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A321787 Irregular triangle read by rows: row n gives the prime factorization of n-th average of twin prime pairs (A014574). 0
2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 3, 2, 1, 1, 0, 0, 0, 0, 1, 2, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 1, 6, 1, 1, 2, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 4, 1, 1, 1, 3, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 1, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Table of n, a(n) for n=1..87.

EXAMPLE

Consider the first pair of twin primes, 3 and 5. Their average is 4, which is the first term in A014574. When 4 is factored into primes it is 2^2, so the first row in this triangle is 2.

For the next pair of twin primes, 5 and 7, the average is 6, and its prime factorization is 2*3, so the second row is 1,1.

Since all the averages are even, each row begins with a term > 0.

The initial rows are:

  2;                 as (3+5)/2    =  4 = 2^2

  1, 1;              as (5+7)/2    =  6 = 2^1 * 3^1

  2, 1;              as (11+13)/2  = 12 = 2^2 * 3^1

  1, 2;              as (17+19)/2  = 18 = 2^1 * 3^2

  1, 1, 1;           as (29+31)/2  = 30 = 2^1 * 3^1 * 5^1

  1, 1, 0, 1;        as (41+43)/2  = 42 = 2^1 * 3^1 * 5^0 * 7^1

  ...

MATHEMATICA

expo[n_] := Module[{f = FactorInteger[n]}, ps = f[[;; , 1]]; es = f[[;; , 2]]; pm = ps[[-1]]; np = PrimePi[pm]; v = Table[0, {np}]; Do[v[[PrimePi[ps[[k]]]]] = es[[k]], {k, 1, Length[ps]}]; v]; mids = Select[Range[500], PrimeQ[# - 1] && PrimeQ[# + 1] &]; Map[expo, mids] // Flatten (* Amiram Eldar, Dec 22 2018 *)

PROG

(PARI) tabf(nn) = {my(p=2, mp); forprime(q=3, nn, if (q-p==2, mp = vecmax(factor(p+1)[, 1]); forprime(pr=1, mp, print1(valuation(p+1, pr), ", ")); print); p=q; ); } \\ Michel Marcus, Dec 23 2018

CROSSREFS

Cf. A014574.

Sequence in context: A159704 A101428 A307223 * A331596 A023586 A023584

Adjacent sequences:  A321784 A321785 A321786 * A321788 A321789 A321790

KEYWORD

nonn,tabf

AUTHOR

Nika Sigua, Nov 18 2018

EXTENSIONS

Edited by N. J. A. Sloane, Dec 22 2018

STATUS

approved

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Last modified April 18 22:45 EDT 2021. Contains 343098 sequences. (Running on oeis4.)