Ray Chandler - cp's OEIS frontend

A365418 Partial sums of A301298.

1, 6, 15, 29, 48, 71, 99, 132, 169, 211, 258, 309, 365, 426, 491, 561, 636, 715, 799, 888, 981, 1079, 1182, 1289, 1401, 1518, 1639, 1765, 1896, 2031, 2171, 2316, 2465, 2619, 2778, 2941, 3109, 3282, 3459, 3641, 3828, 4019, 4215, 4416, 4621, 4831, 5046, 5265, 5489, 5718, 5951, 6189, 6432, 6679, 6931, 7188, 7449

Offset: 0

Ray Chandler, Sep 03 2023

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A301298.

LinearRecurrence[{2, -1, 1, -2, 1}, {1, 6, 15, 29, 48}, 50]

G.f.: -(x^4 + 4*x^3 + 4*x^2 + 4x + 1)/((x - 1)^3*(x^2 + x + 1)).

a(n) = 2*a(n-1) - a(n-2) + a(n-3) - 2*a(n-4) + a(n-5) for n > 5.

A365417 Partial sums of A265036.

Original entry on oeis.org

1, 5, 11, 18, 28, 42, 62, 86, 110, 133, 159, 193, 235, 279, 319, 356, 398, 452, 516, 580, 636, 687, 745, 819, 905, 989, 1061, 1126, 1200, 1294, 1402, 1506, 1594, 1673, 1763, 1877, 2007, 2131, 2235, 2328, 2434, 2568, 2720, 2864, 2984, 3091, 3213, 3367, 3541, 3705, 3841, 3962, 4100, 4274, 4470, 4654

Offset: 0

Ray Chandler, Sep 03 2023

Ray Chandler, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-12,18,-18,12,-5,1).

Cf. A265036.

Join[{1,5,11},LinearRecurrence[{5, -12, 18, -18, 12, -5, 1},{18, 28, 42, 62, 86, 110, 133},50]]

G.f.: (2*x^9 - 6*x^8 + 8*x^7 - 7*x^6 + 2*x^5 + 2*x^4 - 5*x^3 + 2*x^2 - 1)/((x - 1)^3*(x^2 - x + 1)^2).

a(n) = 5*a(n-1) - 12*a(n-2) + 18*a(n-3) - 18*a(n-4) + 12*a(n-5) - 5*a(n-6) + a(n-7) for n > 10.

A362689 Binomial(n+p, n) mod n where p=9.

Original entry on oeis.org

0, 1, 1, 3, 2, 1, 2, 6, 2, 8, 1, 2, 1, 10, 14, 15, 1, 3, 1, 5, 18, 1, 1, 12, 6, 14, 4, 12, 1, 22, 1, 13, 1, 1, 13, 23, 1, 1, 14, 34, 1, 14, 1, 34, 15, 24, 1, 27, 8, 11, 18, 1, 1, 7, 12, 16, 1, 30, 1, 28, 1, 32, 17, 25, 14, 23, 1, 35, 47, 25, 1, 54, 1, 38, 66

Offset: 1

Ray Chandler, Apr 29 2023

nonn

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, order 5806080.

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133875, A133885, A133880, A133890, A133900, A133910, A362686, A362687, A362688.

Mathematica
```
Table[Mod[Binomial[n+9,n],n],{n,90}]
```

a(n)=binomial(n+9,n) mod n.

For n > 5806081, a(n) = 2*a(n-2903040) - a(n-5806080).

A362688 Binomial(n+p, n) mod n where p=8.

Original entry on oeis.org

0, 1, 0, 3, 2, 3, 2, 6, 1, 8, 1, 6, 1, 10, 9, 15, 1, 1, 1, 5, 18, 1, 1, 12, 6, 14, 1, 12, 1, 12, 1, 13, 12, 1, 13, 19, 1, 1, 27, 34, 1, 0, 1, 34, 10, 24, 1, 27, 8, 11, 18, 1, 1, 1, 12, 16, 39, 30, 1, 48, 1, 32, 10, 25, 14, 45, 1, 35, 24, 25, 1, 46, 1, 38, 66

Offset: 1

Ray Chandler, Apr 29 2023

nonn

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, order 645120.

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133875, A133885, A133880, A133890, A133900, A133910, A362686, A362687, A362689.

Mathematica
```
Table[Mod[Binomial[n+8,n],n],{n,90}]
```

a(n)=binomial(n+8,n) mod n.

For n > 645240, a(n) = 2*a(n-322560) - a(n-645120).

A362687 Binomial(n+p, n) mod n where p=7.

Original entry on oeis.org

0, 0, 0, 2, 2, 0, 2, 3, 1, 8, 1, 0, 1, 10, 9, 5, 1, 10, 1, 10, 18, 12, 1, 15, 6, 14, 1, 12, 1, 12, 1, 9, 12, 18, 13, 10, 1, 20, 27, 19, 1, 0, 1, 12, 10, 24, 1, 45, 8, 36, 18, 14, 1, 28, 12, 23, 39, 30, 1, 48, 1, 32, 10, 17, 14, 12, 1, 18, 24, 60, 1, 19, 1

Offset: 1

Ray Chandler, Apr 29 2023

nonn

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, order 10080.

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133875, A133885, A133880, A133890, A133900, A133910, A362686, A362688, A362689.

Mathematica
```
Table[Mod[Binomial[n+7,n],n],{n,90}]
```

a(n)=binomial(n+7,n) mod n.

For n > 10122, a(n) = 2*a(n-5040) - a(n-10080).

A362686 Binomial(n+p, n) mod n where p=6.

Original entry on oeis.org

0, 0, 0, 2, 2, 0, 1, 3, 1, 8, 1, 0, 1, 8, 9, 5, 1, 10, 1, 10, 15, 12, 1, 15, 6, 14, 1, 8, 1, 12, 1, 9, 12, 18, 8, 10, 1, 20, 27, 19, 1, 36, 1, 12, 10, 24, 1, 45, 1, 36, 18, 14, 1, 28, 12, 15, 39, 30, 1, 48, 1, 32, 1, 17, 14, 12, 1, 18, 24, 50, 1, 19, 1, 38

Offset: 1

Ray Chandler, Apr 29 2023

nonn

Ray Chandler, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, order 1440.

Cf. A000040, A133620-A133625, A133630, A038509, A133634-A133636.

Cf. A133875, A133885, A133880, A133890, A133900, A133910, A362687, A362688, A362689.

Table[Mod[Binomial[n+6, n], n], {n, 90}]

a(n)=binomial(n+6,n) mod n.

For n > 1452, a(n) = 2*a(n-720) - a(n-1440).

A331671 Number of Pythagorean triangles with sum of legs n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 2, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1

Offset: 1

Ray Chandler, Feb 26 2020

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A118905, A198390, A254064.

A330174 Number of primitive Pythagorean triangles with sum of legs n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2

Offset: 1

Ray Chandler, Feb 15 2020

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A058529, A198441.

f[n_]:=Block[{ps,sps}, ps=First/@FactorInteger[n];
sps=Select[ps,MemberQ[{1,7},Mod[#,8]]&];
If[sps==ps&&n!=1,2^(Length[ps]-1),0]];  Table[f[n],{n,120}]

A331019 a(n) is the least k such that the denominator(sigma(sigma(kn))/(kn)) equals n.

Original entry on oeis.org

1, 15, 1, 8, 1, 144, 1, 16, 1, 1, 1, 8, 1, 255, 3, 16, 1, 12, 1, 2, 7, 1, 1, 288, 1, 18, 21, 8, 1, 84, 1, 13, 1, 11, 1, 4096, 1, 4, 3, 270, 1, 2448, 1, 2, 3, 1, 1, 16, 1, 420, 3, 1, 1, 124, 3, 16, 3, 128, 1, 616, 1, 85, 3, 16, 1, 1, 1, 8, 3, 1, 1, 32, 1, 64

Offset: 1

Ray Chandler, Jan 09 2020

nonn

a(n) is the least k such that the A318060(k*n) equals n.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A000203 (sigma), A051027 (sigma(sigma)), A318060.

Cf. A019278, A330598, A331033.

a(n) = my(k=n); while (denominator(sigma(sigma(k))/k) != n, k+=n); k/n;

a(n) = A331033(n)/n.

A329148 Composite hypotenuses of primitive Pythagorean triangles (A120961) that are not circumdiameters of non-Pythagorean primitive Heronian triangles (A285579).

Original entry on oeis.org

1073, 1537, 1961, 3277, 4181, 5713, 7289, 7373, 8633, 9193, 9773, 10001, 10397, 11729, 13837, 14393, 14837, 14893, 15397, 16153, 16781, 17777, 17861, 19517, 20513, 20609, 20617, 20737, 21053, 21253, 21473, 21953, 22601, 23141, 23393

Offset: 1

Ray Chandler, Dec 04 2019

nonn

Inspired by comments from Frank M Jackson in A285579.

All of the 1378 terms in the b-file are the product of two distinct Pythagorean primes.

Ray Chandler, Table of n, a(n) for n = 1..1378 (terms < 6*10^6, computed using files from Kurz link)
Sascha Kurz, Lists of primitive Heronian triples, Bayreuth University.

Cf. A120961, A285579, A002144.

cp's OEIS Frontend

User: Ray Chandler

A365418 Partial sums of A301298.

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A365417 Partial sums of A265036.

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A362689 Binomial(n+p, n) mod n where p=9.

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A362688 Binomial(n+p, n) mod n where p=8.

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A362687 Binomial(n+p, n) mod n where p=7.

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A362686 Binomial(n+p, n) mod n where p=6.

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A331671 Number of Pythagorean triangles with sum of legs n.

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A330174 Number of primitive Pythagorean triangles with sum of legs n.

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Mathematica

A331019 a(n) is the least k such that the denominator(sigma(sigma(k*n))/(k*n)) equals n.

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A331019 a(n) is the least k such that the denominator(sigma(sigma(kn))/(kn)) equals n.