A160917
Averages of twin prime pairs which can be represented as a sum of three consecutive of such pair averages.
Original entry on oeis.org
60, 282, 348, 522, 570, 618, 1788, 2112, 4050, 4422, 5880, 6198, 8232, 9678, 10458, 11700, 12072, 12162, 12378, 14010, 16140, 17598, 17838, 21648, 22698, 33348, 36342, 39228, 41610, 43782, 44088, 46272, 48780, 51198, 53088, 56910, 58230
Offset: 1
a(1) = 60 = A014574(7) = 12 + 18 + 30 = A014574(3) + A014574(4) + A014574(5).
a(2) = 282 = A014574(19) = 72 + 102 + 108 = A014574(8) + A014574(9) + A014574(10).
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PrimeNextTwinAverage[n_]:=Module[{k},k=n+1;While[ !PrimeQ[k-1]||!PrimeQ[k+1], k++ ];k];lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],a=n;b=PrimeNextTwinAverage[a]; c=PrimeNextTwinAverage[b];a=a+b+c;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst, a]]],{n,8!}];lst
Module[{m=Mean/@Select[Partition[Prime[Range[10000]],2,1],#[[2]]-#[[1]] == 2&],t},t=Total/@Partition[m,3,1];Intersection[m,t]] (* Harvey P. Dale, Mar 06 2018 *)
A160918
Averages of twin prime pairs that are sums of 4 consecutive averages of twin prime pairs.
Original entry on oeis.org
102, 150, 420, 660, 858, 1020, 2310, 2730, 3120, 3390, 5100, 5502, 5850, 6198, 7758, 12540, 13692, 13998, 15360, 17292, 21840, 23688, 25932, 32832, 41520, 43398, 46092, 49032, 49410, 50892, 58152, 61560, 64920, 70878, 72270, 78138, 88818
Offset: 1
102 is in the sequence because it can be written as 12 + 18 + 30 + 42.
150 is in the sequence because it is 18 + 30 + 42 + 60.
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PrimeNextTwinAverage[n_]:=Module[{k},k=n+1;While[ !PrimeQ[k-1]||!PrimeQ[k+1],k++ ];k];lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],a=n;b=PrimeNextTwinAverage[a];c=PrimeNextTwinAverage[b];d=PrimeNextTwinAverage[c];a=a+b+c+d;If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,a]]],{n,2*8!}];lst
With[{tpms=Mean/@Select[Partition[Prime[Range[10000]],2,1],#[[2]]- #[[1]] ==2&]},Total/@Select[Partition[tpms,4,1],MemberQ[tpms,Total[#]]&]] (* Harvey P. Dale, Apr 27 2012 *)
A160919
Averages of twin prime pairs that are sums of 5 consecutive averages of twin prime pairs.
Original entry on oeis.org
108, 570, 858, 1452, 3330, 6792, 7458, 9420, 9630, 10710, 10890, 13722, 17388, 18120, 25032, 27582, 27792, 34032, 68712, 68898, 72270, 76830, 78978, 81372, 89820, 90402, 95232, 99708, 104472, 119772, 122868, 125790, 138078, 165312
Offset: 1
Averages of twin prime pairs: 4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, ...
108 = 6 + 12 + 18 + 30 + 42, 570 = 72 + 102 + 108 + 138 + 150, ...
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PrimeNextTwinAverage[n_]:=Module[{k},k=n+1; While[ !PrimeQ[k-1]||!PrimeQ[k+1],k++ ];k];lst={};Do[If[PrimeQ[n-1]&&PrimeQ[n+1],a=n;b=PrimeNextTwinAverage[a]; c=PrimeNextTwinAverage[b]; d=PrimeNextTwinAverage[c];e=PrimeNextTwinAverage[d]; a=a+b+c+d+e; If[PrimeQ[a-1]&&PrimeQ[a+1],AppendTo[lst,a]]],{n,3*8!}];lst
Select[Total/@(Partition[Mean/@Select[Partition[Prime[Range[10000]],2,1],#[[2]]-#[[1]]==2&],5,1]),AllTrue[#+{1,-1},PrimeQ]&] (* Harvey P. Dale, Sep 26 2024 *)
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