A153409 -id:A153409

Search: a153409 -id:a153409

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A153410

Middle of 3 consecutive prime numbers, p1, p2, p3, such that p1*p2*p3*d1*d2 = average of twin prime pairs; d1 (delta) = p2 - p1, d2 (delta) = p3 - p2.

+10
4

3, 5, 23, 67, 233, 503, 683, 1013, 1759, 2099, 2797, 3169, 10663, 12391, 12899, 13487, 15149, 18583, 20563, 21881, 25373, 26237, 26681, 33613, 36787, 36943, 41411, 41443, 43573, 61547, 63337, 63841, 68909, 71999, 75721, 76367, 76481, 86677

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

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

EXAMPLE

2*3*5*1*2 = 60 and 60 +- 1 are primes.

3*5*7*2*2 = 420 and 420 +- 1 are primes.

19*23*29*4*6 = 304152 and 304152 +- 1 are primes.

MATHEMATICA

lst={}; Do[p1=Prime[n]; p2=Prime[n+1]; p3=Prime[n+2]; d1=p2-p1; d2=p3-p2; a=p1*p2*p3*d1*d2; If[PrimeQ[a-1]&&PrimeQ[a+1], AppendTo[lst, p2]], {n, 8!}]; lst

cpnQ[{a_, b_, c_}]:=Module[{x=Times@@Join[{a, b, c}, Differences[ {a, b, c}]]}, AllTrue[ x+{1, -1}, PrimeQ]]; Select[Partition[ Prime[Range[ 10000]], 3, 1], cpnQ][[All, 2]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 01 2020 *)

CROSSREFS

Cf. A099349, A153374, A153375, A153376, A153377, A153378, A153379, A153406, A153407, A153408, A153409.

KEYWORD

nonn

AUTHOR

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

STATUS

approved

A153411

Larger of 3 consecutive prime numbers such that p1*p2*p3*d1*d2=average of twin prime pairs; p1,p2,p3 consecutive prime numbers; d1(delta)=p2-p1, d2(delta)=p3-p2.

+10
1

5, 7, 29, 71, 239, 509, 691, 1019, 1777, 2111, 2801, 3181, 10667, 12401, 12907, 13499, 15161, 18587, 20593, 21893, 25391, 26249, 26683, 33617, 36791, 36947, 41413, 41453, 43577, 61553, 63347, 63853, 68917, 72019, 75731, 76369, 76487, 86689

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

2*3*5*1*2=60+-1=primes, 3*5*7*2*2=420+-1=primes, 19*23*29*4*6=304152+-1=primes,...

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

MATHEMATICA

lst={}; Do[p1=Prime[n]; p2=Prime[n+1]; p3=Prime[n+2]; d1=p2-p1; d2=p3-p2; a=p1*p2*p3*d1*d2; If[PrimeQ[a-1]&&PrimeQ[a+1], AppendTo[lst, p3]], {n, 8!}]; lst

tppQ[n_]:=Module[{c=Times@@Join[n, Differences[n]]}, AllTrue[c+{1, -1}, PrimeQ]]; Transpose[Select[Partition[Prime[Range[10^4]], 3, 1], tppQ]] [[3]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 17 2016 *)

CROSSREFS

Cf. A099349, A153374, A153375, A153376, A153377, A153378, A153379, A153406, A153407, A153408, A153409, A153410

KEYWORD

nonn

AUTHOR

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

STATUS

approved

A153413

Smaller of twin prime pairs such that p1*p2+average_of_twin_prime_pair=prime.

+10
1

3, 5, 29, 59, 137, 179, 239, 419, 617, 1049, 1607, 1697, 1787, 2267, 2309, 2729, 3257, 3389, 3527, 3767, 4157, 4217, 4337, 4799, 5639, 5867, 6659, 6689, 6869, 6959, 7487, 7547, 7589, 8537, 8627, 8969, 9629, 9857, 9929, 10457, 11117, 11969, 12539, 13337

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

3*5+4=19 prime, 5*7+6=41 prime, 29*31+30=929 prime, ...

LINKS

Table of n, a(n) for n=1..44.

MATHEMATICA

lst={}; Do[p1=Prime[n]; p2=Prime[n+1]; If[p2-p1==2, a=p1*p2+(p1+1); If[PrimeQ[a], AppendTo[lst, p1]]], {n, 7!}]; lst

CROSSREFS

Cf. A099349, A153374, A153375, A153376, A153377, A153378, A153379, A153406, A153407, A153408, A153409, A153410

KEYWORD

nonn

AUTHOR

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

STATUS

approved

A153414

Larger of twin prime pairs such that p1*p2+average_of_twin_prime_pair=prime.

+10
1

5, 7, 31, 61, 139, 181, 241, 421, 619, 1051, 1609, 1699, 1789, 2269, 2311, 2731, 3259, 3391, 3529, 3769, 4159, 4219, 4339, 4801, 5641, 5869, 6661, 6691, 6871, 6961, 7489, 7549, 7591, 8539, 8629, 8971, 9631, 9859, 9931, 10459, 11119, 11971, 12541, 13339

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

3*5+4=19 prime, 5*7+6=41 prime, 29*31+30=929 prime, ...

LINKS

Harvey P. Dale, Table of n, a(n) for n = 1..1000

MATHEMATICA

lst={}; Do[p1=Prime[n]; p2=Prime[n+1]; If[p2-p1==2, a=p1*p2+(p1+1); If[PrimeQ[a], AppendTo[lst, p2]]], {n, 7!}]; lst

Transpose[Select[Select[Partition[Prime[Range[1600]], 2, 1], Last[#]- First[#] == 2&], PrimeQ[Times@@#+Mean[#]]&]][[2]] (* Harvey P. Dale, Jan 23 2012 *)

CROSSREFS

Cf. A099349, A153374, A153375, A153376, A153377, A153378, A153379, A153406, A153407, A153408, A153409, A153410, A153413

KEYWORD

nonn

AUTHOR

Vladimir Joseph Stephan Orlovsky, Dec 25 2008

STATUS

approved

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