Search: a101043 -id:a101043
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2, 3, 5, 7, 13, 19, 11, 31, 29, 23, 17, 43, 37, 41, 53, 47, 67, 79, 61, 71, 59, 73, 101, 149, 83, 127, 97, 89, 109, 137, 157, 107, 103, 151, 113, 163, 139, 181, 197, 131, 193, 167, 191, 173, 227, 199, 179, 223, 211, 307, 241, 349, 229, 239, 233, 257, 379, 277, 271
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OFFSET
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1,1
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COMMENTS
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The order in which the primes (except 2) first appear in A020483. It is conjectured that all primes are in this sequence.
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A020483
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Least prime p such that p+2n is also prime.
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+10
29
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2, 3, 3, 5, 3, 3, 5, 3, 3, 5, 3, 7, 5, 3, 3, 7, 5, 3, 5, 3, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 7, 5, 3, 5, 3, 3, 7, 5, 3, 5, 3, 7, 5, 3, 13, 7, 5, 3, 5, 3, 3, 5, 3, 3, 5, 3, 19, 13, 11, 13, 7, 5, 3, 5, 3, 7, 5, 3, 3, 11, 11, 7, 5, 3, 3, 7, 5, 3, 7, 5, 3, 5, 3, 7, 5, 3, 7, 5, 3, 3, 11, 11, 7, 5, 3, 3, 5, 3, 3, 13, 11, 31, 7
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OFFSET
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0,1
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COMMENTS
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It is conjectured that a(n) always exists. a(n) has been computed for n < 5 * 10^11, with largest value a(248281210271) = 3307. - Jens Kruse Andersen, Nov 28 2004
If a(n) = a(n+1) = k, then 2*n + k and 2*(n+1) + k are twin primes. - Ya-Ping Lu, Sep 22 2020
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LINKS
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FORMULA
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If a(n) exists, a(n) < 2n, which of course is a great overestimate. - T. D. Noe, Jul 16 2002
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EXAMPLE
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Given n = 2, we see that 2 + 2n = 6 = 2 * 3, but 3 + 2n = 7, which is prime, so a(2) = 3.
Given n = 3, we see that 2 + 2n = 8 = 2^3 and 3 + 2n = 9 = 3^2, but 5 + 2n = 11, which is prime, so a(3) = 5.
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MAPLE
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local p;
p := 2;
while true do
if isprime(p+2*n) then
return p;
end if;
p := nextprime(p) ;
end do:
end proc:
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MATHEMATICA
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Table[j = 1; found = False; While[!found, j++; found = PrimeQ[Prime[j] + 2i]]; Prime[j], {i, 200}]
leastPrimep2n[n_] := Block[{k = 1, p, q = 2 n}, While[p = Prime@k; !PrimeQ[p + q], k++]; p]; Array[leastPrimep2n, 102] (* Robert G. Wilson v, Mar 26 2008 *)
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PROG
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(Haskell)
a020483 n = head [p | p <- a000040_list, a010051' (p + 2 * n) == 1]
(GAP) P:=Filtered([1..10000], IsPrime);;
a:=List(List([0..110], n->Filtered(P, i->IsPrime(i+2*n))), Minimum); # Muniru A Asiru, Mar 26 2018
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CROSSREFS
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It is likely that A054906 is an identical sequence, although this seems to have not yet been proved. - N. J. A. Sloane, Feb 06 2017
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A101042
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a(n) is the smallest positive d such that the n-th prime is the smallest prime p for which p+d is also prime.
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+10
6
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1, 2, 6, 22, 116, 88, 470, 112, 284, 242, 202, 772, 1326, 718, 1334, 1328, 2558, 1762, 1642, 2402, 3274, 1732, 7094, 9512, 7984, 5246, 12688, 10532, 9952, 16766, 7702, 60458, 9974, 25708, 5888, 13528, 10342, 25678, 62156, 69518, 76838, 37666
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OFFSET
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1,2
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COMMENTS
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Except for n=1, A020483(a(n)/2) is the first appearance of the n-th prime. It is conjectured that a(n) always exists. a(386) is the first number which must be above 10^12.
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LINKS
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EXAMPLE
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a(3)=6 because: The 3rd prime is 5. 2+6, 3+6 is composite, 5+6 is prime. 6 is the smallest such number.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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2, 3, 5, 7, 13, 19, 31, 43, 53, 67, 79, 101, 149, 157, 163, 181, 197, 227, 307, 349, 379, 409, 431, 619, 631, 661, 691, 751, 757, 811, 829, 1093, 1117, 1217, 1279, 1423, 1453, 1481, 1531, 1549, 1579, 1759, 1877, 2239, 2273, 2287, 2383, 2447, 2659, 2671, 2707
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OFFSET
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1,1
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COMMENTS
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This sequence (except 2) is also the record size primes in the longer A020483.
Conjecture: lim_{n->infinity} a(n)/n^2 = 1. - Ya-Ping Lu, Sep 24 2020
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LINKS
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PROG
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(Python)
from sympy import isprime, nextprime
m = p_max = 0
while m >= 0:
p = 2
while isprime(p + 2*m) == 0:
p = nextprime(p)
if p > p_max:
print(p)
p_max = p
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A101046
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d such that the smallest prime p for which p+d is also prime is larger than for any smaller d.
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+10
5
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1, 2, 6, 22, 88, 112, 202, 718, 1328, 1642, 1732, 5246, 5888, 10342, 25678, 37666, 59894, 76004, 103102, 108412, 180814, 359662, 651362, 872698, 2373478, 6088792, 7642528, 9244552, 13038352, 13591192, 24318988, 34857778, 55076404, 147838742
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OFFSET
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1,2
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COMMENTS
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The numbers in A101042 which are smaller than all following numbers.
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LINKS
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EXAMPLE
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Consider d=6. The smallest prime p for which p+6 is also prime, is p=5. All numbers below d=6 have a p<5 (or no p at all), so 6 is in the sequence.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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