A357059 Decimal expansion of sum of squares of reciprocals of primes whose distance to the next prime is equal to 4, Sum_{j>=1} 1/A029710(j)^2.

0, 3, 1, 3, 2, 1, 6, 2, 0, 6, 4, 6

Offset

0,2

Comments

Convergence table:

k A029710(k) Sum_{j=1..k} 1/A029710(j)^2

10000000 3285441223 0.031321620645456519799598611681

20000000 7067090263 0.031321620645890982910821292996

30000000 11044597393 0.031321620646019474620358985896

40000000 15153534937 0.031321620646079307404248696076

50000000 19360462153 0.031321620646113421819579063642

60000000 23647877233 0.031321620646135276227114122713

70000000 28000392817 0.031321620646150384406674037099

Links

Table of n, a(n) for n=0..11.

Example

0.031321620646...

Mathematica

aa = {}; Do[g1[2 n] = 0, {n, 1, 1000}]; Do[g2[2 n] = 0, {n, 1, 1000}]; Do[g3[2 n] = 0, {n, 1, 1000}]; Do[g4[2 n] = 0, {n, 1, 1000}]; Do[g[2 n] = 0, {n, 1, 1000}];
w1 = 3; n = 3; Monitor[While[n < 10^10, w2 = NextPrime[w1]; kk = w2 - w1;
  If[kk < 2000, g[kk] = g[kk] + 1; g1[kk] = g1[kk] + N[1/w1, 1000]; g2[kk] = g2[kk] + N[1/w1^2, 1000]; g3[kk] = g3[kk] + N[1/w1^3, 1000]; g4[kk] = g4[kk] + N[1/w1^4, 1000];
If[IntegerQ[g[kk]/1000000], Print[{n, w1, kk, g[kk]}]; If[kk == 4, AppendTo[aa, {n, w1, kk, g[kk], g1[kk], g2[kk], g3[kk], g4[kk]}]]]]; w1 = w2; n++], n]; aa

Crossrefs

Cf. A006512, A023200, A029710, A046132, A065421, A077800, A078437, A085548, A096247, A160910, A194098, A209328, A209329, A242301, A242302, A242303, A242304, A306539, A342714, A356793.

Sequence in context: A271617 A057741 A133571 * A326420 A171899 A355784

Adjacent sequences: A357056 A357057 A357058 * A357060 A357061 A357062

Keyword

nonn,cons,hard,more

Author

Artur Jasinski, Sep 10 2022

Status

approved