A154671 -id:A154671 - cp's OEIS frontend

A154331 Numbers m such that 12 m^2 is the average of a twin prime pair.

1, 3, 4, 6, 11, 13, 17, 20, 29, 39, 94, 108, 136, 154, 158, 172, 214, 227, 245, 256, 262, 283, 288, 290, 308, 315, 328, 357, 358, 371, 403, 413, 414, 420, 475, 491, 510, 522, 536, 543, 546, 556, 559, 561, 581, 585, 622, 630, 633, 647, 666, 669, 676, 699, 735

Offset: 1

M. F. Hasler, Jan 15 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

Cf. A037073, A154671.

[m:m in [1..740]| IsPrime(12*m^2-1) and IsPrime(12*m^2+1)]; // Marius A. Burtea, Jan 23 2020

Select[Range[800],AllTrue[12#^2+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 27 2014 *)

for(i=1,999, isprime(12*i^2+1) & isprime(12*i^2-1) & print1(i","))

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

Original entry on oeis.org

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

A154673 Averages of twin prime pairs k such that k*7 and k/7 are squares.

Original entry on oeis.org

2268, 6300, 30492, 49392, 81648, 100800, 170352, 183708, 197568, 211932, 242172, 363888, 444528, 681408, 847728, 1032192, 1342908, 1694448, 1820700, 2041200, 3049200

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 13 2009

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..100 from Harvey P. Dale)

Cf. A154670, A154671, A154672.

lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n*7)^(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/7)^(1/2);If[Floor[s]==s,AppendTo[lst,n]]],{n,6,10!,6}];lst
Select[Mean/@Select[Partition[Prime[Range[250000]],2,1],#[[2]]-#[[1]] == 2&],AllTrue[{Sqrt[7#],Sqrt[#/7]},IntegerQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 22 2018 *)
Select[7*Range[10^3]^2, And @@ PrimeQ[# + {-1, 1}] &] (* Amiram Eldar, Dec 25 2019 *)

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

cp's OEIS Frontend

A154331 Numbers m such that 12 m^2 is the average of a twin prime pair.

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

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A154673 Averages of twin prime pairs k such that k*7 and k/7 are squares.

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Mathematica

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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Maple

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

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Programs

Mathematica

PARI

Formula

Extensions

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

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Programs

Mathematica

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Formula

Extensions

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).