A159047 -id:A159047 - cp's OEIS frontend

A159048 Primes of the form m*(m+1)/2 + 4.

5, 7, 19, 59, 109, 157, 257, 439, 599, 907, 1039, 1229, 1279, 1489, 1657, 3407, 3659, 4099, 5569, 6907, 7507, 7879, 8389, 9049, 10589, 11329, 11939, 14369, 16657, 17209, 17959, 18149, 18919, 19507, 20507, 22159, 26339, 30139, 31379, 34457, 36319

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 03 2009

nonn

			7=3+4, 19=15+4, 59=55+4, 109=105+4, 157=153+4, 257=253+4, ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000217, A055469, A055472, A159047

Select[Accumulate[Range[500]]+4,PrimeQ]  (* Harvey P. Dale, Apr 23 2011 *)

for(n=1, 500, if(isprime(k=n*(n+1)/2 + 4), print1(k, ", "))) \\ G. C. Greubel, Jul 03 2017

Edited by N. J. A. Sloane, Apr 06 2009

A159049 Primes of the form (5+ a triangular number A000217).

Original entry on oeis.org

5, 11, 41, 71, 83, 281, 383, 1181, 1601, 2351, 2633, 3491, 3833, 4283, 5783, 6221, 6791, 8783, 10301, 10883, 11633, 12251, 14033, 15581, 18341, 26111, 26801, 30881, 31883, 34721, 38231, 41333, 41621, 42491, 43961, 46061, 47591, 53633, 60383

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 03 2009

			11 is in the list because it is A000217(3)+5 and a prime. 41=36+5= A000217(8)+5 is a prime. 71=66+5=A000217(11)+5 is a prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A055469, A055472, A159047, A159048

s=0;lst={};Do[s+=n;p=s+5;If[PrimeQ[p],AppendTo[lst,p]],{n,0,7!}];lst
Select[Accumulate[Range[0,500]]+5,PrimeQ] (* Harvey P. Dale, Jul 08 2017 *)

for(n=1,500, if(isprime(k=n*(n+1)/2 + 5), print1(k, ", "))) \\ G. C. Greubel, Jul 13 2017

Definition rephrased, R. J. Mathar, Apr 05 2009

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}]]]

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A159048 Primes of the form m*(m+1)/2 + 4.

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A159049 Primes of the form (5+ a triangular number A000217).

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Extensions

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

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Author

Keywords

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Crossrefs

Programs

Mathematica