A110494 -id:A110494 - cp's OEIS frontend

A110493 Largest prime p such that p^2 divides binomial(2n,n), or 0 if binomial(2n,n) is squarefree.

0, 0, 0, 2, 0, 3, 2, 2, 3, 2, 2, 2, 2, 5, 5, 3, 3, 3, 5, 5, 3, 2, 2, 5, 5, 7, 7, 7, 2, 2, 2, 2, 7, 7, 7, 3, 2, 2, 5, 7, 7, 7, 3, 5, 5, 3, 7, 7, 7, 5, 3, 3, 3, 3, 2, 2, 3, 2, 2, 3, 3, 11, 11, 11, 11, 11, 5, 5, 5, 5, 5, 5, 11, 11, 11, 11, 11, 3, 5, 5, 3, 7, 7, 11, 11, 13, 13, 13, 13, 13, 13, 5, 5, 5, 11, 11

Offset: 0

T. D. Noe, Jul 22 2005

Binomial(2n,n) is squarefree for only n = 0, 1, 2, 4. Sequence A059097 lists n such that a(n) = 0 or 2. The plot shows the quadratic nature of this sequence. Sequence A110494 makes the quadratic behavior clearer.

Granville and Ramaré show that if n >= 2082 then a(n) >= sqrt(n/5). - Robert Israel, Sep 04 2019

			a(5) = 3 because binomial(10,5) = 252 = (2^2)(3^2)(7).

T. D. Noe, Table of n, a(n) for n = 0..10000
T. D. Noe, Plot of A110493
A. Granville and O. Ramaré, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients, Mathematika 43 (1996), 73-107, [DOI].

Cf. A110494 (least k such that prime(n)^2 divides binomial(2k, k)).

Cf. A059097, A110494.

f:= proc(n) local F;
  F:= select(t -> t[2]>=2, ifactors(binomial(2*n,n))[2]);
  if F = [] then 0 else max(map(t -> t[1],F)) fi
end proc:
map(f, [$0..100]); # Robert Israel, Sep 04 2019

Table[f=FactorInteger[Binomial[2n, n]]; s=Select[f, #[[2]]>1&]; If[s=={}, 0, s[[-1,1]]], {n, 0, 100}]

a(0) prepended by T. D. Noe, Mar 27 2014

A239622 Conjecturally, the irregular triangle of numbers k such that prime(n)^2 is the largest squared prime divisor of binomial(2k,k).

Original entry on oeis.org

0, 1, 2, 4, 3, 6, 7, 9, 10, 11, 12, 21, 22, 28, 29, 30, 31, 36, 37, 54, 55, 57, 58, 110, 171, 784, 786, 5, 8, 15, 16, 17, 20, 35, 42, 45, 50, 51, 52, 53, 56, 59, 60, 77, 80, 133, 134, 135, 136, 156, 157, 158, 159, 160, 161, 170, 210, 211, 212, 400, 401, 402, 651, 652, 785

Offset: 0

T. D. Noe, Mar 27 2014

Row 0 lists the numbers k such that binomial(2k,k) is squarefree. Sequence A110494 lists the first term of each row; A239623 lists the conjectured last term; A239624 lists the conjectured length of each row.

			The irregular triangle begins:
0, 1, 2, 4
3, 6, 7, 9,..., 784, 786
5, 8, 15, 16,..., 652, 785
13, 14, 18, 19,..., 445, 2080
25, 26, 27, 32,..., 783, 902
61, 62, 63, 64,..., 2033, 2034

T. D. Noe, Rows n = 0..40 of irregular triangle, flattened

Cf. A059097 (union of first two rows), A110493, A110494, A239623, A239624.

b = 1; t = Table[b = b*(4 - 2/n); last = 0; Do[If[Mod[b, p^2] == 0, last = p], {p, Prime[Range[PrimePi[Sqrt[2*n]]]]}]; last, {n, 20000}]; t = Join[{0}, t]; Table[Flatten[Position[t, p]] - 1, {p, Join[{0}, Prime[Range[20]]]}]

A239623 Conjecturally, the largest k such that prime(n)^2 is the largest squared prime divisor of binomial(2k,k).

Original entry on oeis.org

4, 786, 785, 2080, 902, 2034, 2079, 1086, 2081, 2090, 1652, 2562, 3905, 8185, 4987, 3507, 5562, 2713, 3584, 4191, 8285, 9319, 12237, 12117, 12248, 9311, 8180, 8399, 9308, 20123, 11977, 11683, 12261, 14365, 15403, 20114, 16867, 19938, 19559, 20316, 24706

Offset: 0

T. D. Noe, Mar 27 2014

nonn

The last number in row n of A239622. The 0th term is the largest number k such that binomial(2k,k) is squarefree. The first 41 terms were checked by computing binomial(2k,k) for k <= 10^5. See the plot in A110493.

Vincenzo Librandi, Table of n, a(n) for n = 0..125

Cf. A110493, A110494, A239622, A239624.

b = 1; t = Table[b = b*(4 - 2/n); last = 0; Do[If[Mod[b, p^2] == 0, last = p], {p, Prime[Range[PrimePi[Sqrt[2*n]]]]}]; last, {n, 25000}]; t = Join[{0}, t]; Table[Flatten[Position[t, p]][[-1]] - 1, {p, Join[{0}, Prime[Range[20]]]}]

A155185 Primes in A155175.

Original entry on oeis.org

5, 13, 113, 1741, 5101, 8581, 9941, 21841, 26681, 47741, 82013, 481181, 501001, 1009621, 2356621, 2542513, 3279361, 3723721, 4277813, 7757861, 8124481, 13204661, 25311613, 30772013, 44170601, 48619661, 51521401, 52541501, 54236113, 60731221, 72902813

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

Hypotenuse C (prime numbers only) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, ar=a*b/2; s=a+b+c, s-+1 are primes. p=1,q=2,a=3,b=4,c=5=prime,s=12-+1primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178, A155180, A088483, A001844, A096891, A066885, A099776, A110494, A081589

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],If[PrimeQ[c],AppendTo[lst,c]]],{n,8!}];lst (* corrected by Ray Chandler, Feb 11 2020 *)

Sequence corrected by Ray Chandler, Feb 11 2020

A239624 Conjecturally, the number of numbers k such that prime(n)^2 is the largest squared prime divisor of binomial(2k,k).

Original entry on oeis.org

4, 23, 38, 50, 51, 54, 65, 70, 107, 127, 127, 165, 155, 150, 239, 287, 280, 179, 336, 314, 230, 453, 423, 600, 612, 419, 246, 454, 455, 892, 1117, 624, 916, 432, 1115, 363, 934, 1061, 763, 1073, 1203, 524, 1523, 559, 1278, 735, 2221, 1987, 929, 475, 1179, 1605

Offset: 0

T. D. Noe, Mar 27 2014

nonn

The 0th term is the largest number k such that binomial(2k,k) is squarefree.

Cf. A110493, A110494, A239622, A239623.

b = 1; t = Table[b = b*(4 - 2/n); last = 0; Do[If[Mod[b, p^2] == 0, last = p], {p, Prime[Range[PrimePi[Sqrt[2*n]]]]}]; last, {n, 20000}]; t = Join[{0}, t]; Table[Length[Position[t, p]], {p, Join[{0}, Prime[Range[20]]]}]

A155186 Primes in A155171.

Original entry on oeis.org

2, 7, 29, 101, 107, 197, 227, 457, 647, 829, 1549, 1627, 2221, 2309, 2347, 2521, 2677, 2801, 3181, 3299, 3529, 3541, 3557, 3739, 3769, 4231, 4549, 4871, 4987, 5651, 5827, 5881, 6037, 6079, 6637, 6827, 7517, 7639, 7937, 9787, 11621, 12041, 12329, 13009

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

Numbers p (prime numbers only) of primitive Pythagorean triangles such that perimeters are Averages of twin prime pairs, q=p+1, a=q^2-p^2, c=q^2+p^2, b=2*p*q, s=a+b+c, s-+1 are primes.

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178, A155180, A088483, A001844, A096891, A066885, A099776, A110494, A081589, A155185, A019389, A062090, A050150

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],If[PrimeQ[p],AppendTo[lst,p]]],{n,8!}];lst

A155187 Prime numbers q of primitive Pythagorean triangles such that perimeters are averages of twin prime pairs, p+1=q(prime), a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.

Original entry on oeis.org

2, 3, 11, 71, 227, 491, 683, 1103, 1187, 2591, 3923, 4271, 4931, 6737, 7193, 7703, 8093, 8753, 8963, 9173, 9377, 10271, 13043, 13451, 13997, 15233, 15443, 15803, 15887, 17957, 18701, 19961, 20681, 21701, 22031, 22073, 24371, 24473, 24683

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 21 2009

nonn

p=1, q=2(prime), a=3, b=4, c=5, s=12-+1 primes, ...

Cf. A020882, A020886, A020884, A020883, A024364, A024406, A155171, A155173, A155174, A155175, A155176, A155177, A155178, A155180, A088483, A001844, A096891, A066885, A099776, A110494, A081589, A155185, A019389, A062090, A050150, A155186, A068231, A129517, A054574, A132281.

lst={};Do[p=n;q=p+1;a=q^2-p^2;c=q^2+p^2;b=2*p*q;ar=a*b/2;s=a+b+c;If[PrimeQ[s-1]&&PrimeQ[s+1],If[PrimeQ[q],AppendTo[lst,q]]],{n,8!}];lst

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A110493 Largest prime p such that p^2 divides binomial(2n,n), or 0 if binomial(2n,n) is squarefree.

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A239622 Conjecturally, the irregular triangle of numbers k such that prime(n)^2 is the largest squared prime divisor of binomial(2k,k).

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A239623 Conjecturally, the largest k such that prime(n)^2 is the largest squared prime divisor of binomial(2k,k).

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A155185 Primes in A155175.

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A239624 Conjecturally, the number of numbers k such that prime(n)^2 is the largest squared prime divisor of binomial(2k,k).

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A155186 Primes in A155171.

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A155187 Prime numbers q of primitive Pythagorean triangles such that perimeters are averages of twin prime pairs, p+1=q(prime), a=q^2-p^2, c=q^2+p^2, b=2*p*q, ar=a*b/2; s=a+b+c, s-+1 are primes.

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A155187 Prime numbers q of primitive Pythagorean triangles such that perimeters are averages of twin prime pairs, p+1=q(prime), a=q^2-p^2, c=q^2+p^2, b=2pq, ar=a*b/2; s=a+b+c, s-+1 are primes.