A099776 -id:A099776 - cp's OEIS frontend

A144396 The odd numbers greater than 1.

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133

Offset: 1

Paul Curtz, Oct 03 2008

Last number of the n-th row of the triangle described in A142717.
If negated, these are also the values at local minima of the sequence A141620.
a(n) is the shortest leg of the n-th Pythagorean triple with consecutive longer leg and hypotenuse. The n-th such triple is given by (2n+1,2n^2+2n, 2n^2+2n+1), so that the longer legs are A046092(n) and the hypotenuses are A099776(n). - Ant King, Feb 10 2011
Numbers k such that the symmetric representation of sigma(k) has a pair of bars as its ends (cf. A237593). - Omar E. Pol, Sep 28 2018
Numbers k such that there is a prime knot with k crossings and braid index 2. (IS this true with "prime" removed?) - Charles R Greathouse IV, Feb 14 2023

Muniru A Asiru, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1)

Complement of A004275 and of A004277.
Cf. A000217, A002378, A005408, A007395, A046092, A099776, A237593, A254858.
Essentially the same as A140139, A130773, A062545, A020735, A005818.

List([3,5..200],n->n); # Muniru A Asiru, Sep 28 2018

a144396 = (+ 1) . (* 2)
a144396_list = [3, 5 ..] -- Reinhard Zumkeller, Feb 09 2015

seq(n,n=3..200,2); # Muniru A Asiru, Sep 28 2018

Range[3, 200, 2] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2012 *)

a(n)=2*n+1 \\ Charles R Greathouse IV, Feb 14 2023

a(n) = A005408(n+1) = A000290(n+1) - A000290(n).
G.f.: x*(3-x)/(1-x)^2. - Jaume Oliver Lafont, Aug 30 2009
a(n) = A254858(n-1,2). - Reinhard Zumkeller, Feb 09 2015

Edited by R. J. Mathar, May 21 2009

A087484 Least hypotenuse of a primitive Pythagorean triangle with inradius n.

Original entry on oeis.org

5, 13, 17, 41, 37, 29, 65, 145, 101, 53, 145, 65, 197, 85, 73, 545, 325, 125, 401, 97, 109, 173, 577, 185, 677, 229, 785, 137, 901, 157, 1025, 2113, 205, 365, 169, 185, 1445, 445, 265, 233, 1765, 205, 1937, 241, 221, 629, 2305, 617, 2501, 733, 409, 305, 2917, 845

Offset: 1

Lekraj Beedassy, Oct 23 2003

nonn

Least hypotenuse which is 2n short of the sum of the legs of a primitive Pythagorean triangle.

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

Cf. A020887, A099776 (largest hypotenuse).

More terms from Ray Chandler, Oct 25 2003
Name simplified by Ray Chandler, Jan 26 2020
Deleted a link to a bad web site - N. J. A. Sloane, Jan 26 2020

A108769 Numbers m such that m^2 + (m+1)^2 is a semiprime.

Original entry on oeis.org

3, 6, 8, 10, 11, 13, 15, 16, 18, 20, 26, 27, 31, 33, 37, 38, 40, 43, 44, 45, 48, 51, 52, 54, 55, 56, 57, 59, 62, 63, 64, 67, 68, 73, 74, 75, 76, 77, 80, 81, 83, 89, 92, 94, 98, 105, 107, 111, 112, 113, 114, 117, 120, 123, 124, 129, 131, 133, 134, 138, 140, 141, 142, 143

Offset: 1

Jason Earls, Jun 25 2005

Numbers m such that A099776(m) is a semiprime. - Michel Marcus, Nov 17 2022

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Cf. A001358, A027861, A099776.

a:= proc(n) local k; for k from 1+`if`(n=1, 0, a(n-1))
      while (t-> isprime(t) or numtheory[bigomega](t)
      >2)(2*k*(k+1)+1) do od: k
    end:
seq(a(n), n=1..70);  # Alois P. Heinz, Aug 01 2019

Select[Range[1000], PrimeOmega[#^2 + (#+1)^2] == 2&] (* Jean-François Alcover, Nov 17 2022 *)

isok(m) = bigomega(m^2 + (m+1)^2) == 2; \\ Michel Marcus, Nov 17 2022

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

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

A322135 Table of truncated square pyramid numbers, read by antidiagonals.

Original entry on oeis.org

1, 4, 5, 9, 13, 14, 16, 25, 29, 30, 25, 41, 50, 54, 55, 36, 61, 77, 86, 90, 91, 49, 85, 110, 126, 135, 139, 140, 64, 113, 149, 174, 190, 199, 203, 204, 81, 145, 194, 230, 255, 271, 280, 284, 285, 100, 181, 245, 294, 330, 355, 371, 380, 384, 385, 121, 221, 302

Offset: 1

Allan C. Wechsler, Nov 27 2018

The n-th row contains n numbers: n^2, n^2 + (n-1)^2, ..., n^2 + (n-1)^2 + ... + 1^2.
All numbers that appear in the table are listed in ascending order at A034705.
All numbers that appear twice or more are listed at A130052.
The left column is A000290 (the squares).
The top row is A000330 (the square pyramidal numbers).
The columns are A000290, A099776 (or a tail of A001844), a tail of A005918 or A120328, a tail of A027575, a tail of A027578, a tail of A027865, ...
The first two rows are A000330 and a tail of A168599, but subsequent rows are not currently in the OEIS, and are all tails of A000330 minus various constants.
The main diagonal is A050410.

			The 17th term is entry 2 on antidiagonal 6, so we sum two terms: 6^2 + 5^2 = 61.
Table begins:
   1   5  14  30  55  91 140 204 ...
   4  13  29  54  90 139 203 ...
   9  25  50  86 135 199 ...
  16  41  77 126 190 ...
  25  61 110 174 ...
  36  85 149 ...
  49 113 ...
  64 ...
  ...

See comments; also cf. A000330, A059255.

T[n_,k_] = Sum[(n+i)^2, {i,0,k-1}]; Table[T[n-k+1, k], {n,1,10},  {k,1,n}] // Flatten (* Amiram Eldar, Nov 28 2018 *)
f[n_] := Table[SeriesCoefficient[-((y (y (1 + y) + x (1 - 2 y - 3 y^2) + x^2 (1 - 3 y + 4 y^2)))/((-1 + x)^3 (-1 + y)^4)) , {x, 0,
i + 1 - j}, {y, 0, j}], {i, n, n}, {j, 1, n}]; Flatten[Array[f, 10]] (* Stefano Spezia, Nov 28 2018 *)

T(n,k) = n^2 + (n+1)^2 + ... + (n+k-1)^2 = A000330(n + k - 1) - A000330(n - 1) = T(n, k) = k*n^2 + (k^2 - k)*n + (1/3*k^3 - 1/2*k^2 + 1/6*k)

G.f.: -y*(y*(1 + y) + x*(1 - 2*y - 3*y^2) + x^2*(1 - 3*y + 4*y^2))/((- 1 + x)^3*(- 1 + y)^4). - Stefano Spezia, Nov 28 2018

A350757 a(1)=1; for n>1, a(n) is the smallest number k > a(n-1) such that a(n-1) + k is not a square.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

J. Lowell, Jan 13 2022

nonn

Complement of A099776.

			5 is not a term because 4 + 5 = 9 = 3^2.

Cf. A099776.

a[1] = 1; a[n_] := a[n] = Module[{k = a[n - 1] + 1}, While[IntegerQ[Sqrt[a[n - 1] + k]], k++]; k]; Array[a, 100] (* Amiram Eldar, Jan 14 2022 *)

lista(nn) = {my(x=1, list=List(x)); for (n=2, nn, my(k=x+1); while (issquare(x+k), k++); listput(list, k); x = k;); list;} \\ Michel Marcus, Jan 14 2022

from math import isqrt
def A350757(n): return n+(m:=isqrt(n>>1))-int(n<=m*((m<<1)+1)+1) if n>1 else 1 # Chai Wah Wu, Oct 01 2024

For n>1, a(n) = n+m if n>m(2m+1)+1 and a(n) = n+m-1 otherwise where m = floor(sqrt(n/2)). - Chai Wah Wu, Oct 01 2024

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

cp's OEIS Frontend

A144396 The odd numbers greater than 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A087484 Least hypotenuse of a primitive Pythagorean triangle with inradius n.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A108769 Numbers m such that m^2 + (m+1)^2 is a semiprime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A155185 Primes in A155175.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A155186 Primes in A155171.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A322135 Table of truncated square pyramid numbers, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A350757 a(1)=1; for n>1, a(n) is the smallest number k > a(n-1) such that a(n-1) + k is not a square.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

Formula

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.

Views

Author

Keywords

Comments

Crossrefs

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.