A088165 -id:A088165 - cp's OEIS frontend

A240278 Primes p which are floor of Root-Mean-Square (RMS) of prime(n), prime(n+1) and prime(n+2).

3, 5, 13, 19, 43, 47, 53, 83, 89, 103, 109, 131, 157, 167, 173, 193, 211, 229, 233, 257, 263, 313, 349, 353, 359, 373, 383, 389, 409, 443, 449, 463, 503, 563, 593, 607, 643, 647, 653, 677, 683, 691, 709, 733, 797, 823, 859, 883, 919, 941, 947, 971, 977, 983, 1013

Offset: 1

K. D. Bajpai, Apr 03 2014

nonn

			11, 13 and 17 are consecutive primes: sqrt(( 11^2 + 13^2 + 17^2)/3) = 13.89244399: floor(13.89244399) = 13, which is prime and appears in the sequence.
17, 19 and 23 are consecutive  primes: sqrt(( 17^2 + 19^2 + 23^2)/3) = 19.82422760: floor(19.82422760) = 19, which is prime and appears in the sequence.
41, 43 and 47 are consecutive  primes: sqrt(( 41^2 + 43^2 + 47^2)/3) = 43.73785546: floor(43.73785546) = 43, which is prime and appears in the sequence.

K. D. Bajpai, Table of n, a(n) for n = 1..3075

Cf. A000040, A075471, A088165.

a := proc(n) local c, b, d, e; c:=ithprime(n); b:=ithprime(n+1); d:=ithprime(n+2); e:=floor(sqrt((c^2+b^2+d^2)/3)); if isprime(e) then RETURN(e); fi; end: seq(a(n), n=1..500);

Select[Floor[RootMeanSquare[#]]&/@Partition[Prime[Range[200]],3,1],PrimeQ] (* Harvey P. Dale, Mar 23 2018 *)

A094666 Composite NSW numbers.

Original entry on oeis.org

1393, 8119, 47321, 275807, 1607521, 54608393, 318281039, 1855077841, 10812186007, 367296043199, 2140758220993, 12477253282759, 72722761475561, 423859315570607, 2470433131948081, 14398739476117879, 83922003724759193

Offset: 1

Lekraj Beedassy, Jun 07 2004

nonn

J. A. Sellers, Infinitely Many Composite NSW Numbers: An Inductive Proof
J. A. Sellers and H. Williams, On the Infinitude of Composite NSW Numbers

Cf. A002315, A088165.

a[0] = 1; a[1] = 7; a[n_] := a[n] = 6a[n - 1] - a[n - 2]; a /@ Select[ Range[23], !PrimeQ[ a[ # ]] &] (* Robert G. Wilson v, Jun 09 2004 *)
nxt[{a_,b_}]:={b,6b-a}; Select[NestList[nxt,{1,7},30][[;;,1]],CompositeQ] (* Harvey P. Dale, Mar 11 2023 *)

More terms from Robert G. Wilson v, Jun 09 2004

A164986 Numbers of the form 2p^2 = q^2 + 1, where p and q are primes.

Original entry on oeis.org

50, 1682, 3971273138702695316402, 367680737852094722224630791187352516632102802

Offset: 1

Rick L. Shepherd, Sep 03 2009

nonn

A079704 INTERSECT A002522. Subsequence of A088920 (Solutions k to the Diophantine equation k = 2n^2 = m^2+1): those terms for which associated m in A002315 and n in A001653 are both prime.
Corresponding p are prime Pell numbers (prime denominators of continued fraction convergents to sqrt(2)).
Corresponding q are prime numerators of the continued fraction convergents to sqrt(2).
Corresponding p, q, p^2, q^2, (p,q), (q,p), etc., form subsequences of many other OEIS sequences; see cross-references.
Any further terms are too large to include here.

			a(1) = 50 as 50 = 2*5^2 = 7^2 + 1, where 5 and 7 are prime.

Cf. A088920, A118612, A086397, A086395, A002315 (NSW numbers), A088165 (prime NSW numbers = prime RMS numbers (A140480)), A001653, A000129 (Pell numbers), A086383, A101411, A079704, A002522, A008843, A104683, A163742, etc.

a(n) = 2*(A118612(n+1))^2 = (A086397(n+1))^2 + 1.

A292082 Primes p such that (p^2 - 1) / 2 is a square (A000290).

Original entry on oeis.org

3, 17, 577, 665857

Offset: 1

Jaroslav Krizek, Sep 12 2017

Corresponding values of squares: 4, 144, 166464, 221682772224.
Subsequence of A257553.
Conjecture: sequence is finite.
Numbers k such that (k^2 - 1) / 2 is a square are given by A001541, of which the only prime terms are 3, 17, 577, and 665857 (see Alexander Adamchuk's Nov 24 2006 Comments entry there), so a(4) = 665857 is the last term of this sequence. - Jon E. Schoenfield, Nov 20 2017

			Number 3 is in the sequence because (3^2 - 1) / 2 = 4 (square).

Cf. A088165 (primes p such that (p^2 + 1) / 2 is a square).

Cf. A000290, A002315, A257553.

[n: n in [3..1000000] | IsPrime(n) and IsSquare((n^2-1) / 2)];

Select[Prime[Range[55000]],IntegerQ[Sqrt[(#^2-1)/2]]&] (* Harvey P. Dale, Mar 10 2019 *)

cp's OEIS Frontend

A240278 Primes p which are floor of Root-Mean-Square (RMS) of prime(n), prime(n+1) and prime(n+2).

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Maple

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A094666 Composite NSW numbers.

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Extensions

A164986 Numbers of the form 2p^2 = q^2 + 1, where p and q are primes.

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Formula

A292082 Primes p such that (p^2 - 1) / 2 is a square (A000290).

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Comments

Examples

Crossrefs

Programs

Magma

Mathematica