A356793 -id:A356793 - cp's OEIS frontend

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

Artur Jasinski, Sep 10 2022

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

			0.031321620646...

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

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

A357483 Decimal expansion of sum of squares of reciprocals of primes whose distance to the next prime is equal to 6, Sum_{j>=1} 1/A031924(j)^2.

Original entry on oeis.org

0, 0, 4, 7, 5, 7, 2, 8, 6, 9, 7, 5

Offset: 0

Artur Jasinski, Sep 30 2022

			0.004757286975...

Cf. A031924.

Cf. A085548, A160910, A242301, A356793, A357059.

A357528 Decimal expansion of Sum_{j>=1} 1/A031926(j)^2.

Original entry on oeis.org

0, 0, 0, 1, 8, 3, 9, 3, 0, 8, 5, 1, 7

Offset: 0

Artur Jasinski, Oct 02 2022

			0.000183930851...

Cf. A031926.

Cf. A085548, A160910, A242301, A356793, A357059, A357483.

A380276 Decimal expansion of infinite sum of reciprocals Sum_{n>=1} 1/A357934(n).

Original entry on oeis.org

4, 7, 7, 9, 9, 9, 3

Offset: 0

Artur Jasinski, Jan 18 2025

Sum of reciprocals of product of two distinct lesser twin primes.

			0.47799932... = 1/15 + 1/33 + 1/51 + 1/55 + 1/85 + 1/87 + ...

Cf. A001359, A342714, A356793, A357934.

Cf. A209328.

Equals (A342714^2-A356793)/2.

Equals Sum_{m=2} 1/(A001359(m)*A001359(n)).

cp's OEIS Frontend

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.

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A357483 Decimal expansion of sum of squares of reciprocals of primes whose distance to the next prime is equal to 6, Sum_{j>=1} 1/A031924(j)^2.

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A357528 Decimal expansion of Sum_{j>=1} 1/A031926(j)^2.

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A380276 Decimal expansion of infinite sum of reciprocals Sum_{n>=1} 1/A357934(n).

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