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COMMENTS
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Prime q is sum of even semiprimes 2*p (A100484) and prime(i+2) (i = 1, 2, ...) (A000040).
Equivalent conditions: q = 2*p + prime(i+2) <=> p = (q - prime(i+2))/2 <=> (q - prime(i+2))/p = 2.
Algorithm: Q = prime(i+2) + 2*N, search first N with digsum(N) = digsum(Q);
(a) if Q AND N prime: p = N;
(b) if N composite: for n(k) = N + 9*k, q(k) = prime(i+2) + 2*(N + 9*k)
(i) digsum(n(k)) = digsum(q(k)),
(ii) n(k) has to be prime, let k = 1, 2, ... (Dirichlet's theorem on arithmetic progressions),
(iii) q(k) has to be prime (k = 1, 2, ... , again Dirichlet's theorem on arithmetic progressions);
Smallest k with (i), (ii) and (iii) gives term of sequence p = p(i).
Each prime > 3 is so "associated" uniquely with a pair (p,q) of primes.
List of triples (prime(i+2), p, q) is given in link for 0 <= i <= 100.
Pairs of primes (p, q) classified according to their digsum:
2: (2,11) (11,101) (2,101)
4: (13,31) (31,103) (31,211)
5: (5,23) (5,41) (23,113) (5,113) (41,311)
7: (151,313) (7,43) (7,61) (61,223) (7,151) (61,313) (7,241) (7,331)
8: (17,53) (17,71) (71,251) (53,233) (17,233) (107,521)
10: (73,163) (37,127) (19,109) (17,107) (19,127) (37,181) (127,433) (37,271) (19,271) (181,613) (19,307) (37,433)
11: (47,137) (83,227) (83,317) (29,281) (461,1163) (83,443) (353,1019) (443,1217)
13: (67,157) (67,193) (229,571) (139,409) (283,733) (67,373) (157,571) (139,571) (409,1129) (157,661)
14: (59,257) (167,491) (149,419) (149,509) (167,617) (491,1319)
16: (97,277) (97,367) (97,457) (79,439) (97,547)
17: (197,557) (89,359) (89,449)
Note: prime(i+2) is used in the sequence for 2 reasons: prime(i) could not be used for i=1 because q would be 2*p + prime(1) = 2*p + 2 that is not prime; prime(i+1) could not be used for i=1, because q would be 2*p + prime(2) = 2*p + 3, but numbers q with digsum(q) = digsum(p) are necessarily of form q = 9*k + 6 that is not prime.
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