A154673 -id:A154673 - cp's OEIS frontend

A154672 Numbers n = 5k^2 such that n - 1 and n + 1 are (twin) primes (thus k=6m).

180, 1620, 8820, 35280, 87120, 151380, 302580, 380880, 691920, 737280, 808020, 1393920, 5020020, 5767380, 7712820, 9604980, 10281780, 11160180, 12450420, 12736080, 14723280, 15138000, 17186580, 17860500, 18663120, 18779220, 19129680, 21300480

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 13 2009

nonn

Original definition: Averages of twin prime pairs n such that n*5 and n/5 are squares.

Obviously, n*5 is a square iff n/5 is a square, say k^2. But n=5k^2 can't be the average of a twin prime pair unless it's a multiple of 6, thus k=6m and n=5*36*m^2. - M. F. Hasler, Apr 11 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000290, A014574, A154670, A154671, A154673, A154674, A154675, A154676.

lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n*5)^(1/2);If[Floor[s]==s,AppendTo[lst,n]]],{n,6,10!,6}];lst (*...and/or...*) lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n/5)^(1/2);If[Floor[s]==s,AppendTo[lst,n]]],{n,6,10!,6}];lst

forstep(k=0,1e4,6, isprime(k^2*5+1) & isprime(k^2*5-1) & print1(k^2*5,",")) \\ M. F. Hasler, Apr 11 2009

A154672 = 5*A000290 intersect A014574 = 180*A000290 intersect A014574. - M. F. Hasler, Apr 11 2009

Edited and extended by M. F. Hasler, Apr 11 2009

A154676 Numbers n = 103k^2 such that (n-1,n+1) is a twin prime pair (thus k = 6m).

Original entry on oeis.org

2317500, 12047292, 26163648, 43250112, 47347452, 61704828, 168228252, 333720000, 351755712, 426127068, 513127872, 840143808, 979638768, 998790588, 1089276912, 1330434108, 1357220700, 1388809152, 1694467008, 1927570428, 1986835392, 2035992348, 2136108348, 2858437872, 3070594800, 3241626300, 3903322608

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 14 2009

Original definition: Averages of twin prime pairs n such that n*103 and n/103 are squares.
All terms are of the form 3708*k^2. - Zak Seidov, Jan 15 2009
Obviously n*103 is a square iff n/103 is a square, say k^2. But n=103k^2 can't be the average of a twin prime pair unless it's a multiple of 6, thus k=6m and n=103*36*m^2. - M. F. Hasler, Apr 11 2009

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A154670, A154671, A154672, A154673, A154674, A154675.

select(t -> isprime(t+1) and isprime(t-1), [seq(3708*i^2, i=1..2000)]); # Robert Israel, Mar 13 2019

lst={}; Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n*103)^(1/2); If[Floor[s]==s,AppendTo[lst,n]]],{n,9!,2*11!,6}]; lst (*...and/or...*) lst={}; Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n/103)^(1/2); If[Floor[s]==s,AppendTo[lst,n]]],{n,9!,2*11!,6}]; lst
Select[3708*Range[1200]^2,AllTrue[#+{1,-1},PrimeQ]&] (* Harvey P. Dale, May 15 2025 *)

forstep(k=0,1e4,6, isprime(k^2*103+1) & isprime(k^2*103-1) & print1(k^2*103,",")) \\ M. F. Hasler, Apr 11 2009

Edited and extended by M. F. Hasler, Apr 11 2009

A154674 Averages of twin prime pairs k such that k*11 and k/11 are squares.

Original entry on oeis.org

990000, 1940400, 2227500, 6187500, 35640000, 54212400, 57182400, 74933100, 159677100, 164745900, 167310000, 194040000, 289485900, 380318400, 424205100, 496742400, 664101900, 684773100, 902919600, 976100400, 982327500, 1007433900

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 13 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A154670, A154671, A154672, A154673, A154774.

lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n*11)^(1/2);If[Floor[s]==s,AppendTo[lst,n]]],{n,6,11!,6}];lst...and/or... lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n/11)^(1/2);If[Floor[s]==s,AppendTo[lst,n]]],{n,6,11!,6}];lst
Select[11*Range[10^4]^2, And @@ PrimeQ[# + {-1, 1}] &] (* Amiram Eldar, Dec 25 2019 *)

for(i=1,999, isprime(9900*i^2+1) && isprime(9900*i^2-1) && print1(9900*i^2",")) \\ M. F. Hasler, Jan 15 2009

a(n) = 9900*A154774(n)^2. - M. F. Hasler, Jan 15 2009

More terms from M. F. Hasler, Jan 15 2009

A154675 Averages of twin prime pairs k such that k*13 and k/13 are squares.

Original entry on oeis.org

1872, 7488, 11700, 825552, 990288, 1123668, 1629108, 3146832, 3302208, 4680000, 6627348, 7667712, 8783892, 15502032, 16017300, 16365492, 17252352, 25407252, 32617728, 42401268, 42966612, 54100800, 66163968, 71182800, 73019700

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 13 2009

nonn

Terms of this sequence must be of the form 13(6m)^2, the values for m are listed in A154775. - M. F. Hasler, Jan 15 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A154670, A154671, A154672, A154673, A154674, A154775.

lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n*13)^(1/2);If[Floor[s]==s,AppendTo[lst,n]]],{n,6,11!,6}];lst
(* ... and/or ... *)
lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n/13)^(1/2);If[Floor[s]==s,AppendTo[lst,n]]],{n,6,11!,6}];lst
Select[13*Range[10^3]^2, And @@ PrimeQ[# + {-1, 1}] &] (* Amiram Eldar, Dec 25 2019 *)

for(i=1,499, isprime(468*i^2+1) && isprime(468*i^2-1) && print1(468*i^2",")) \\ M. F. Hasler, Jan 15 2009

a(n) = 468 A154775(n)^2 - M. F. Hasler, Jan 15 2009

More terms from M. F. Hasler, Jan 15 2009

A154773 Numbers n such that 252n^2 is the average of a twin prime pair.

Original entry on oeis.org

3, 5, 11, 14, 18, 20, 26, 27, 28, 29, 31, 38, 42, 52, 58, 64, 73, 82, 85, 90, 110, 125, 138, 156, 167, 180, 212, 234, 248, 297, 299, 303, 305, 308, 312, 314, 317, 319, 334, 336, 348, 361, 365, 371, 372, 377, 379, 414, 451, 465, 478, 499, 508, 509, 535, 554, 564

Offset: 1

M. F. Hasler, Jan 15 2009

nonn

Inspired by Zak Seidov's post to the SeqFan list, cf. link: This yields A154673 as 252 a(n)^2. Indeed, if N/7 is a square, then N=7m^2 and this can't be the average of a twin prime pair unless m=6a.

Zak Seidov, "A154676", Jan 15 2009

Cf. A037073, A154331, A154772.

Select[Range[600],And@@PrimeQ[252#^2+{1,-1}]&] (* Harvey P. Dale, Dec 13 2012 *)

for(i=1,999, isprime(252*i^2+1) & isprime(252*i^2-1) & print1(i","))

a(n) = sqrt(A154673(n)/252).

cp's OEIS Frontend

A154672 Numbers n = 5*k^2 such that n - 1 and n + 1 are (twin) primes (thus k=6*m).

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A154676 Numbers n = 103*k^2 such that (n-1,n+1) is a twin prime pair (thus k = 6*m).

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A154674 Averages of twin prime pairs k such that k*11 and k/11 are squares.

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A154675 Averages of twin prime pairs k such that k*13 and k/13 are squares.

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A154773 Numbers n such that 252n^2 is the average of a twin prime pair.

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Mathematica

PARI

Formula

A154672 Numbers n = 5k^2 such that n - 1 and n + 1 are (twin) primes (thus k=6m).

A154676 Numbers n = 103k^2 such that (n-1,n+1) is a twin prime pair (thus k = 6m).