A073445 - OEIS

A073445

Second differences of A002808, the sequence of composites.

0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, 1, 0, -1, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = c(n+2)-2*c(n+1)+c(n), where c(n)=A002808(n).

a(n) = A073783(n+1) - A073783(n). - Reinhard Zumkeller, Jan 10 2013

EXAMPLE

From Gus Wiseman, Oct 10 2024: (Start)

The composite numbers (A002808) are:

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, ...

with first differences (A073783):

2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, ...

with first differences (A073445):

0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 0, 0, 1, -1, ...

(End)

MATHEMATICA

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] Table[(c[w+2]-2*c[w+1])+c[w]), {w, 1, 1000}]

(* second program *)

Differences[Select[Range[100], CompositeQ], 2] (* Gus Wiseman, Oct 08 2024 *)

PROG

(Haskell)

a073445 n = a073445_list !! (n-1)

a073445_list = zipWith (-) (tail a073783_list) a073783_list

-- Reinhard Zumkeller, Jan 10 2013

(Python)

from sympy import primepi

def A073445(n):

def iterfun(f, n=0):

m, k = n, f(n)

while m != k: m, k = k, f(k)

return m

return (a:=iterfun(f:=lambda x:n+primepi(x)+1, n))-((b:=iterfun(lambda x:f(x)+1, a))<<1)+iterfun(lambda x:f(x)+2, b) # Chai Wah Wu, Oct 03 2024

CROSSREFS

Also first differences of A054546.

For first differences we had A073783 (ones A375929), run-lengths A376680.

Positions of zeros are A376602.

Positions of nonzeros are A376603.

Positions of ones are A376651, negative-ones A376652.

A002808 lists the composite numbers.

A064113 lists positions of adjacent equal prime gaps.

A333254 gives run-lengths of differences between consecutive primes.

Other second differences: A036263 (prime), A376590 (squarefree), A376596 (prime-power), A376604 (Kolakoski).

Cf. A076259, A174965, A251092, A376562, A376593, A376599.

Sequence in context: A134452 A327515 A327532 * A285589 A179081 A113217

Adjacent sequences: A073442 A073443 A073444 * A073446 A073447 A073448

KEYWORD

sign,easy

AUTHOR

Labos Elemer, Aug 01 2002

STATUS

approved