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A178548 a(n) is the smallest prime p such that q = 2 * p + prime(n+2) is prime AND A007953(p) = A007953(q) where A007953 is sum of digits. 2
13, 2, 151, 5, 73, 17, 67, 7, 5, 17, 31, 47, 7, 37, 67, 83, 23, 19, 17, 11, 97, 19, 2, 61, 5, 37, 71, 229, 53, 139, 7, 59, 31, 83, 167, 197, 283, 97, 127, 89, 61, 149, 37, 17, 149, 29, 7, 41, 19, 67, 461, 181, 157, 97, 19, 89, 83, 79, 167, 139, 107, 409, 353, 7, 443, 491 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
EXAMPLE
i=1: 2*13 + prime(3) = 26+5 = 31 = prime(11), digsum(13) = digsum(31) = 4, 13 is first term.
i=2: 2*2 + prime(4) = 4+7 = 11 = prime(5), digsum(2) = digsum(11) = 2, 2 is 2nd term.
i=3: 2*151 + prime(5) = 302+11 = 313 = prime(65), digsum(151) = digsum(313) = 7, 151 is 3rd term.
MATHEMATICA
f[n_]:= Block[{p = 2}, While[q = 2*p +Prime[n+2]; !PrimeQ[q] ||
(Total[IntegerDigits[p]] != Total[IntegerDigits[q]]), p = NextPrime[p]];
Array[f, 100] (* G. C. Greubel, Feb 08 2019 *)
PROG
(PARI) a(n) = {my(p=2, q=prime(n+2)+2*p); while (!(isprime(q) && (sumdigits(p) == sumdigits(q))), p = nextprime(p+1); q = prime(n+2) + 2*p); p; } \\ Michel Marcus, Feb 09 2019
CROSSREFS
Cf. A000040, A007953 (digsum), A100484, A178561 (primes q).
Sequence in context: A154355 A078421 A185808 * A098222 A114457 A010220
KEYWORD
base,nonn,less
AUTHOR
Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 29 2010
EXTENSIONS
Edited by Michel Marcus, Feb 28 2013
a(37) and a(62) corrected by Michel Marcus, Feb 09 2019
STATUS
approved

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Last modified February 29 03:50 EST 2024. Contains 370401 sequences. (Running on oeis4.)