A101778 -id:A101778 - cp's OEIS frontend

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

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]

A100555 Smallest square that is equal to the sum of n not-necessarily-distinct primes plus 1.

Original entry on oeis.org

1, 4, 9, 9, 9, 16, 16, 16, 25, 25, 25, 25, 25, 36, 36, 36, 36, 36, 49, 49, 49, 49, 49, 49, 49, 64, 64, 64, 64, 64, 64, 64, 81, 81, 81, 81, 81, 81, 81, 81, 81, 100, 100, 100, 100, 100, 100, 100, 100, 100, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 121, 144, 144

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)=9 because 3^2=1+2+3+3.
a(4)=9 because 3^2=1+2+2+2+2.

Cf. A100498, A101776-A101778.

Corrected and extended by Ray Chandler, Jan 10 2005

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.

Original entry on oeis.org

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

cp's OEIS Frontend

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

A100555 Smallest square that is equal to the sum of n not-necessarily-distinct primes plus 1.

Views

Author

Keywords

Examples

Crossrefs

Extensions

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