A154331 -id:A154331 - cp's OEIS frontend

A154671 Averages of twin prime pairs k such that k*3 and k/3 are squares.

12, 108, 192, 432, 1452, 2028, 3468, 4800, 10092, 18252, 106032, 139968, 221952, 284592, 299568, 355008, 549552, 618348, 720300, 786432, 823728, 961068, 995328, 1009200, 1138368, 1190700, 1291008, 1529388, 1537968, 1651692, 1948908

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 13 2009

nonn

			12*3 = 36 = 6^2, 12/3 = 4 = 2^2.

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

Cf. A154670.

lst={}; Do[If[PrimeQ[n-1]&&PrimeQ[n+1],s=(n*3)^(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/3)^(1/2); If[Floor[s]==s,AppendTo[lst,n]]],{n,6,10!,6}]; lst
Select[Mean/@Select[Partition[Prime[Range[150000]],2,1],#[[2]]-#[[1]] == 2&],AllTrue[{Sqrt[#/3],Sqrt[3#]},IntegerQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 06 2015 *)

for(i=1,999, isprime(12*i^2+1) && isprime(12*i^2-1) && print1(12*i^2",")) \\ M. F. Hasler, Jan 15 2009

a(n) = 12*A154331(n)^2. - 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)

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).

A154774 Numbers n such that 9900n^2 is the average of a twin prime pair.

Original entry on oeis.org

10, 14, 15, 25, 60, 74, 76, 87, 127, 129, 130, 140, 171, 196, 207, 224, 259, 263, 302, 314, 315, 319, 333, 337, 377, 389, 451, 454, 470, 491, 508, 518, 549, 568, 574, 589, 592, 624, 629, 636, 690, 696, 729, 748, 753, 769, 770, 771, 781, 791, 802, 833, 844

Offset: 1

M. F. Hasler, Jan 15 2009

nonn

Inspired by Zak Seidov's post to the SeqFan list, cf. link: This yields A154674 as 9900 a(n)^2. Indeed, if N/11 is a square, then N=11 m^2 and this can't be the average of a twin prime pair unless m=30a (considering N+1 mod 2,3,5 and N-1 mod 5).

Zak Seidov, "A154676", Jan 15 2009

Cf. A037073, A154331, A154772-A154773.

Select[Range[1000],AllTrue[9900#^2+{1,-1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 19 2019 *)

for(i=1,999, isprime(9900*i^2+1) & isprime(9900*i^2-1) & print1(i","))

a(n) = sqrt(A154674(n)/9900).

A154775 Numbers k such that 13(6k)^2 is the average of a twin prime pair.

Original entry on oeis.org

2, 4, 5, 42, 46, 49, 59, 82, 84, 100, 119, 128, 137, 182, 185, 187, 192, 233, 264, 301, 303, 340, 376, 390, 395, 422, 438, 446, 471, 472, 494, 518, 527, 570, 598, 609, 611, 633, 667, 688, 714, 716, 726, 728, 733, 744, 831, 837, 865, 875, 896, 926, 940, 948

Offset: 1

M. F. Hasler, Jan 15 2009

nonn

Inspired by Zak Seidov's post to the SeqFan list, cf. link: This yields A154675 as 468 a(n)^2. Indeed, if N/13 is a square, then N=13 k^2 and this can't be the average of a twin prime pair unless k=6m.

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

Cf. A014574, A037073, A154331, A154772, A154773, A154774, A154675.

okQ[n_]:=Module[{av=468n^2},PrimeQ[av-1]&&PrimeQ[av+1]]; Select[Range[1000],okQ] (* Harvey P. Dale, Jan 21 2011 *)

for(i=1,999, isprime(468*i^2+1) & isprime(468*i^2-1) & print1(i","))

a(n) = sqrt(A154675(n)/468).

cp's OEIS Frontend

A154671 Averages of twin prime pairs k such that k*3 and k/3 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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A154773 Numbers n such that 252n^2 is the average of a twin prime pair.

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A154774 Numbers n such that 9900n^2 is the average of a twin prime pair.

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A154775 Numbers k such that 13*(6*k)^2 is the average of a twin prime pair.

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A154775 Numbers k such that 13(6k)^2 is the average of a twin prime pair.