A057060 -id:A057060 - cp's OEIS frontend

A057061 a(n) = number of the column of (R(i,j)) that contains prime(n), where R(i,j) is the rectangle with antidiagonals 1; 2,3; 4,5,6; ...

2, 1, 2, 4, 5, 3, 5, 3, 6, 8, 6, 9, 5, 3, 9, 3, 8, 6, 12, 8, 6, 13, 9, 3, 9, 5, 3, 14, 12, 8, 10, 6, 17, 15, 5, 3, 15, 9, 5, 18, 12, 10, 20, 18, 14, 12, 21, 9, 5, 3, 21, 15, 13, 3, 20, 14, 8, 6, 24, 20, 18, 8, 19, 15, 13, 9, 21, 15, 5, 3, 26, 20, 12, 6

Offset: 1

Clark Kimberling, Jul 30 2000

nonn

The rectangle R(i,j) has this corner:
   1,  2,  4,  7, 11, 16, 22, 29, ...
   3,  5,  8, 12, 17, 23, 30, 38, ...
   6,  9, 13, 18, 24, 31, 39, 48, ...
  10, 14, 19, 25, 32, 40, 49, 59, ...
  15, 20, 26, 33, 41, 50, 60, 71, ...
  21, 27, 34, 42, 51, 61, 72, 84, ...
  28, 35, 43, 52, 62, 73, 85, 98, ...

			The 8th prime, 19, is in column 3, so a(8) = 3.

Cf. A000027, A000040, A004736, A185787.

See A057060 for primes in rows.

f(n) = 1 + binomial(1 + floor(1/2 + sqrt(2*n)), 2) - n; \\ A004736
a(n) = f(prime(n)); \\ Michel Marcus, Feb 24 2023

a(n) = A004736(prime(n)). - Michel Marcus, Feb 24 2023

Edited by Clark Kimberling, Feb 12 2013

A185510 Array of primes in the natural number array A000027, by antidiagonals.

Original entry on oeis.org

2, 7, 3, 11, 5, 13, 29, 17, 31, 19, 37, 23, 139, 59, 41, 67, 47, 193, 109, 71, 61, 79, 107, 409, 157, 83, 97, 43, 137, 173, 499, 257, 281, 331, 73, 53, 191, 233, 823, 439, 383, 601, 127, 113, 199, 211, 353, 1381, 599, 1181, 709, 197, 179, 829, 101, 277, 467, 1543, 907, 1601, 1087, 283, 239, 1549, 163, 89

Offset: 1

Clark Kimberling, Jan 29 2011

Start with the natural number array A000027:
...2.....4....7...11...16...22...29...
...5.....8...12...17...23...30...38...
...9....13...18...24...31...39...48...
..14...19...25...32...40...49...59...
..20...26...33...41...50...60...71...
..27...34...42...51...61...72...84...
..35...43...52...62...73...85...98...
Row n of A185510 shows the primes in row n of A000027:
...7....11...29...37....67....79...137...(A055469)
...5....17...23...47...107...173...233...(A055472)
.31...139..193..409...499...823..1381...(A159047)
.59...109..157..257...439...599...907...(A159048)
.71....83..281..383..1181..1601..2351...(A159049)
.97...331..601..709..1087..1231..2707...
.73...127..197..283..307...503...673...
Conjecture:  Every row contains infinitely many primes.
Every prime occurs exactly once; that is, every prime is uniquely expressible as (1/2)(n^2 + (2k-1)n + (k-2)(k-1)) for some positive integers n and k.  We assume as true the conjecture that each row is infinite. - Clark Kimberling, Mar 10 2020

Cf. A000027, A000040, A185511, A057060, A057061.

f[n_,k_]:=n+(k+n-2)(k+n-1)/2;
TableForm[Map[Select[#,PrimeQ]&, Table[f[n,k],{n,1,20}, {k,1,100}]]]

cp's OEIS Frontend

A057061 a(n) = number of the column of (R(i,j)) that contains prime(n), where R(i,j) is the rectangle with antidiagonals 1; 2,3; 4,5,6; ...

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