A154672 -id:A154672 - cp's OEIS frontend

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

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

A154772 Numbers m such that 180 m^2 is the average of a twin prime pair.

Original entry on oeis.org

1, 3, 7, 14, 22, 29, 41, 46, 62, 64, 67, 88, 167, 179, 207, 231, 239, 249, 263, 266, 286, 290, 309, 315, 322, 323, 326, 344, 350, 353, 354, 372, 392, 421, 444, 454, 458, 496, 505, 553, 560, 561, 571, 585, 613, 636, 647, 661, 669, 682, 745, 788, 790, 791, 815

Offset: 1

M. F. Hasler, Jan 15 2009

nonn

Inspired by Z. Seidov's post to the SeqFan list, cf. link. This yields A154672 as 180 a(n)^2. Indeed, if N is such that N/5 is a square, then M=5m^2 and this can't by the average of a twin prime pair unless m=6a.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Zak Seidov, "A154676", Jan 15 2009

Cf. A014574, A037073, A154331, A154672.

Select[Range[10^3], And @@ PrimeQ[180#^2 + {-1, 1}] &] (* Amiram Eldar, Dec 25 2019 *)

for(i=1,999, isprime(180*i^2+1) & isprime(180*i^2-1) & print1(i","))

a(n) = sqrt(A154672(n)/180)

cp's OEIS Frontend

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

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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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A154772 Numbers m such that 180 m^2 is the average of a twin prime pair.

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