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Search: a101043 -id:a101043
Displaying 1-5 of 5 results found. page 1
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A101044 Primes corresponding to A101043 (which is A101042 sorted). +20
5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The order in which the primes (except 2) first appear in A020483. It is conjectured that all primes are in this sequence.
LINKS
Table of n, a(n) for n=1..59.
J. K. Andersen, Prime gaps (not necessarily consecutive).
Mike Oakes, Ed Pegg Jr, Jens Kruse Andersen, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004. [Cached copy]
CROSSREFS
Cf. A020483, A101042, A101043, A101045, A101046.
KEYWORD
nonn
AUTHOR
Jens Kruse Andersen, Nov 28 2004
STATUS
approved
A020483 Least prime p such that p+2n is also prime. +10
29
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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
LINKS
T. D. Noe, Table of n, a(n) for n = 0..10000
Jens Kruse Andersen, Prime gaps (not necessarily consecutive), Yahoo! group "primenumbers", Nov 26 2004.
Jens Kruse Andersen, Mike Oakes, Ed Pegg Jr, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004.
FORMULA
If a(n) exists, a(n) < 2n, which of course is a great overestimate. - T. D. Noe, Jul 16 2002
a(n) = A087711(n) - n. - Zak Seidov, Nov 28 2007
a(n) = A020484(n) - 2n. - Zak Seidov, May 29 2014
a(n) = 2 if and only if n = 0. - Alonso del Arte, Mar 14 2018
EXAMPLE
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.
MAPLE
A020483 := proc(n)
local p;
p := 2;
while true do
if isprime(p+2*n) then
return p;
end if;
p := nextprime(p) ;
end do:
end proc:
seq(A020483(n), n=0..40); # R. J. Mathar, Sep 23 2016
MATHEMATICA
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 *)
PROG
(PARI) a(n)=forprime(p=2, , if(isprime(p+2*n), return(p))) \\ Charles R Greathouse IV, Mar 19 2014
(Haskell)
a020483 n = head [p | p <- a000040_list, a010051' (p + 2 * n) == 1]
-- Reinhard Zumkeller, Nov 29 2014
(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
CROSSREFS
Cf. A087711, A101042, A101043, A101044, A101045, A101046.
Cf. A101045, A239392 (record values).
Cf. A000040, A010051, A020484, A237055.
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
KEYWORD
nonn
AUTHOR
David W. Wilson
EXTENSIONS
a(0)=2 added by N. J. A. Sloane, Apr 25 2015
STATUS
approved
A101042 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. +10
6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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.
LINKS
Table of n, a(n) for n=1..42.
J. K. Andersen, Prime gaps (not necessarily consecutive).
Mike Oakes, Ed Pegg Jr, Jens Kruse Andersen, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004. [Cached copy]
EXAMPLE
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.
CROSSREFS
Cf. A020483, A101043, A101044, A101045, A101046.
KEYWORD
nonn
AUTHOR
Jens Kruse Andersen, Nov 28 2004
STATUS
approved
A101045 Record size primes in A101044. +10
6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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
LINKS
Table of n, a(n) for n=1..51.
J. K. Andersen, Prime gaps (not necessarily consecutive).
Taras Goy and Mark Shattuck, Determinant Formulas of Some Hessenberg Matrices with Jacobsthal Entries, Applications and Appl. Math.: An Int'l J. (2021) Vol. 16, Issue 1, Art. 10.
Mike Oakes, Ed Pegg Jr, and Jens Kruse Andersen, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004. [Cached copy]
PROG
(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
m += 1 # Ya-Ping Lu, Sep 24 2020
CROSSREFS
Cf. A020483, A101042, A101043, A101044, A101046.
KEYWORD
nonn
AUTHOR
Jens Kruse Andersen, Nov 28 2004
STATUS
approved
A101046 d such that the smallest prime p for which p+d is also prime is larger than for any smaller d. +10
5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The numbers in A101042 which are smaller than all following numbers.
LINKS
Table of n, a(n) for n=1..34.
J. K. Andersen, Prime gaps (not necessarily consecutive).
Mike Oakes, Ed Pegg Jr, Jens Kruse Andersen, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004. [Cached copy]
EXAMPLE
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.
CROSSREFS
Cf. A020483, A101042, A101043, A101044, A101045.
KEYWORD
nonn
AUTHOR
Jens Kruse Andersen, Nov 28 2004
STATUS
approved
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Last modified August 30 07:09 EDT 2024. Contains 375532 sequences. (Running on oeis4.)