A131559 Indices of records in A078571.

1, 3, 7, 8, 14, 41, 270, 277, 1595, 5899, 6320, 6668, 117221, 180037, 295677, 587152, 703625, 4397877, 12151726, 29573235

Offset

1,2

Comments

Numbers k such that total number of prime factors (with multiplicity) of the average of k-th twin prime pair is a record.

The records are 1, 2, 3, 4, 5, 7, 9, 10, 12, 14, 15, 17, 19, ... and the associated twin primes recorded in A086827. - R. J. Mathar, Oct 24 2007

Links

Table of n, a(n) for n=1..20.

Formula

a(n) = {k such that A001222(A014574(k)) is a record)}.

a(n) = {k such that A001222{(A001359(k+1) + A006512(k+1)}/2) is a record)}.

a(n) = {k such that A001222(2*A040040(k)) is a record)}.

a(n) = {k such that A001222(A054735(k+1)/2) is a record)}.

a(n) = {k such that A001222(A111046(k+1)/4) is a record)}.

Example

a(1) = 1 because A078571(1) = 2, since the average of twin prime pair (3,5) is 4 = 2^2 is a semiprime with two prime factors (with multiplicity).
a(2) = 3 because A078571(3) = smallomega(12) = 3.
a(3) = 7 because A078571(7) = smallomega(60) = 4.
a(4) = 8 because A078571(8) = smallomega(72) = 5.
a(5) = 14 because A078571(14) = smallomega(192) = 7.
a(6) = 41 because A078571(41) = smallomega(1152) = smallomega(2^7 * 3^2) = 9.

Prog

(PARI) lista(nn) = {my(r=0, c=0, p=2); forprime(q=3, nn, if(q-p==2, c++; if(bigomega(p+1)>r, r=bigomega(p+1); print1(c, ", "))); p=q); } \\ Jinyuan Wang, Apr 01 2020

Crossrefs

Cf. A001221, A001359, A006512, A014574, A078571.

Sequence in context: A093722 A309745 A002381 * A366783 A051211 A105173

Adjacent sequences: A131556 A131557 A131558 * A131560 A131561 A131562

Keyword

nonn,more

Author

Jonathan Vos Post, Oct 02 2007

Extensions

More terms from R. J. Mathar, Oct 24 2007

a(13)-a(20) from Jinyuan Wang, Apr 01 2020

Status

approved