A273897 -id:A273897 - cp's OEIS frontend

A140079 Numbers n such that n and n+1 have 5 distinct prime factors.

254540, 310155, 378014, 421134, 432795, 483405, 486590, 486794, 488565, 489345, 507129, 522444, 545258, 549185, 558789, 558830, 567644, 577940, 584154, 591260, 598689, 627095, 634809, 637329, 663585, 666995, 667029, 678755, 687939, 690234

Offset: 1

Artur Jasinski, May 07 2008

nonn

For the smallest number r such that r and r+1 have n distinct prime factors, see A093548.
Goldston, Graham, Pintz, & Yildirim prove that this sequence is infinite. - Charles R Greathouse IV, Jun 02 2016
Subsequence of the variant A321505 defined with "at least 5" instead of "exactly 5" distinct prime factors. See A321495 for the differences. - M. F. Hasler, Nov 12 2018
The subset of numbers where n and n+1 are also squarefree gives A318964. - R. J. Mathar, Jul 15 2023

Seiichi Manyama, Table of n, a(n) for n = 1..10000
D. A. Goldston, S. W. Graham, J. Pintz, C. Y. Yildirim., Small gaps between almost primes, the parity problem and some conjectures of Erdos on consecutive integers, arXiv:0803.2636 [math.NT], 2008.

Cf. A074851, A140077, A140078, A273897.
Cf. A093548.
Equals A321505 \ A321495.

a = {}; Do[If[Length[FactorInteger[n]] == 5 && Length[FactorInteger[n + 1]] == 5, AppendTo[a, n]], {n, 1, 100000}]; a (*Artur Jasinski*)
Transpose[SequencePosition[Table[If[PrimeNu[n]==5,1,0],{n,700000}],{1,1}]][[1]] (* The program uses the SequencePosition function from Mathematica version 10 *) (* Harvey P. Dale, Jul 25 2015 *)

is(n)=omega(n)==5 && omega(n+1)==5 \\ Charles R Greathouse IV, Jun 02 2016

{k: k in A051270 and k+1 in A051270}. - R. J. Mathar, Jul 19 2023

A273898 Sum of the abscissae of the first descents of all bargraphs of semiperimeter n (n>=2).

Original entry on oeis.org

1, 3, 9, 27, 81, 244, 739, 2251, 6895, 21232, 65703, 204245, 637573, 1997892, 6282635, 19820580, 62716923, 198997349, 633015543, 2018391204, 6449819095, 20652628601, 66256638509, 212939343591, 685497649231, 2210217592624, 7136781993563, 23076554161563

Offset: 2

Emeric Deutsch, Jun 06 2016

nonn

A descent in a bargraph is a maximal sequence of adjacent down steps.

			a(4)=9 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1],[1,2],[2,1],[2,2],[3] and the corresponding pictures give the values 3,2,1,2,1 for the abscissae of their first descents.

Alois P. Heinz, Table of n, a(n) for n = 2..1000
M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

Cf. A082582, A273897.

g := ((1-4*z+3*z^2-(1-2*z)*Q)*(1/2))/z^3: Q := sqrt(1-4*z+2*z^2+z^4): gser := series(g,z = 0,40): seq(coeff(gser, z, n), n = 2 .. 35);
# second Maple program:
a:= proc(n) option remember; `if`(n<4, [0$2, 1, 3][n+1],
     ((2*(14*n^2+6+13*n))*a(n-1)-(2*(7*n^2-6-4*n))*a(n-2)
     +12*a(n-3) -(n-4)*(3+7*n)*a(n-4))/((n+3)*(7*n-4)))
    end:
seq(a(n), n=2..40);  # Alois P. Heinz, Jun 07 2016

a[n_] := a[n] = If[n<4, {0, 0, 1, 3}[[n+1]], ((2*(14*n^2+6+13*n))*a[n-1] - (2*(7*n^2-6-4*n))*a[n-2] + 12*a[n-3] - (n-4)*(3+7*n)*a[n-4])/((n+3)*(7*n - 4))]; Table[a[n], {n, 2, 40}] (* Jean-François Alcover, Dec 02 2016 after Alois P. Heinz *)

G.f.:  g(z)=(1-4z+3z^2-(1-2z)Q)/(2z^3), where Q = sqrt(1-4z+2z^2+z^4).
a(n) = Sum(k*A273897(n,k), k>=1).
a(n) = A082582(n+2)-2*A082582(n+1).
D-finite with recurrence (n+3)*a(n) +2*(-3*n-4)*a(n-1) +2*(5*n-2)*a(n-2) +4*(-n+2)*a(n-3) +(n-3)*a(n-4) +2*(-n+5)*a(n-5)=0. - R. J. Mathar, Jul 24 2022

cp's OEIS Frontend

A140079 Numbers n such that n and n+1 have 5 distinct prime factors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula