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User talk:M. F. Hasler/Sequences/A008470

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Decades containing prime triplets, i.e. such that 10 a(n) + {1, 3, 7, 9} contains at least 3 primes.

edit sect.0 Already in OEIS as A008470 - cf Cross-References below.

Sequence

1, 4, 7, 10, 13, 19, 22, 31, 43, 46, 61, 64, 82, 85, 88, 103, 106, 109, 130, 142, 145, 148, 160, 166, 169, 178, 187, 199, 208, 238, 268, 271, 316, 325, 346, 367, 376, 391, 400, 409, 415, 421, 451, 472, 478, 493, 523, 541, 544, 547, 550, 565, 574, 586, 670, 682, 712, 721, 745, 748, 754, 775, 787, 808, 823, 829, 862, 883, 886, 904, 934, 943, 946, 985, ...

Offset

1,3

Comment

  • (From the initial post by Rupert Wood, Dec 31, 2009) (START)
A decadal prime triplet is a set matching one of the four types
{10K+1, 10K+3, 10K+7}, {10K+1, 10K+3, 10K+9}, {10K+1, 10K+7, 10K+9}, {10K+3, 10K+7, 10K+9},
all members of course being prime. Consecutive occurrences have a gap of 30 (i.e. the K value is incremented by 3), and in this context quadruplets are permitted as well.
There is an instance of 4 consecutive triplets at 5413 5417 5419; 5441 5443 5449; 5471 5477 5479; 5501 5503 5507.
By choosing suitable values modulo some small prime divisors, it appears to be possible to have up to 11 consecutive occurrences of such triplets. There is not even another case of 4 up to about 25000, from a quick check. (END)
  • The decade K must be of the form 3m+1, since for K=3m, 10K+3 and 10K+9 cannot be prime and for K=3m+2, 10K+1 and 10K+7 cannot be prime.
  • The corresponding m-values (i.e. [a(n)/3]) are:
0, 1, 2, 3, 4, 6, 7, 10, 14, 15, 20, 21, 27, 28, 29, 34, 35, 36, 43, 47, 48, 49, 53, 55, 56, 59, 62, 66, 69, 79, 89, 90, 105, 108, 115, 122, 125, 130, 133, 136, 138, 140, 150, 157, 159, 164, 174, 180, 181, 182, 183, 188, 191, 195, 223, 227, 237, 240, 248, 249, 251, 258, 262, 269, 274, 276, 287, 294, 295, 301, 311, 314, 315, 328, ...
The positive integers not appearing here are: 5, 8, 9, 11, 12, 13, 16, 17, 18, 19, 22, 23, 24, 25, 26, 30, 31, 32, 33, 37, 38, 39, 40, 41, 42, 44, 45, 46, 50, 51, 52, 54, 57, 58, 60, 61, 63, 64, 65, 67, 68, 70, 71, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100,...

From this it is seen that although missing m's seem to be an exception for very small values, the situation soon turns the other way round for larger values (only 12 decades between K=900 to 1200 have triplets, only 11 decades from 2700 to 3000, only 6 decades from K=5700 to 6000).


Links

Formula

K in A171251 <=> PrimePi(10K+10) > primePi(10K)+2.

Example

  • The first occurrence of 5 consecutive triplets is:

11, 13, 17 (or 19) ; 41, 43, 47 ; 71, 73, 79 ; 101, 103, 107 (or 109) ; 131, 137, 139. This corresponds to decades 1,4,7,10,13; i.e. the first 5 terms of this sequence. The next occurrence of 4 consecutive triples starts with decade K=541 (cf. original post), and the next occurrence of 5 consecutive triples starts with decade K=910052463685 (found by J.K.Andersen),

Program

  • (PARI) A171251(max,min=1)=for(K=min,nMax, primepi(10*K+10)-primepi(10*K)>2 & print1(K", "))

Crossrefs

  • A008470: At least 3 out of 10m+1, 10m+3, 10m+7, 10m+9 are primes. My notes: ../A008470
  • A008471: Exactly 3 out of 10m+1, 10m+3, 10m+7, 10m+9 are primes. My notes: ../A008471
  • A238713: Least member of decadal prime triplets. Notes: ../A238713
  • ../A171253 Largest member of decadal prime triplet
  • ../A171254 Decades containing exactly 3 primes
  • ../A171255 Least prime in a decade containing exactly 3 primes
  • ../A171256 Run lengths of decadal prime triplets
  • ../A171257 Least prime of a run of 3 or more consecutive decadal prime triplets

Keyword

nonn,new

Author

M. F. Hasler (MHasler(AT)univ-ag.fr), Jan 03 2010


decadal prime triplets

NOT IN OEIS: none of the following possibilities are in OEIS

(Note: The PARI code is simplistic and for illustration purpose only)

Least member of decadal prime triplets:

(PARI) 

for(d=1,999,primepi(10*(d+1))-primepi(10*d) >2|next; print1(nextprime(d*10+1)","))

11,41,71,101,131,191,223,311,431,461,613,641,821,853,881,1031,1061,1091,1301,1423,1451, 1481,1601,1663,1693,1783,1871,1993,2081,2381,2683,2711,3163,3251,3461,3671,3761,3911,4001, 4091,4153,4211,4513,4721,4783,4931,5231,5413,5441,5471,5501,5651,5741,5861,6701,6823,7121, 7211,7451,7481,7541,7753,7873,8081,8231,8291,8623,8831,8861,9041,9341,9431,9461,9851,

Exactly 3, not 4 in a decade:

for(d=1,999,primepi(10*(d+1))-primepi(10*d) == 3|next; print1(nextprime(d*10+1)","))

41,71,131,223,311,431,461,613,641,853,881,1031,1061,1091,1301,1423,1451,1601,1663,1693,1783,1993, 2381,2683,2711,3163,3671,3761,3911,4001,4091,4153,4211,4513,4721,4783,4931,5231,5413,5441,5471,5501, 5741,5861,6701,6823,7121,7211,7451,7481,7541,7753,7873,8081,8231,8291,8623,8831,8861,9041,9341,9461,9851,

To check the "gap of 30" statement:

print number of primes in each of the following 10 decades 

for(d=1,99, primepi(10*(d+1))-primepi(10*d) > 2|next; print([nextprime(d*10+1),vector(10,i,primepi(10*(d+i))-primepi(10*(d+i-1)))]))

[ 11, [ 4,  2, 2,  3,  2, 2,  3,  2, 1,  4 ]]
[ 41, [ 3,  2, 2,  3,  2, 1,  4,  1, 1,  3 ]]
[ 71, [ 3,  2, 1,  4,  1, 1,  3,  1, 2,  2 ]]
[101, [ 4,  1, 1,  3,  1, 2,  2,  2, 1,  4 ]]
[131, [ 3,  1, 2,  2,  2, 1,  4,  0, 1,  3 ]]
[191, [ 4,  0, 1,  3,  2, 1,  2,  2, 2,  2 ]]
[223, [ 3,  2, 1,  2,  2, 2,  2,  1, 1,  3 ]] /* the first not to start at 10K+1 */
[311, [ 3,  0, 2,  2,  2, 1,  2,  2, 1,  2 ]]
[431, [ 3,  2, 1,  3,  1, 1,  2,  2, 0,  2 ]]
[461, [ 3,  1, 1,  2,  2, 0,  2,  0, 2,  1 ]]
[613, [ 3,  0, 1,  3,  2, 1,  2,  1, 1,  2 ]]
[641, [ 3,  2, 1,  2,  1, 1,  2,  1, 1,  2 ]]
[821, [ 4,  1, 0,  3,  1, 1,  3,  0, 1,  2 ]]
[853, [ 3,  1, 1,  3,  0, 1,  2,  1, 1,  2 ]]
[881, [ 3,  0, 1,  2,  1, 1,  2,  1, 1,  2 ]]
...

Note:

  • if the decade is divisible by 3, then trailing digits cannot be 3 or 9, => no triplet possible
  • if the decade is = 1 mod 3, then trailing digits cannot be 2,5,8 -> no restriction
  • if the decade is = 2 mod 3, trailing digits cannot be 1,(4),7 => no triplet possible

Thus, triplets are only possible for K=3n+1.

Number of consecutive decadal prime triplets :

count the maximal i such that in decade K, K+3, ..., K+3(i-1) there is a triplet

  • for any K
  • for K with i>1
  • rather consider "runs" (else any i>1 will be followed by i-1, i-2,...) and after an i>1, skip to the next run. (Needs a 2nd sequence giving the 1st prime of each run.)

Runs of decadal prime triplets

{d=10; p=primepi(d); i=0; while( po=p, p=primepi( d+=10 ); p>2+po & i++ & (p=primepi(d+=20)) & next; i | next; print1(i",");i>=3 & print1("/*",[nextprime(d-10-30*i),precprime(d-30)]"*/ ");i=0;)}

5,/*[11, 139]*/2,1,2,2,3,/*[821, 887]*/3,/*[1031, 1097]*/ 1,3,/*[1423, 1489]*/1,2,1,1,1,1,1,2,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,4,/*[5413, 5507]*/1,1,1,1,1,1,1,2,1,1, 1,1,1,1,1,2,1,1,2,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1, 3,/*[13691, 13759]*/2,1,1, ...


The first run is indeed of length 5: 11,13,17,19, 23,29,31,37, 41,43,47, 53,59,61,67, 71,73,79, 83,89,97, 101,103,107,109, 113,127, 131,137,139, 149,151,157,163,167, 


default(primelimit,10^7); /*  LIMIT=10^6, but print only 2's and above*/
{d=10; p=primepi(d); i=0; while( po=p, p=primepi( d+=10 ); p>2+po & i++  & (p=primepi(d+=20)) & next;
i | next; i>1 & print1(i",");i>=3 & print1("/*",[nextprime(d-10-30*i),precprime(d-30)]"*/");i=0;)}

5,/*[11, 139]*/2,2,2,3,/*[821, 887]*/ 3,/*[1031, 1097]*/ 3,/*[1423, 1489]*/ 2,2,4,/*[5413, 5507]*/ 2,2,2,2, 3,/*[13691, 13759]*/ 2,2,2,2,2,2,2,2, 2,2,2,2,2,2,2,2,2, 2,2,2,2,2,2,2,2,2,2, 2,2,2,2,2,2,2,2,2, 3,/*[140831, 140897]*/ 2,2,2,2,2,2, 3,/*[220873, 220939]*/ 2,2,2,2, 3,/*[266023, 266089]*/ 2,2, 3,/*[283571, 283639]*/ 2,2,2,2,2,2,2,2,2, 2,2,2,2,2,2,2,2,2,2,2,3,/*[464741, 464809]*/ 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2,2,2,2, 2,2,2,2,2,2,2,2, 2,2,2,2,2,2,2,2, 2,2,2,2,2,2,2,2, 2,2,2,2,2,2,2,2,2,2,2,2, 3,/*[1596311, 1596379]*/ 2,2,2,2,2, 2,2,3,/*[1660661, 1660727]*/ 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2, 3,/*[1966813, 1966879]*/ 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,3,/*[2655403, 2655469]*/ 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,3,/*[3303341, 3303409]*/ 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2, 3,/*[5191331, 5191397]*/ 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,3, /*[5485393, 5485457]*/ 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,3,/*[8125511, 8125577]*/ 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, 2,2,2,2,2, [*** stop at primelimit 10^7]

This suggests the sequence: start (end) of runs of length > 3 [11, 139] [821, 887] [1031, 1097] [1423, 1489] [5413, 5507] [13691, 13759] [140831, 140897], [220873, 220939], [ 266023, 266089], [283571, 283639], [464741, 464809], [1596311, 1596379], [1660661, 1660727], [1966813, 1966879], [2655403, 2655469], [3303341, 3303409], [5485393, 5485457], [8125511, 8125577]

Search for consecutive occurrences

  • if searching for runs of 4, one should increase in steps of 4 decades
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