentagonal number.

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A342491 a(n) = f(x)+f(y)+f(z), where (x,y,h) is the n-th Pythagorean triple listed in (A046083, A046084, A009000), and f(m)=A176774(m) is the smallest polygonality of m.

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A342858 a(n) is the least integer h such that there exists a Pythagorean triple (x, y, h) that satisfies f(x)+f(y)+f(h)=n where f(m)=A176774(m) is the smallest polygonality of m; a(n) = 0 if no such h exists.

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A342469 Positive numbers that are triangular, square or pentagonal.

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