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

Author

Artur Jasinski, Sep 10 2022

Keywords

• nonn
• cons
• hard
• more

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

Examples

			0.031321620646...
		

Crossrefs

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

Programs

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