A057061 -id:A057061

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A057060

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

+10
2

1, 2, 2, 1, 1, 3, 2, 4, 2, 1, 3, 1, 5, 7, 2, 8, 4, 6, 1, 5, 7, 1, 5, 11, 6, 10, 12, 2, 4, 8, 7, 11, 1, 3, 13, 15, 4, 10, 14, 2, 8, 10, 1, 3, 7, 9, 1, 13, 17, 19, 2, 8, 10, 20, 4, 10, 16, 18, 1, 5, 7, 17, 7, 11, 13, 17, 6, 12, 22, 24, 2, 8, 16, 22, 1, 5, 11

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The rectangle 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, ...

LINKS

Table of n, a(n) for n=1..77.

FORMULA

a(n) = A002260(prime(n)). - Kevin Ryde, Feb 12 2023

EXAMPLE

The 8th prime, 19, is in row 4, so a(8) = 4.

MATHEMATICA

s = Flatten[Table[Range[n], {n, 1, 40}]];

Table[s[[Prime[n]]], {n, 1, 100}]

PROG

(PARI) f(n) = n-binomial((sqrtint(8*n)+1)\2, 2); \\ A002260

a(n) = f(prime(n)); \\ Michel Marcus, Feb 24 2023

CROSSREFS

Cf. A000027, A000040, A002260, A185787.

See A057061 for primes in columns.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 30 2000

EXTENSIONS

Edited by Clark Kimberling, Feb 13 2023

STATUS

approved

A185510

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

+10
2

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Start with the natural number array A000027:

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...

Row n of A185510 shows the primes in row n of A000027:

2....7....11...29...37....67....79...137...(A055469)

3....5....17...23...47...107...173...233...(A055472)

13..31...139..193..409...499...823..1381...(A159047)

19..59...109..157..257...439...599...907...(A159048)

41..71....83..281..383..1181..1601..2351...(A159049)

61..97...331..601..709..1087..1231..2707...

43..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

LINKS

Table of n, a(n) for n=1..66.

MATHEMATICA

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}]]]

CROSSREFS

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

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jan 29 2011

STATUS

approved

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