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%I #6 May 27 2018 09:54:51
%S 21,34,49,76,97,122,145,169,211,241,274,313,337,361,401,449,481,513,
%T 553,577,609,649,689,745,793,817,841,865,889,963,1033,1073,1105,1153,
%U 1201,1233,1281,1321,1361,1409,1441,1489,1537,1561,1585,1641,1737,1801,1825
%N Number of positive solutions of the Diophantine x*p+y*q=(p+q)^3, where p=n-th prime, q=(n+1)-th prime.
%H Harvey P. Dale, <a href="/A135595/b135595.txt">Table of n, a(n) for n = 1..1000</a>
%e a(1)=21 because Diophantine 2x+3y=(2+3)^3 has 21 positive solutions {x, y}:
%e {1, 41}, {4, 39}, {7, 37}, {10, 35}, {13, 33}, {16, 31}, {19, 29}, {22, 27}, {25, 25}, {28, 23}, {31, 21}, {34, 19}, {37, 17}, {40, 15}, {43, 13}, {46, 11}, {49, 9}, {52, 7}, {55, 5}, {58, 3}, {61, 1};
%e a(2)=34 because Diophantine 3x+5y=(3+5)^3 has 34 positive solutions {x, y}:
%e {4, 100}, {9, 97}, {14, 94}, {19, 91}, {24, 88}, {29, 85}, {34, 82}, {39, 79}, {44, 76}, {49, 73}, {54, 70}, {59, 67}, {64, 64}, {69, 61}, {74, 58}, {79, 55}, {84, 52}, {89, 49}, {94, 46}, {99, 43}, {104, 40}, {109, 37}, {114, 34}, {119, 31}, {124, 28}, {129, 25}, {134, 22}, {139, 19}, {144, 16}, {149, 13}, {154, 10}, {159, 7}, {164, 4}, {169, 1}.
%t Table[With[{p=Prime[n],q=Prime[n+1]},Floor[(p+q)^2/p]+Floor[(p+q)^2/q]+1],{n,1,100}]
%t Floor[Total[#]^2/#[[1]]]+Floor[Total[#]^2/#[[2]]]+1&/@Partition[ Prime[ Range[ 50]],2,1] (* _Harvey P. Dale_, May 27 2018 *)
%K nonn
%O 1,1
%A _Zak Seidov_, Feb 25 2008
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