A079015 -id:A079015 - cp's OEIS frontend

A079024 Let p and q be two prime numbers, not necessarily consecutive, such that q - p = 2n; a(n) is the number of distinct partitions of 2n into even numbers so that each partition corresponds to a consecutive prime difference pattern (k-tuple) and p<=A000230(n). Multiple occurrences of a partition are not counted.

1, 2, 3, 5, 5, 12, 9, 17, 30, 29, 32, 79, 64, 70, 236, 116, 48, 342, 375, 359, 633, 310, 852, 846, 644, 354, 1048, 1191, 635, 1664, 539, 1127, 3971, 1656, 3022, 984, 3894, 2399, 4439, 6431, 2765, 10256, 1818, 5427, 10251, 8153, 9119, 7083, 6456, 5033

Offset: 1

Labos Elemer, Jan 24 2003

nonn

In case of partitions enumerated in A079022-A079024 permutation if parts is relevant since may correspond to different possible consecutive prime-difference patterns.

			Only those and distinct partitions are counted which appear not later than prime A000230(n); n=7, d=14, A000230(7)=113, number of solutions to p+14=q, - both p and q are primes and p<=113 - is 9. This 9 distinct partitions and their introducing primes are as follows:3[2244], 5[24242], 17[2462], 23[626], 29[2642], 47[662], 83[68], 89[842], 113[14]=A000230(7).

Cf. A000230, A079015-A079024.

A372085 Primes starting a sequence of 6 consecutive primes with gaps 2, 4, 8, 16, 32.

Original entry on oeis.org

6824897, 10132607, 12674657, 13699457, 14148047, 27353237, 43918997, 44152307, 50608007, 53944337, 60426257, 60825827, 61325057, 68721047, 68933717, 72069707, 78577817, 82108127, 82334297, 87020177, 88226777, 97013927, 102043757, 106053917, 122271557, 140859707, 146049047, 161788787, 162036227

Offset: 1

Zak Seidov and Robert Israel, Apr 17 2024

nonn

First differs from A079015 at a(25) = 122271557.

All terms == 7 (mod 10).

			a(3) = 12674657 is a term because 12674657 is prime and the next five primes are 12674657 + 2 = 12674659, 12674659 + 4 = 12674663, 12674663 + 8 = 12674671, 12674671 + 16 = 12674687 and 12674687 + 32 = 12674719.

Robert Israel, Table of n, a(n) for n = 1..1000
Index entries for primes, gaps between

Cf. A079015, A372248 (gaps 2,4,8,16).

p:= 2: state:= 1: count:= 0: Res:= NULL:
while count < 100 do
  q:= nextprime(p);
  if q - p = 2^state then
     state:= state+1;
     if state = 6 then
       count:= count+1; Res:= Res,q-62;
     fi;
  else state:= 1
  fi;
  p:= q;
od:
Res;

A079023 Let p and q be two prime numbers, not necessarily consecutive, such that q - p = 2n; then a(n) is the number of partitions of 2n into even numbers so that each partition corresponds to a consecutive prime difference pattern (k-tuple) and p <= A000230(n).

Original entry on oeis.org

1, 2, 6, 9, 14, 24, 11, 56, 46, 45, 46, 109, 82, 97, 287, 124, 51, 390, 507, 434, 691, 332, 1105, 898, 676, 359, 1080, 1259, 659, 1688, 540, 1146, 4081, 1672, 3081, 985, 3975, 2423, 4460, 6512, 2779, 10324, 1820, 5458, 10273, 8196, 9177, 7085, 6462, 5037

Offset: 1

Labos Elemer, Jan 24 2003

nonn

Partitions are counted with multiplicity and may overlap.

			Only those partitions are counted that appear not later than prime A000230(n); n=7, d=14, A000230(7)=113; the number of solutions to p+14=q, with p and q both prime and p <= 113, is 11. These 11 (not necessarily distinct) partitions and their initial primes are as follows: 3[22424], 5[24242], 17[2462], 23[626], 29[2642], 47[662], 53[626], 59[2642], 83[68], 89[842], 113[14]=A000230(7).

Cf. A000230, A079015-A079024.

{for(n=1,50, c=0; p=2; done=0; until(done, if(isprime(p+2*n), c++; if(nextprime(p+1)-p==2*n, done=1; print1(c,","))); p=nextprime(p+1)))} \\ Rick L. Shepherd

Corrected and extended by Rick L. Shepherd, Sep 08 2003

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A372085 Primes starting a sequence of 6 consecutive primes with gaps 2, 4, 8, 16, 32.

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A079023 Let p and q be two prime numbers, not necessarily consecutive, such that q - p = 2n; then a(n) is the number of partitions of 2n into even numbers so that each partition corresponds to a consecutive prime difference pattern (k-tuple) and p <= A000230(n).

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