editing

approved

0, 0, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 3, 1, 1, 2, 3, 0, 1, 2, 0, 3, 1, 0, 2, 2, 0, 0, 1, 1, 2, 3, 0, 1, 3, 0, 2, 0, 0, 2, 1, 0, 1, 2, 0, 3, 0, 0, 4, 2, 1, 1, 1, 1, 3, 4, 1, 1, 3, 1, 0, 2, 1, 1, 3, 0, 0, 2, 3, 3, 3, 1, 1, 3, 3, 2, 3, 1, 1, 5, 0, 1, 4, 2, 1, 1, 0, 2, 6, 1, 1, 2, 0, 1, 3, 0, 1, 3, 3

a(8) = 1 since 2*8 = 5 + 11 with 5, 11, prime(5) - 5 + 1 = 7 and prime(11) + 11 + 1 = 43 all prime.

p[n_] := PrimeQ[n] && PrimeQ[Prime[n] - n + 1];

q[n_] := PrimeQ[n] && PrimeQ[Prime[n] + n + 1];

a[n_] := Sum[If[p[k] && q[2n2 n - k], 1, 0], {k, 1, 2n 2 n - 1}];

Table[a[n], {n, 1, 100}]

proposed

editing

editing

proposed

Parts (i) and (ii) are stronger than Goldbach's conjecture (A045917) and Lemoine's conjecture (A046927) respectively.

a(8) = 1 since 2*8 = 5 + 11 with 5, 11, prime(5) - 5 + 1 = 7 and prime(11) + 11 + 1 = 43 all prime.

Cf. A000040, A045917, A046927, A234695, A235187, A235189.

Conjecture: (i) a(n) > 0 for all n >= 2480.

This is stronger than Goldbach's conjecture(ii) If n > 4368 then 2*n+1 can be written as 2*p + q with p and q terms of the sequence A234695.

Parts (i) and (ii) are stronger than Goldbach's conjecture and Lemoine's conjecture respectively.

p[n_]:=PrimeQ[n]&&PrimeQ[Prime[n]-n+1]

Cf. A000040, A234695, A235187, A235189.

a(n) = sigma(n^3), where sigma(k) is the sum of all positive divisors of k.

Number of ways to write 2*n = p + q with p, q, prime(p) - p + 1 and prime(q) + q + 1 all prime.

0, 0, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 3, 1, 1, 2, 3, 0, 1, 2, 0, 3, 1, 0, 2, 2, 0, 0, 1, 1, 2, 3, 0, 1, 3, 0, 2, 0, 0, 2, 1, 0, 1, 2, 0, 3, 0, 0, 4, 2, 1, 1, 1, 1, 3, 4, 1, 1, 3, 1, 0, 2, 1, 1, 3, 0, 0, 2, 3, 3, 3, 1, 1, 3, 3, 2, 3, 1, 1, 5, 0, 1, 4, 2, 1, 1, 0, 2, 6, 1, 1, 2, 0, 1, 3, 0, 1, 15, 40, 127, 156, 600, 400, 1023, 1093, 2340, 1464, 5080, 2380, 6000, 6240, 8191, 5220, 16395, 7240, 19812, 16000, 21960, 12720, 40920, 19531, 35700, 29524, 50800, 25260, 93600, 30784, 65535, 58560, 78300, 62400, 138811, 52060, 108600, 95200, 1595883, 3

1,29

Conjecture: All the terms are pairwise distincta(n) > 0 for all n >= 2480.

This is stronger than Goldbach's conjecture.

Zhi-Wei Sun, <a href="/A235330/b235330.txt">Table of n, a(n) for n = 1..10000</a>

sigma p[n_]:=DivisorSigmaPrimeQ[n]&&PrimeQ[Prime[1, n]-n+1]

Tableq[n_]:=PrimeQ[n]&&PrimeQ[sigmaPrime[n^3], {+n, +1, 40}]

a[n_]:=Sum[If[p[k]&&q[2n-k], 1, 0], {k, 1, 2n-1}]

Table[a[n], {n, 1, 100}]

Cf. A000203A000040.

allocated for Zhi-Wei Sun

a(n) = sigma(n^3), where sigma(k) is the sum of all positive divisors of k.

1, 15, 40, 127, 156, 600, 400, 1023, 1093, 2340, 1464, 5080, 2380, 6000, 6240, 8191, 5220, 16395, 7240, 19812, 16000, 21960, 12720, 40920, 19531, 35700, 29524, 50800, 25260, 93600, 30784, 65535, 58560, 78300, 62400, 138811, 52060, 108600, 95200, 159588

1,2

Conjecture: All the terms are pairwise distinct.

sigma[n_]:=DivisorSigma[1, n]

Table[sigma[n^3], {n, 1, 40}]

Cf. A000203.

allocated

nonn

Zhi-Wei Sun, Jan 05 2014

approved

editing

allocated for Zhi-Wei Sun

allocated

approved