A135595 Number of positive solutions of the Diophantine x*p+y*q=(p+q)^3, where p=n-th prime, q=(n+1)-th prime.

21, 34, 49, 76, 97, 122, 145, 169, 211, 241, 274, 313, 337, 361, 401, 449, 481, 513, 553, 577, 609, 649, 689, 745, 793, 817, 841, 865, 889, 963, 1033, 1073, 1105, 1153, 1201, 1233, 1281, 1321, 1361, 1409, 1441, 1489, 1537, 1561, 1585, 1641, 1737, 1801, 1825

Offset

1,1

Links

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

Example

a(1)=21 because Diophantine 2x+3y=(2+3)^3 has 21 positive solutions {x, y}:
{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};
a(2)=34 because Diophantine 3x+5y=(3+5)^3 has 34 positive solutions {x, y}:
{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}.

Mathematica

Table[With[{p=Prime[n], q=Prime[n+1]}, Floor[(p+q)^2/p]+Floor[(p+q)^2/q]+1], {n, 1, 100}]
Floor[Total[#]^2/#[[1]]]+Floor[Total[#]^2/#[[2]]]+1&/@Partition[ Prime[ Range[ 50]], 2, 1] (* Harvey P. Dale, May 27 2018 *)

Crossrefs

Sequence in context: A146910 A328862 A167397 * A082483 A075735 A053409

Adjacent sequences: A135592 A135593 A135594 * A135596 A135597 A135598

Keyword

nonn

Author

Zak Seidov, Feb 25 2008

Status

approved