A100555 -id:A100555 - cp's OEIS frontend

A101777 Triangle read by rows where n-th row is the lexicographically least set of n not-necessarily-distinct primes summing to A100555(n)-1.

3, 3, 5, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 5, 7, 2, 2, 2, 2, 2, 2, 2, 3, 7, 2, 2, 2, 2, 2, 2, 2, 2, 3, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 7

Offset: 1

Ray Chandler, Jan 10 2005

			Triangle begins:
  {3}
  {3,5}
  {2,3,3}
  {2,2,2,2}
  {2,2,2,2,7}
  {2,2,2,2,2,5}
  {2,2,2,2,2,2,3}
  {2,2,2,2,2,2,5,7}
  {2,2,2,2,2,2,2,3,7}
  {2,2,2,2,2,2,2,2,3,5}
  {2,2,2,2,2,2,2,2,2,3,3}
  {2,2,2,2,2,2,2,2,2,2,2,2}
  {2,2,2,2,2,2,2,2,2,2,2,2,11}
  {2,2,2,2,2,2,2,2,2,2,2,3,3,7}
  {2,2,2,2,2,2,2,2,2,2,2,2,2,2,7}

Cf. A100555, A101776, A101778.

Row 8 of table corrected at the suggestion of Jinyuan Wang by Ray Chandler, Jan 28 2020

A100498 Smallest square that is equal to the sum of n distinct primes plus 1.

Original entry on oeis.org

1, 4, 9, 16, 36, 49, 64, 81, 100, 121, 144, 169, 256, 289, 324, 361, 400, 441, 576, 625, 676, 729, 900, 961, 1024, 1089, 1296, 1369, 1444, 1521, 1764, 1849, 1936, 2025, 2304, 2401, 2500, 2601, 2916, 3025, 3136, 3481, 3600, 3721, 3844, 4225, 4356, 4489

Offset: 0

Giovanni Teofilatto, Dec 31 2004

nonn

			a(1)=4 because 2^2=1+3;
a(2)=9 because 3^2=1+3+5;
a(3)=16 because 4^2=1+3+5+7.

Cf. A100555, A101771-A101775.

Extended by Ray Chandler, Jan 10 2005

A101776 Smallest k such that k^2 is equal to the sum of n not-necessarily-distinct primes plus 1.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13

Offset: 0

Ray Chandler, Jan 10 2005

nonn

Pattern appears to be: one 1, one 2, three 3's, three 4's, ..., (2k+1) (2k+1)'s, (2k+1) (2k+2)'s.

It appears that a(n) is also the number of pixels in C_{n}, a pixelated arc of circle x^2 + y^2 = n, defined as the set of the (x, y), ordered pairs of nonnegative integers, such that (x^2 + y^2 = n) or ((x^2 + y^2 < n) and ((x+1)^2 + y^2 > n or x^2 + (y+1)^2 > n)). - Luc Rousseau, Dec 30 2019

Jinyuan Wang, Table of n, a(n) for n = 0..1000

Cf. A100555, A101777, A101778.

iMax[k_,n_]:=PrimePi[k^2-2*n+1]
f[k_,n_]:=IntegerPartitions[k^2-1,{n},Table[Prime[i],{i,1,iMax[k,n]}]]
a[n_]:=Module[{k=1},While[f[k,n]=={},k++];k]
Table[a[n],{n,0,100}]
(* Luc Rousseau, Dec 30 2019 *)

a(n) = ceil(sqrt(2*n+1)); \\ Jinyuan Wang, Jan 28 2020

a(n) = sqrt(A100555(n)).

a(n) = ceiling(sqrt(2*n+1)). - Mohammad K. Azarian, Jun 15 2016 [Proof: for any k > 1 and 1 <= m <= 2*k, a(2*k^2-2*k+m) = 2*k because (2*k-1)^2 < 2*(2*k^2-2*k+m) + 1 and (2*k)^2 = 2*(2*k^2-6*k+3*m+1) + 3*(4*k-2*m-1) + 1; a(2*k^2+m) = 2*k + 1 because (2*k)^2 < 2*(2*k^2+m) + 1 and (2*k+1)^2 = 2*(2*k^2-4*k+3*m) + 3*(4*k-2*m) + 1. Therefore, a(n) = ceiling(sqrt(2*n+1)) for n >= 5. Note that the formula is also correct for n < 5, hence a(n) = ceiling(sqrt(2*n+1)). - Jinyuan Wang, Jan 28 2020]

A101778 Last term in each row of triangle referenced in A101777.

Original entry on oeis.org

3, 5, 3, 2, 7, 5, 3, 7, 7, 5, 3, 2, 11, 7, 7, 5, 3, 13, 11, 7, 7, 5, 3, 2, 13, 13, 11, 7, 7, 5, 3, 17, 13, 13, 11, 7, 7, 5, 3, 2, 19, 17, 13, 13, 11, 7, 7, 5, 3, 19, 19, 17, 13, 13, 11, 7, 7, 5, 3, 2, 23, 19, 19, 17, 13, 13, 11, 7, 7, 5, 3, 23, 23, 19, 19, 17, 13, 13, 11, 7, 7, 5, 3, 2, 23

Offset: 1

Ray Chandler, Jan 10 2005

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A100555, A101776, A101777.

A020482(k) = forprime(q=2, k, if(isprime(2*k-q), return(2*k-q)));
a(n) = {my(r=(ceil(sqrt(2*n+1)))^2-2*n+3); if(r%2==0, r=A020482(r/2), if(isprime(r-2), r-=2, r=A020482(r\2))); r; } \\ Jinyuan Wang, Jan 29 2020

a(n) = A101777(A000217(n)).

cp's OEIS Frontend

A101777 Triangle read by rows where n-th row is the lexicographically least set of n not-necessarily-distinct primes summing to A100555(n)-1.

Views

Author

Keywords

Examples

Crossrefs

Extensions

A100498 Smallest square that is equal to the sum of n distinct primes plus 1.

Views

Author

Keywords

Examples

Crossrefs

Extensions

A101776 Smallest k such that k^2 is equal to the sum of n not-necessarily-distinct primes plus 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A101778 Last term in each row of triangle referenced in A101777.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula