A306196
Irregular triangle read by rows where row n lists the primes 2n - k, with 1 < k < 2n-1, and if k is composite also 2n - p has to be prime for some prime divisor p of k.
Original entry on oeis.org
2, 3, 2, 3, 5, 3, 5, 7, 2, 5, 7, 2, 3, 5, 7, 11, 3, 5, 7, 11, 13, 3, 5, 7, 11, 13, 2, 3, 5, 7, 11, 13, 17, 2, 3, 5, 7, 11, 13, 17, 19, 2, 3, 5, 7, 11, 13, 17, 19, 2, 3, 5, 7, 11, 13, 17, 19, 23, 3, 5, 11, 13, 17, 23, 2, 7, 11, 13, 17, 19, 23, 2, 3, 5, 11, 13, 17, 19, 23, 29
Offset: 2
Row 2 = [2] because 2*2 = 2 + 2;
Row 3 = [3] because 2*3 = 3 + 3;
Row 4 = [2,3,5] because 2*4 - 2 = 6 = 2*3 and 2*4 = 3 + 5;
Row 5 = [3,5,7] because 2*5 = 3 + 7 = 5 + 5.
The table starts:
2;
3;
2, 3, 5;
3, 5, 7;
2, 5, 7;
2, 3, 5, 7, 11;
3, 5, 7, 11, 13;
3, 5, 7, 11, 13;
2, 3, 5, 7, 11, 13, 17;
2, 3, 5, 7, 11, 13, 17, 19;
2, 3, 5, 7, 11, 13, 17, 19;
2, 3, 5, 7, 11, 13, 17, 19, 23;
3, 5, 11, 13, 17, 23;
2, 7, 11, 13, 17, 19, 23;
2, 3, 5, 11, 13, 17, 19, 23, 29;
-
isok(k,n) = {if (isprime(2*n-k), pf = factor(k)[,1]; for (j=1, #pf, if (isprime(2*n-pf[j]), return (1));););}
row(n) = {my(v = []); for (k=1, 2*n, if (isok(k,n), v = concat(v, 2*n-k))); vecsort(v);} \\ Michel Marcus, Mar 02 2019
A377842
a(n) = q - 2*p, where q is the greatest prime such that p=2*n - q is also prime.
Original entry on oeis.org
-2, -3, -1, 1, -3, 5, 7, 3, 11, 13, 9, 17, 13, 9, 23, 25, 21, 17, 31, 27, 35, 37, 33, 41, 37, 33, 47, 43, 39, 53, 55, 51, 47, 61, 57, 65, 67, 63, 59, 73, 69, 77, 73, 69, 83, 79, 75, 41, 91, 87, 95, 97, 93, 101, 103, 99, 107, 103, 99, 83, 91, 87, 71, 121, 117, 125, 121, 117, 131, 133
Offset: 2
-
Table[Module[{p = 2, q},
While[True, q = 2 n - p; If[PrimeQ[p] && PrimeQ[q], Break[]];
p = NextPrime[p]]; q - 2 p], {n, 2, 100}]
-
a(n) = my(q=precprime(2*n)); while (!isprime(2*n - q), q = precprime(q-1)); q - 2*(2*n-q); \\ Michel Marcus, Dec 12 2024
A380545
Cumulative sum of the smallest prime in the minimal Goldbach partition for 2*n, n>=2.
Original entry on oeis.org
2, 5, 8, 11, 16, 19, 22, 27, 30, 33, 38, 41, 46, 53, 56, 59, 64, 71, 74, 79, 82, 85, 90, 93, 98, 105, 108, 113, 120, 123, 126, 131, 138, 141, 146, 149, 152, 157, 164, 167, 172, 175, 180, 187, 190, 195, 202, 221, 224, 229, 232, 235, 240, 243, 246, 251, 254, 259
Offset: 2
For n = 2, 4 = 2 + 2, the smallest prime p_1 = 2, so a(2) = A020481(2) = 2 = 2.
For n = 3, 6 = 3 + 3, the smallest prime p_2 = 3, so a(3) = a(2) + A020481(3) = 2 + 3 = 5.
For n = 4, 8 = 3 + 5, the smallest prime p_3 = 3, so a(4) = a(3) + A020481(4) = 5 + 3 = 8.
-
GoldbachMinPrimeCumSum[N_] := If[N < 4, {}, Accumulate[Table[Select[Prime[Range[PrimePi[n]]], PrimeQ[n - #] &, 1][[1]], {n, 4, N, 2}]]]
Comments