A099088 -id:A099088 - cp's OEIS frontend

A113501 Indices of prime NSW numbers A088165.

1, 2, 3, 9, 14, 23, 29, 81, 128, 210, 468, 473, 746, 950, 3344, 4043, 4839, 14376, 39521, 64563, 72984, 82899, 84338, 85206, 86121, 139160

Offset: 1

Eric W. Weisstein, Jan 09 2006

Very closely related to indices of prime Pell-Lucas numbers (A099088).

a(27) > 221400. - Robert Price, Mar 29 2019

			NSW(1) = 7, NSW(2) = 41, NSW(3) = 239, NSW(9) = 9369319, ...

H. Li, T. MacHenry, Permanents and Determinants, Weighted Isobaric Polynomials, and Integer Sequences, J. Int. Seq. 16 (2013) #13.3.5, example 47.
Eric Weisstein's World of Mathematics, NSW Number
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A002315, A088165, A099088.

nsw = LinearRecurrence[{6, -1}, {7, 41}, 1000]; Position[nsw, ?(PrimeQ[#] &)] // Flatten (* _Amiram Eldar, Dec 07 2018 *)

isok(n) = my(w=3+quadgen(32)); isprime(imag((1+w)*w^n)); \\ Michel Marcus, Dec 07 2018

a(19)-a(20) from Eric W. Weisstein, May 22 2006
a(21) from Eric W. Weisstein, Aug 29 2006
a(22) from Eric W. Weisstein, Nov 11 2006
a(23) from Eric W. Weisstein, Nov 26 2006
a(24) from Eric W. Weisstein, Dec 10 2006
a(25) from Eric W. Weisstein, Jan 25 2007
a(26) from Robert Price, Dec 07 2018

A086395 Primes found among the numerators of the continued fraction rational approximations to sqrt(2).

Original entry on oeis.org

3, 7, 17, 41, 239, 577, 665857, 9369319, 63018038201, 489133282872437279, 19175002942688032928599, 123426017006182806728593424683999798008235734137469123231828679

Offset: 1

Cino Hilliard, Sep 06 2003, Jul 30 2004, Oct 02 2005

nonn

Or, starting with the fraction 1/1, the prime numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and twice bottom to get the new top. Or, A001333(n) is prime.
The transformation of fractions is 1/1 -> 3/2 -> 7/5 -> 17/12 -> 41/29 -> ... A001333(n)/A000129(n). - R. J. Mathar, Aug 18 2008
Is this sequence infinite?

Prime Obsession, John Derbyshire, Joseph Henry Press, April 2004, p 16.

G. C. Greubel, Table of n, a(n) for n = 1..18

Cf. A001333, A086383, A257553.

Select[Numerator[Convergents[Sqrt[2],250]],PrimeQ] (* Harvey P. Dale, Oct 19 2011 *)

\Continued fraction rational approximation of numeric constants f. m=steps. cfracnumprime(m,f) = { default(realprecision,3000); cf = vector(m+10); x=f; for(n=0,m, i=floor(x); x=1/(x-i); cf[n+1] = i; ); for(m1=0,m, r=cf[m1+1]; forstep(n=m1,1,-1, r = 1/r; r+=cf[n]; ); numer=numerator(r); denom=denominator(r); if(ispseudoprime(numer),print1(numer,",")); ) }

primenum(n,k,typ) = \yp = 1 num, 2 denom. print only prime num or denom. { local(a,b,x,tmp,v); a=1;b=1; for(x=1,n, tmp=b; b=a+b; a=k*tmp+a; if(typ==1,v=a,v=b); if(isprime(v),print1(v","); ) ); print(); print(a/b+.) }

a(n) = A001333(A099088(n)). - R. J. Mathar, Feb 01 2024

Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar

A005850 Primes p such that the NSW number A002315((p-1)/2) is prime.

Original entry on oeis.org

3, 5, 7, 19, 29, 47, 59, 163, 257, 421, 937, 947, 1493, 1901, 6689, 8087, 9679, 28753, 79043, 129127, 145969, 165799, 168677, 170413, 172243

Offset: 1

N. J. A. Sloane

Some of the larger entries may only correspond to probable primes.

Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 290.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 248.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. Newman, D. Shanks, and H. Williams, Simple groups of square order and interesting sequence of primes, Acta Arithmetica (1980), Volume: 38, Issue: 2, page 129-140
Eric Weisstein's World of Mathematics, Pell Number
Eric Weisstein's World of Mathematics, Pythagoras's Constant

Cf. A002315, A088165.

A099088 is a closely related sequence.

max = 10000 (* computation is very slow beyond this limit *); nc = Numerator[Convergents[Sqrt[2], max]]; Reap[Do[If[PrimeQ[n], If[PrimeQ[nc[[n]]], Print[n]; Sow[n]]] , {n, 3, max}]][[2, 1]] (* Jean-François Alcover, Oct 22 2012, after David Applegate *)

is(n)=my(w=3+quadgen(32)); isprime(n) && n>2 && ispseudoprime(imag((1+w)*w^(n\2))) \\ Charles R Greathouse IV, Oct 19 2012

A088165(n) mod a(n) = 1. - Altug Alkan, Mar 17 2016

6689, 8087, 9679 reported by Warut Roonguthai on the PrimeForm mailing list.
28753 found by Andrew Walker (ajw01(AT)uow.edu.au), Jul 12 2001.
129127, 145969, 165799, 168677, 170413, 172243 found by Eric W. Weisstein, May 22 2006 - Jan 25 2007 [from Mike Oakes, Mar 29 2009]
Name corrected by David Applegate, Jean-François Alcover and Charles R Greathouse IV, Oct 19 2012

A228916 Indices of primes in sequence A108300.

Original entry on oeis.org

1, 3, 9, 15, 39, 225, 231, 363, 687, 1299, 1335, 1809, 2367, 12735

Offset: 1

Vaclav Kotesovec, Sep 08 2013

Conjecture: for n>1 a(n) is a multiple of 3.
The indices greater than 363 yield probable primes. - Vaclav Kotesovec, Oct 28 2013
Next term > 100000. - Tyler Busby, Mar 29 2024

Cf. A108300, A209493, A001605, A001606, A096650, A099088.

seq=RecurrenceTable[{a[n]==3*a[n-1]+a[n-2],a[0]==1,a[1]==5},a,{n,1,1000}]; Select[Range[1000],PrimeQ[seq[[#]]]&]

A229287 Least k such that the numerator of the continued fraction [1,..,1,k] (n 1s) is prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 4, 2, 1, 2, 1, 11, 4, 2, 1, 8, 3, 9, 6, 2, 1, 6, 4, 5, 4, 2, 1, 8, 3, 53, 12, 4, 7, 14, 3, 13, 4, 20, 3, 2, 1, 21, 8, 2, 1, 66, 5, 17, 16, 9, 3, 10, 4, 41, 4, 20, 15, 20, 3, 43, 6, 33, 15, 22, 12, 63, 36, 20, 3, 98, 37, 25, 30, 21, 17, 20

Offset: 1

Clark Kimberling, Sep 19 2013

nonn

			The numerators of the continued fraction [1,1,1,1,k] for k=1,2 are 8,13; 8 is not prime and 13 is, so a(4) = 2.

Cf. A099088, A086395, A229288.

z = 160; c[n_, k_] := Join[ContinuedFraction[GoldenRatio, n], {k}]; x[n_, k_] := Numerator[FromContinuedFraction[c[n, k]] ]; t[n_] := Table[x[n, k], {k, 1, z}]; u = Table[First[Select[t[n], PrimeQ]], {n, 1, z}]; Flatten[Table[Position[t[n], u[[n]]], {n, 1, z}]]

A229288 Least k such that the numerator of the continued fraction [2,..,2,k] (n 2s) is prime.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 2, 15, 2, 13, 6, 3, 1, 5, 3, 1, 5, 13, 12, 55, 6, 97, 6, 9, 2, 1, 5, 39, 28, 7, 14, 31, 11, 83, 21, 51, 2, 31, 30, 51, 4, 7, 3, 1, 40, 37, 21, 27, 2, 95, 15, 9, 14, 5, 2, 1, 11, 107, 6, 51, 18, 31, 9, 13, 13, 13, 9, 81, 7, 90, 13, 19

Offset: 1

Clark Kimberling, Sep 19 2013

nonn

			The numerators of the continued fraction [2,2,2,2,2,k] for k=1,2,3 are 99,169,239;  the first two are not prime and 239 is, so a(5) = 3.

Cf. A099088, A086395, A229287.

z = 160; c[n_, k_] := Join[ContinuedFraction[1 + Sqrt[2], n], {k}]; x[n_, k_] := Numerator[FromContinuedFraction[c[n, k]]]; t[n_] := Table[x[n, k], {k, 1, z}]; u = Table[First[Select[t[n], PrimeQ]], {n, 1, z}]; Flatten[Table[Position[t[n], u[[n]]], {n, 1, z}]]

A331234 Triangular numbers having exactly 9 divisors.

Original entry on oeis.org

36, 1225, 1413721, 7885505171090778556470578126753302097454601, 67594562493730400324395236678194231988753004665644702944024074038452164931450549832074201

Offset: 1

Jon E. Schoenfield, Jan 12 2020

nonn

Any number having an odd number of divisors is a square, so each term in this sequence is a term of A001110 (numbers that are both triangular and square). Since A001110(k) = (A000129(k)*A001333(k))^2, A001110(k) will have exactly 9 divisors iff A000129(k) and A001333(k) are both prime (i.e., k is in both A096650 and A099088); the first 5 values of k at which this occurs are 2, 3, 5, 29, and 59.

Conjecture: a(5) is the final term of this sequence.

			Writing the k-th triangular number A000217(k) as T(k):
a(1) = T(8) = 8*9/2 = 36 = 2^2 * 3^2;
a(2) = T(49) = 49*50/2 = 1225 = 5^2 * 7^2;
a(3) = T(1681) = 1681*1682/2 = 1413721 = 29^2 * 41^2.
Factorization of larger known terms:
a(4) = 44560482149^2 * 63018038201^2;
a(5) = 13558774610046711780701^2 * 19175002942688032928599^2.

Intersection of A000217 (triangular numbers) and A030627 (numbers with exactly 9 divisors).
Triangular numbers having exactly k divisors: A068443 (k=4), A292989 (k=6), A330809 (k=8).
Cf. A063440 (number of divisors of n-th triangular number), A242585 (number of divisors of the n-th positive number that is both triangular and square).
Cf. A001110, A000129, A001333, A096650, A099088.

A331399 Numbers k such that A000129(k) and A001333(k) are both prime.

Original entry on oeis.org

2, 3, 5, 29, 59

Offset: 1

Jon E. Schoenfield, Jan 16 2020

This sequence is the intersection of A096650 and A099088.

The k-th square triangular number A001110(k) = (A000129(k)*A001333(k))^2 has exactly 9 divisors iff k is in this sequence, so if a(5) is the final term of this sequence, then there are only 5 triangular numbers that have exactly 9 divisors (cf. A331234).

Cf. A001110 (numbers that are both triangular and square), A000129 (Pell numbers), A001333 (numerators of continued fraction convergents to sqrt(2); equivalently, prime companion Pell numbers, divided by 2), A096650 (indices of prime Pell numbers), A099088 (indices of prime companion Pell numbers, divided by 2), A331234 (triangular numbers having exactly 9 divisors).

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A113501 Indices of prime NSW numbers A088165.

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A086395 Primes found among the numerators of the continued fraction rational approximations to sqrt(2).

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A005850 Primes p such that the NSW number A002315((p-1)/2) is prime.

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A228916 Indices of primes in sequence A108300.

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A229287 Least k such that the numerator of the continued fraction [1,..,1,k] (n 1s) is prime.

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A229288 Least k such that the numerator of the continued fraction [2,..,2,k] (n 2s) is prime.

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A331234 Triangular numbers having exactly 9 divisors.

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A331399 Numbers k such that A000129(k) and A001333(k) are both prime.

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