A022009
Initial members of prime septuplets (p, p+2, p+6, p+8, p+12, p+18, p+20).
Original entry on oeis.org
11, 165701, 1068701, 11900501, 15760091, 18504371, 21036131, 25658441, 39431921, 45002591, 67816361, 86818211, 93625991, 124716071, 136261241, 140117051, 154635191, 162189101, 182403491, 186484211, 187029371, 190514321, 198453371
Offset: 1
- Dana Jacobsen, Table of n, a(n) for n = 1..10000 (first 1000 terms from Matt C. Anderson)
- Matt C. Anderson, table of prime k-tuplets.
- Tony Forbes and Norman Luhn, Patterns of prime k-tuplets & the Hardy-Littlewood constants.
- Norman Luhn, 1 million terms, zipped archive.
- Vladimir Shevelev and Peter J. C. Moses, Constellations of primes generated by twin primes, arXiv:1610.03385 [math.NT], 2016.
- Eric Weisstein's World of Mathematics, Prime Constellation.
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[p: p in PrimesUpTo(2*10^8) | forall{p+r: r in [2,6,8,12,18,20] | IsPrime(p+r)}]; // Vincenzo Librandi, Oct 01 2015
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Transpose[Select[Partition[Prime[Range[10400000]],7,1],Differences[#] == {2,4,2,4,6,2}&]][[1]] (* Harvey P. Dale, Jul 13 2014 *)
Select[Prime[Range[2 10^8]], Union[PrimeQ[# + {2, 6, 8, 12, 18, 20}]] == {True} &] (* Vincenzo Librandi, Oct 01 2015 *)
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nextcomposite(n)=if(n<4, return(4)); n=ceil(n); if(isprime(n), n+1, n)
is(n)=if(n%30!=11 || !isprime(n) || !isprime(n+2), return(0)); my(p=n, q=n+2, k=2, f); while(p!=q && q-p<7, f=if(isprime(k++), nextprime, nextcomposite); p=f(p+1); q=f(q+1)); p==q \\ Charles R Greathouse IV, Sep 30 2016
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select( {is_A022009(n)=n%210==11&&!foreach([20,18,12,8,6,2,0],d,isprime(n+d)||return)}, [11+k*210|k<-[0..10^5]]) \\ M. F. Hasler, Aug 04 2021
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use ntheory ":all"; say for sieve_prime_cluster(1,1e9, 2,6,8,12,18,20); # Dana Jacobsen, Sep 30 2015
A022010
Initial members of prime septuplets (p, p+2, p+8, p+12, p+14, p+18, p+20).
Original entry on oeis.org
5639, 88799, 284729, 626609, 855719, 1146779, 6560999, 7540439, 8573429, 17843459, 19089599, 24001709, 42981929, 43534019, 69156539, 74266259, 79208399, 80427029, 84104549, 87988709, 124066079, 128469149, 144214319, 157131419, 208729049, 218033729
Offset: 1
a(100) = 2526962939, a(1000) = 80752495919, a(10000) = 2010407120789, a(100000) = 42609827234069, a(1000000) = 822249634821059. See illustration for asymptotic behavior. - _Hugo Pfoertner_, Jun 15 2020
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[p: p in PrimesUpTo(3*10^8) | forall{p+r: r in [2, 8, 12, 14, 18, 20] | IsPrime(p+r)}]; // Vincenzo Librandi, Oct 01 2015
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Select[Prime[Range[2 10^8]], Union[PrimeQ[# + {2, 8, 12, 14, 18, 20}]] == {True} &] (* Vincenzo Librandi, Oct 01 2015 *)
Select[Partition[Prime[Range[12021000]],7,1],Differences[#]=={2,6,4,2,4,2}&][[All,1]] (* or *) Select[Range[179,219*10^6,210], AllTrue[ #+{0,2,8,12,14,18,20},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 04 2019 *)
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forprime(p=2, 10^30, if (isprime(p+2) && isprime(p+8) && isprime(p+12) && isprime(p+14) && isprime(p+18) && isprime(p+20), print1(p", "))) \\ Altug Alkan, Oct 01 2015. [This can be made 2x faster by inserting "p%210==179 &&" before or after "if(". - M. F. Hasler, Aug 04 2021]
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use ntheory ":all"; say for sieve_prime_cluster(1,1e9, 2,8,12,14,18,20); # Dana Jacobsen, Sep 30 2015
A257124
Initial members of prime septuplets.
Original entry on oeis.org
11, 5639, 88799, 165701, 284729, 626609, 855719, 1068701, 1146779, 6560999, 7540439, 8573429, 11900501, 15760091, 17843459, 18504371, 19089599, 21036131, 24001709, 25658441, 39431921, 42981929, 43534019, 45002591, 67816361, 69156539, 74266259, 79208399, 80427029, 84104549, 86818211, 87988709, 93625991, 124066079
Offset: 1
Initial members of all of the first prime k-tuplets:
Cf.
A343637 (distance from 10^n to the next septuplet).
A343635
10^n + a(n) is the least (n+1)-digit prime member of a prime 5-tuple, or a(n) = 0 if no such number exists.
Original entry on oeis.org
4, 1, 1, 481, 5727, 1107, 8851, 18027, 5457, 408807, 57931, 358531, 274587, 256497, 6111627, 67437, 3246567, 1638811, 8224977, 11687221, 24556351, 3129657, 15602131, 571381, 23034391, 110598987, 26716321, 31722117, 39711931, 5046777, 81054327, 1346611, 44656587
Offset: 0
a(0) = 4 because {5, 7, 11, 13, 17} is the smallest prime 5-tuple and it starts with the single-digit prime 10^0 + a(0) = 5 = A022006(1).
a(1) = 1 because 10^1 + 1 = 11 = A022006(2) is the 2-digit prime to start a prime 5-tuple {11, 13, 17, 19, 23}, again of the first type.
a(2) = 1 and a(3) = 481 because 10^2 + 1 = 101 = A022006(3) and 10^3 + 481 = 1481 = A022006(4) are the smallest 3-digit, resp. 4-digit, initial members of a prime 5-tuple, both again of the first type.
a(4) = 5727 because 10^4 + 5727 = 15727 = A022007(6) is the smallest 5-digit initial member of a prime 5-tuple, now of the second type.
It appears that for all n > 0, a(n) < 10^n, so that the primes are of the form 10...0XXX where XXX = a(n) and 0...0 stands for a string of zero or more digits 0.
Cf.
A022006 and
A022007 (initial members of prime 5-tuples of first and second type).
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apply( {A343635(n,q=[1..4],i=0)=forprime(p=10^n,, (q[1+i]+12==q[i++]=p) && return(p-12-10^n); i>3 && i=0)}, [0..15]) \\ Shorter but slightly slower (?)
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apply( {A343635(n, i=ispseudoprime, q)=forprime(p=10^n,, i(p+12) && i(p+6) && (p+6 > q=nextprime(p+2)) && i(q+6) && return(p-10^n))}, [0..15])
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from sympy import nextprime
def a(n):
p = nextprime(10**n)
q = nextprime(p); r = nextprime(q); s = nextprime(r); t = nextprime(s)
while p < 10**(n+1):
if t - p == 12: return p - 10**n
p, q, r, s, t = q, r, s, t, nextprime(t)
return 0
print([a(n) for n in range(14)]) # Michael S. Branicky, Jul 24 2021
Showing 1-4 of 4 results.
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