A101695 -id:A101695 - cp's OEIS frontend

A115057 Number of (2n+1)-almost primes less than or equal to (n-th n-almost prime) * ((n+1)-th (n+1)-almostprime).

2, 5, 11, 17, 25, 30, 45, 67, 74, 82, 95, 111, 141, 177, 193, 208, 211, 223, 257, 277, 288, 353, 431, 453, 481, 509, 528, 540, 563, 619, 672, 700, 725, 745, 804, 857, 905, 1003, 1077, 1127, 1199, 1268, 1281, 1321, 1354, 1379, 1423, 1517, 1607, 1660, 1714, 1748

Offset: 1

Jonathan Vos Post, Oct 08 2007

Numbers k such that Pi(2n-1, (n-th n-almost prime) * ((n+1)-th (n+1)-almostprime)) = Pi(2n-1, A101695(n)*A101695(n+1)) = (2n-1)-AlmostPrime(k).

Robert G. Wilson v, Table of n, a(n) for n = 1..228.
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A101695.

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[ PrimePi[n / Times @@ Prime[ Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]] (* Eric W. Weisstein, Feb 07 2006 *);
lst={ (* the list of entries in A101695 *) }; lsu = {}; Do[a = AlmostPrimePi[2 n + 1, lst[[n]]*lst[[n + 1]]]; AppendTo[lsu, a]; Print[{n, a}], {n, 228}] (* Robert G. Wilson v, Oct 08 2007 *)

A131175 Table, read by rows, of coefficients of characteristic polynomials of almost prime matrices.

Original entry on oeis.org

1, -2, 1, -8, 1, -26, -4, 1, -66, -36, 1, -174, -196, 1, -398, -676, 1, -878, -3044, 1, -2174, -6852, -192, 1, -4862, -18628, -704, 1, -10494, -45508, -1216, 1, -22014, -141252, -6336, 1, -47614, -315332, -10432, 1, -100862, -858052, -55488, 1, -225278, -1878980, -245952

Offset: 1

Jonathan Vos Post, Sep 24 2007

Because the first column of A is a column vector of powers of 2, the determinant (for n>1) is always 0. Hence the rank is always (for n>1) less than n. A[n.n] = n-th n-almost prime A101695. The second column of the table is the negative of the trace of the matrices.

			A_1 = [2], with determinant = 2 and characteristic polynomial = x-2, with coefficients (1, -2) so a(a) = 1 and a(2) = -2.
A_2 =
[2.3]
[4.6]
with determinant = 0, polynomial x^2 - 8x, so the coefficients are (1, -8), hence a(3) = 1 and a(4) = -8.
A_3 =
[2..3..5]
[4..6..9]
[8.12.18]
with determinant = 0, polynomial = x^3 - 26x^2, -4x, so coefficients are (1, -26, -4), hence a(5) = 1, a(6) = -26, a(7) = -4.

Cf. A000040, A001358, A014612, A014613, A014614, A101695.

A078840 := proc(n,m) local p,k ; k := 1 ; p := 2^n ; while k < m do p := p+1 ; while numtheory[bigomega](p) <> n do p := p+1 ; od; k := k+1 ; od: RETURN(p) ; end: A131175 := proc(nrow,showall) local A,row,col,pol,T,a ; A := linalg[matrix](nrow,nrow) ; for row from 1 to nrow do for col from 1 to nrow do if row = col then A[row,col] := x-A078840(row,col) ; else A[row,col] := -A078840(row,col) ; fi ; od: od: pol := linalg[det](A) ; T := [] ; for col from nrow to 0 by -1 do a := coeftayl(pol,x=0,col) ; if a <> 0 or showall then T := [op(T),a] ; fi ; od; RETURN(T) ; end: for n from 1 to 15 do print(op(A131175(n,false))) ; od: # R. J. Mathar, Oct 26 2007

Row n of the table consists of the coefficients of x^n, x^n-1, ... of the characteristic polynomial of the n X n matrix A whose first row is the first n primes (1-almost primes) (A000040), 2nd row is the first n semiprimes (2-almost primes) A001358, 3rd row is the first n 3-almost primes A014612.

Corrected and extended by R. J. Mathar, Oct 26 2007

A215776 Second-largest prime factor of the n-th number that is a product of exactly n primes.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 3, 3, 3, 2, 3, 5, 2, 3, 3, 3, 3, 2, 5, 5, 2, 3, 3, 2, 3, 7, 3, 3, 3, 5, 5, 5, 3, 2, 3, 2, 5, 5, 3, 3, 3, 7, 2, 3, 3, 3, 7, 5, 2, 5, 5, 5, 3, 2, 3, 5, 3, 7, 3, 5, 2, 5, 5, 3, 3, 2, 3, 7, 3, 3, 3, 3, 5, 7, 2, 5, 7, 11, 2, 7, 3, 5, 5, 5, 3, 3, 3

Offset: 1

Jonathan Vos Post, Aug 23 2012

nonn

This is to A215405 as 2nd largest prime factor is to largest (greatest) prime factor. Technically, the prime numbers are "1-almost prime."

			a(2) = 2 because the 2nd number that is a product of exactly 2 primes
(semiprime) is 6 = 2*3, so 2 is the 2nd largest of those two prime factors.
a(4) = 2 because the 4th number that is a product of exactly 4 primes is 40 = 2*2*2*5, so 2 is the 2nd largest of those two distinct prime factors {2,5}. This requires clarity in "distinct prime factors" versus merely "prime factors."
a(87) = 3 because the 87th number that is a product of 87 primes is 5048474222710691433572990976 = 2^84 3^2 29, and 3 is the 2nd largest prime factor.

Cf. A087039, A101695, A215405.

A215776 := proc(n)
    A087039(A101695(n)) ;
end proc: # R. J. Mathar, Sep 14 2012

a(n) = A087039(A101695(n)).

Corrected by R. J. Mathar, Sep 14 2012

More terms from Lars Blomberg, Mar 02 2016

A215567 Numbers k with the property that k and the k-th number which is a product of k primes have the largest prime divisor and with the same multiplicity.

Original entry on oeis.org

3, 13, 20, 42, 44, 45, 56, 60, 86, 87, 112, 119, 121

Offset: 1

Juri-Stepan Gerasimov, Aug 16 2012

			13 and the 13th 13-almost prime = 13*2^12 = 53248 have same largest prime factor = 13, with the same multiplicity, so 13 is a term.

Cf. A001222, A101695, A215405.

AlmostPrimePi[k_Integer,n_]:=...;AlmostPrime[k_,n_]:=...;
fQ[n_]:=FactorInteger[n][[-1,1]]==FactorInteger[AlmostPrime[n,n]][[-1,1]];k=1;lst={};
While[k<150,If[fQ@k,AppendTo[lst,k];Print@k];k++];lst  (* Robert G. Wilson v, Aug 27 2012 *)

a(9)-a(13) from Robert G. Wilson v, Aug 27 2012

Definition clarified by N. J. A. Sloane, Sep 04 2012

cp's OEIS Frontend

A115057 Number of (2n+1)-almost primes less than or equal to (n-th n-almost prime) * ((n+1)-th (n+1)-almostprime).

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A131175 Table, read by rows, of coefficients of characteristic polynomials of almost prime matrices.

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A215776 Second-largest prime factor of the n-th number that is a product of exactly n primes.

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A215567 Numbers k with the property that k and the k-th number which is a product of k primes have the largest prime divisor and with the same multiplicity.

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