A073783 -id:A073783 - cp's OEIS frontend

A002808 The composite numbers: numbers n of the form x*y for x > 1 and y > 1.

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88

Offset: 1

N. J. A. Sloane

The natural numbers 1,2,... are divided into three sets: 1 (the unit), the primes (A000040) and the composite numbers (A002808).
The number of composite numbers <= n (A065855) = n - pi(n) (A000720) - 1.
n is composite iff sigma(n) + phi(n) > 2n. This is a nice result of the well known theorem: For all positive integers n, n = Sum_{d|n} phi(d). For the proof see my contribution to puzzle 76 of Carlos Rivera's Primepuzzles. - Farideh Firoozbakht, Jan 27 2005, Jan 18 2015
The composite numbers have the semiprimes A001358 as primitive elements.
A211110(a(n)) > 1. - Reinhard Zumkeller, Apr 02 2012
A060448(a(n)) > 1. - Reinhard Zumkeller, Apr 05 2012
A086971(a(n)) > 0. - Reinhard Zumkeller, Dec 14 2012
Composite numbers n which are the product of r=A001222(n) prime numbers are sometimes called r-almost primes. Sequences listing r-almost primes are: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011
a(n) = A056608(n) * A160180(n). - Reinhard Zumkeller, Mar 29 2014
Degrees for which there are irreducible polynomials which are reducible mod p for all primes p, see Brandl. - Charles R Greathouse IV, Sep 04 2014
An integer is composite if and only if it is the sum of strictly positive integers in arithmetic progression with common difference 2: 4 = 1 + 3, 6 = 2 + 4, 8 = 3 + 5, 9 = 1 + 3 + 5, etc. - Jean-Christophe Hervé, Oct 02 2014
This statement holds since k+(k+2)+...+k+2(n-1) = n*(n+k-1) = a*b with arbitrary a,b (taking n=a and k=b-a+1 if b>=a). - M. F. Hasler, Oct 04 2014
For n > 4, these are numbers n such that n!/n^2 = (n-1)!/n is an integer (see A056653). - Derek Orr, Apr 16 2015
Let f(x) = Sum_{i=1..x} Sum_{j=2..i-1} cos((2*Pi*x*j)/i). It is known that the zeros of f(x) are the prime numbers. So these are the numbers n such that f(n) > 0. - Michel Lagneau, Oct 13 2015
Numbers n that can be written as solutions of the Diophantine equation n = (x+2)(y+2) where {x,y} in N^2, pairs of natural numbers including zero (cf. Mathematica code and Davis). - Ron R Spencer and Bradley Klee, Aug 15 2016
Numbers n with a partition (containing at least two summands) so that its summands also multiply to n. If n is prime, there is no way to find those two (or more) summands. If n is composite, simply take a factor or several, write those divisors and fill with enough 1's so that they add up to n. For example: 4 = 2*2 = 2+2, 6 = 1*2*3 = 1+2+3, 8 = 1*1*2*4 = 1+1+2+4, 9 = 1*1*1*3*3 = 1+1+1+3+3. - Juhani Heino, Aug 02 2017

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
A. E. Bojarincev, Asymptotic expressions for the n-th composite number, Univ. Mat. Zap. 6:21-43 (1967). - In Russian.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 127.
Martin Davis, "Algorithms, Equations, and Logic", pp. 4-15 of S. Barry Cooper and Andrew Hodges, Eds., "The Once and Future Turing: Computing the World", Cambridge 2016.
R. K. Guy, Unsolved Problems Number Theory, Section A.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.
D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 66.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 51.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..17737 [composites up to 20000]
Rolf Brandl, Integer polynomials that are reducible modulo all primes, Amer. Math. Monthly 93 (1986), pp. 286-288.
C. K. Caldwell, Composite Numbers
Felix Huber, Illustration for a(1)-a(12)
Laurentiu Panaitopol, Some properties of the series of composed [sic] numbers, Journal of Inequalities in Pure and Applied Mathematics 2:3 (2001).
Carlos Rivera, Puzzle 76, z(n)=sigma(n) + phi(n) - 2n, The Prime Puzzles and Problems Connection.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 1962 64-94
Eric Weisstein's World of Mathematics, Composite Number
Index entries for "core" sequences
Index entries for sequences from "Goedel, Escher, Bach"

Complement of A008578. - Omar E. Pol, Dec 16 2016
Cf. A000040, A018252, A065090, A136527.
Cf. A073783 (first differences), A073445 (second differences).
Boustrophedon transforms: A230954, A230955.
Cf. A163870 (nontrivial divisors).
Related sequences:
Primes (p) and composites (c): A000040, A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

a002808 n = a002808_list !! (n-1)
a002808_list = filter ((== 1) . a066247) [2..]
-- Reinhard Zumkeller, Feb 04 2012

[n: n in [2..250] | not IsPrime(n)]; // G. C. Greubel, Feb 24 2024

t := []: for n from 2 to 20000 do if isprime(n) then else t := [op(t),n]; fi; od: t; remove(isprime,[$3..89]); # Zerinvary Lajos, Mar 19 2007
A002808 := proc(n) option remember ; local a ; if n = 1 then 4; else for a from procname(n-1)+1 do if not isprime(a) then return a; end if; end do ; end if; end proc; # R. J. Mathar, Oct 27 2009

Select[Range[2,100], !PrimeQ[#]&] (* Zak Seidov, Mar 05 2011 *)
With[{nn=100},Complement[Range[nn],Prime[Range[PrimePi[nn]]]]] (* Harvey P. Dale, May 01 2012 *)
Select[Range[100], CompositeQ] (* Jean-François Alcover, Nov 07 2021 *)

A002808(n)=for(k=0,primepi(n),isprime(n++)&&k--);n \\ For illustration only: see below. - M. F. Hasler, Oct 31 2008

A002808(n)= my(k=-1); while(-n + n += -k + k=primepi(n),); n \\ For n=10^4 resp. 3*10^4, this is about 100 resp. 500 times faster than the former; M. F. Hasler, Nov 11 2009

forcomposite(n=1, 1e2, print1(n, ", ")) \\ Felix Fröhlich, Aug 03 2014

for(n=1, 1e3, if(bigomega(n) > 1, print1(n, ", "))) \\ Altug Alkan, Oct 14 2015

from sympy import primepi
def A002808(n):
    m, k = n, primepi(n) + 1 + n
    while m != k:
        m, k = k, primepi(k) + 1 + n
    return m # Chai Wah Wu, Jul 15 2015, updated Apr 14 2016

from sympy import isprime
def ok(n): return n > 1 and not isprime(n)
print([k for k in range(89) if ok(k)]) # Michael S. Branicky, Nov 07 2021

next_A002808=lambda n: next(n for n in range(n,n*5)if not isprime(n)) # next composite >= n > 0; next_A002808(n)==n <=> iscomposite(n). - M. F. Hasler, Mar 28 2025
is_A002808=lambda n:not isprime(n) and n>1 # where isprime(n) can be replaced with: all(n%d for d in range(2, int(n**.5)+1))
# generators of composite numbers:
A002808_upto=lambda stop=1<<59: filter(is_A002808, range(2,stop))
A002808_seq=lambda:(q:=2)and(n for p in primes if (o:=q)<(q:=p) for n in range(o+1,p)) # with, e.g.: primes=filter(isprime,range(2,1<<59)) # M. F. Hasler, Mar 28 2025

[n for n in (2..250) if not is_prime(n)] # G. C. Greubel, Feb 24 2024

a(n) = pi(a(n)) + 1 + n, where pi is the prime counting function.
a(n) = A136527(n, n).
A000005(a(n)) > 2. - Juri-Stepan Gerasimov, Oct 17 2009
A001222(a(n)) > 1. - Juri-Stepan Gerasimov, Oct 30 2009
A000203(a(n)) < A007955(a(n)). - Juri-Stepan Gerasimov, Mar 17 2011
A066247(a(n)) = 1. - Reinhard Zumkeller, Feb 05 2012
Sum_{n>=1} 1/a(n)^s = Zeta(s)-1-P(s), where P is prime zeta. - Enrique Pérez Herrero, Aug 08 2012
n + n/log n + n/log^2 n < a(n) < n + n/log n + 3n/log^2 n for n >= 4, see Panaitopol. Bojarincev gives an asymptotic version. - Charles R Greathouse IV, Oct 23 2012
a(n) > n + A000720(n) + 1. - François Huppé, Jan 08 2025

Deleted an incomplete and broken link. - N. J. A. Sloane, Dec 16 2010

A375706 First differences of non-perfect-powers.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2

Offset: 1

Gus Wiseman, Aug 31 2024

nonn

Non-perfect-powers (A007916) are numbers without a proper integer root.

			The 5th non-perfect-power is 7, and the 6th is 10, so a(5) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..100000

For prime-powers (A000961) we have A057820.
For perfect powers (A001597) we have A053289.
For nonprime numbers (A002808) we have A073783.
For squarefree numbers (A005117) we have A076259.
First differences of A007916.
For nonsquarefree numbers (A013929) we have A078147.
For non-prime-powers (A024619) we have A375708.
Positions of 1s are A375740, complement A375714.
Runs of non-perfect-powers:
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (same as A045542 with 8 removed)
- sum: A375705
Cf. A000040, A046933, A052410, A303707, A305630, A305631, A323054, A323088, A323090, A375736.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Differences[Select[Range[100],radQ]]

up_to = 112;
A375706list(up_to) = { my(v=vector(up_to), pk=2, k=2, i=0); while(i<#v, k++; if(!ispower(k), i++; v[i] = k-pk; pk = k)); (v); };
v375706 = A375706list(up_to);
A375706(n) = v375706[n]; \\ Antti Karttunen, Jan 19 2025

from itertools import count
from sympy import mobius, integer_nthroot, perfect_power
def A375706(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x, k)[0]-1) for k in range(2, x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return next(i for i in count(m+1) if not perfect_power(i))-m # Chai Wah Wu, Sep 09 2024

a(n) = A007916(n+1) - A007916(n).

More terms from Antti Karttunen, Jan 19 2025

A027833 Distances between successive 2's in sequence A001223 of differences between consecutive primes.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 3, 6, 2, 5, 2, 6, 2, 2, 4, 3, 5, 3, 4, 5, 12, 2, 6, 9, 6, 5, 4, 3, 4, 20, 2, 2, 4, 4, 19, 2, 3, 2, 4, 8, 11, 5, 3, 3, 3, 10, 5, 4, 2, 17, 3, 6, 3, 3, 9, 9, 2, 6, 2, 6, 5, 6, 2, 3, 2, 3, 9, 4, 7, 3, 7, 20, 4, 7, 6, 5, 3, 7, 3, 20, 2, 14, 4, 10, 2, 3, 6, 4, 2, 2, 7, 2, 6, 3

Offset: 1

Jean-Marc MALASOMA (Malasoma(AT)entpe.fr)

nonn

a(n) = number of primes p such that A014574(n) < p < A014574(n+1). - Thomas Ordowski, Jul 20 2012
Conjecture: a(n) < log(A014574(n))^2. - Thomas Ordowski, Jul 21 2012
Conjecture: All positive integers are represented in this sequence. This is verified up to 184, by searching up to prime indexes of ~128000000. The rate of filling-in the smallest remaining gap among the integers, and the growth in the maximum value found, both slow down considerably relative to a fixed quantity of twin prime incidences examined in each pass. The maximum value found was 237. - Richard R. Forberg, Jul 28 2016
All positive integers below 312 are in this sequence. - Charles R Greathouse IV, Aug 01 2016
From Gus Wiseman, Jun 11 2024: (Start)
Also the length of the n-th maximal antirun of prime numbers > 3, where an antirun is an interval of positions at which consecutive terms differ by more than 2. These begin:
   5
   7  11
  13  17
  19  23  29
  31  37  41
  43  47  53  59
  61  67  71
  73  79  83  89  97 101
(End)

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

First differences of A029707 and A155752 = A029707 - 1. M. F. Hasler, Jul 24 2012
Positions of first appearances are A373401, sorted A373402.
Functional neighbors: A001359, A006512, A251092 or A175632, A373127 (firsts A373128, sorted A373200), A373403, A373405, A373409.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A025157, A038664, A071148,  A176246, A350842, A373406.

A027833 := proc(n)
    local plow,phigh ;
    phigh := A001359(n+1) ;
    plow := A001359(n) ;
    numtheory[pi](phigh)-numtheory[pi](plow) ;
end proc:
seq(A027833(n),n=1..100) ; # R. J. Mathar, Jan 20 2025

Differences[Flatten[Position[Differences[Prime[Range[500]]],2]]] (* Harvey P. Dale, Nov 17 2018 *)
Length/@Split[Select[Range[4,10000],PrimeQ[#]&],#1+2!=#2&]//Most (* Gus Wiseman, Jun 11 2024 *)

n=1; p=5; forprime(q=7,1e3, if(q-p==2, print1(n", "); n=1, n++); p=q) \\ Charles R Greathouse IV, Aug 01 2016

def A027833(n) :
   a = [ ]
   st = 2
   for i in (3..n) :
      if (nth_prime(i+1)-nth_prime(i) == 2) :
         a.append(i-st)
         st = i
   return(a)
A027833(496) # Jani Melik, May 15 2014

A375708 First differences of non-prime-powers (exclusive, so 1 is not a prime-power).

Original entry on oeis.org

5, 4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1

Offset: 1

Gus Wiseman, Aug 31 2024

nonn

Non-prime-powers (exclusive) are listed by A361102.

Warning: For this sequence, 1 is not a prime-power but is a non-prime-power.

			The 6th non-prime-power (exclusive) is 15, and the 7th is 18, so a(6) = 3.

For prime-powers (A000961, A246655) we have A057820, gaps A093555.
For perfect powers (A001597) we have A053289.
For nonprime numbers (A002808) we have A073783.
For squarefree numbers (A005117) we have A076259.
First differences of A361102, inclusive A024619.
Positions of 1's are A375713.
If 1 is considered a prime power we have A375735.
Runs of non-prime-powers:
- length: A110969
- first: A373676
- last: A373677
- sum: A373678
A000040 lists all of the primes, differences A001223.
A007916 lists non-perfect-powers, differences A375706.
A013929 lists the nonsquarefree numbers, differences A078147.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Non-prime-power antiruns: A373679, min A373575, max A255346, length A373672.
Cf. A046933, A061399, A176246, A251092, A375709.

Differences[Select[Range[100],!PrimePowerQ[#]&]]

from itertools import count
from sympy import primepi, integer_nthroot, primefactors
def A375708(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return next(i for i in count(m+1) if len(primefactors(i))>1)-m # Chai Wah Wu, Sep 09 2024

A073445 Second differences of A002808, the sequence of composites.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Aug 01 2002

			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)

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

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.

a073445 n = a073445_list !! (n-1)
a073445_list = zipWith (-) (tail a073783_list) a073783_list
-- Reinhard Zumkeller, Jan 10 2013

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x]; Table[c[w+2]-2*c[w+1]+c[w], {w, 200}]
(* second program *)
Differences[Select[Range[100],CompositeQ],2] (* Gus Wiseman, Oct 08 2024 *)

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

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

A095195 T(n,0) = prime(n), T(n,k) = T(n,k-1)-T(n-1,k-1), 0<=k

Original entry on oeis.org

2, 3, 1, 5, 2, 1, 7, 2, 0, -1, 11, 4, 2, 2, 3, 13, 2, -2, -4, -6, -9, 17, 4, 2, 4, 8, 14, 23, 19, 2, -2, -4, -8, -16, -30, -53, 23, 4, 2, 4, 8, 16, 32, 62, 115, 29, 6, 2, 0, -4, -12, -28, -60, -122, -237, 31, 2, -4, -6, -6, -2, 10, 38, 98, 220, 457, 37, 6, 4, 8, 14, 20, 22, 12

Offset: 1

Reinhard Zumkeller, Jun 22 2004

T(n,0)=A000040(n); T(n,1)=A001223(n-1) for n>1; T(n,2)=A036263(n-2) for n>2; T(n,n-1)=A007442(n) for n>1.

Row k of the array (not the triangle) is the k-th differences of the prime numbers. - Gus Wiseman, Jan 11 2025

			Triangle begins:
   2;
   3,  1;
   5,  2,  1;
   7,  2,  0, -1;
  11,  4,  2,  2,  3;
  13,  2, -2, -4, -6, -9;
Alternative: array form read by antidiagonals:
     2,   3,   5,   7,  11,  13,  17,  19,  23,  29,  31,...
     1,   2,   2,   4,   2,   4,   2,   4,   6,   2,   6,...
     1,   0,   2,  -2,   2,  -2,   2,   2,  -4,   4,  -2,...
    -1,   2,  -4,   4,  -4,   4,   0,  -6,   8,  -6,   0,...
     3,  -6,   8,  -8,   8,  -4,  -6,  14, -14,   6,   4,...
    -9,  14, -16,  16, -12,  -2,  20, -28,  20,  -2,  -8,...
    23, -30,  32, -28,  10,  22, -48,  48, -22,  -6,  10,..,
   -53,  62, -60,  38,  12, -70,  96, -70,  16,  16, -12,...
   115,-122,  98, -26, -82, 166,-166,  86,   0, -28,  28,...
  -237, 220,-124, -56, 248,-332, 252, -86, -28,  56, -98,...
   457,-344,  68, 304,-580, 584,-338,  58,  84,-154, 308,...

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A036262-A036271.
Cf. A140119 (row sums).
Below, the inclusive primes (A008578) are 1 followed by A000040. See also A075526.
Rows of the array (columns of the triangle) begin: A000040, A001223, A036263.
Column n = 1 of the array is A007442, inclusive A030016.
The version for partition numbers is A175804, see A053445, A281425, A320590.
First position of 0 is A376678, inclusive A376855.
Absolute antidiagonal-sums are A376681, inclusive A376684.
The inclusive version is A376682.
For composite instead of prime we have A377033, see A377034-A377037.
For squarefree instead of prime we have A377038, nonsquarefree A377046.
Column n = 2 of the array is A379542.
Cf. A002808, A064113, A065890, A073783, A258026, A293467, A333254, A376683, A377051.

a095195 n k = a095195_tabl !! (n-1) !! (k-1)
a095195_row n = a095195_tabl !! (n-1)
a095195_tabl = f a000040_list [] where
   f (p:ps) xs = ys : f ps ys where ys = scanl (-) p xs
-- Reinhard Zumkeller, Oct 10 2013

A095195A := proc(n,k) # array, k>=0, n>=0
    option remember;
    if n =0 then
        ithprime(k+1) ;
    else
        procname(n-1,k+1)-procname(n-1,k) ;
    end if;
end proc:
A095195 := proc(n,k) # triangle, 0<=k=1
        A095195A(k,n-k-1) ;
end proc: # R. J. Mathar, Sep 19 2013

T[n_, 0] := Prime[n]; T[n_, k_] /; 0 <= k < n := T[n, k] = T[n, k-1] - T[n-1, k-1]; Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Feb 01 2017 *)
nn=6;
t=Table[Differences[Prime[Range[nn]],k],{k,0,nn}];
Table[t[[j,i-j+1]],{i,nn},{j,i}] (* Gus Wiseman, Jan 11 2025 *)

A375735 First differences of non-prime-powers (inclusive).

Original entry on oeis.org

4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1

Offset: 1

Gus Wiseman, Sep 04 2024

nonn

Inclusive means 1 is a prime-power but not a non-prime-power.

Non-prime-powers (inclusive) are listed by A024619.

			The 5th non-prime-power (inclusive) is 15, and the 6th is 18, so a(5) = 3.

For perfect powers (A001597) we have the latter terms of A053289.
For nonprime numbers (A002808) we have the latter terms of A073783.
For squarefree numbers (A005117) we have the latter terms of A076259.
First differences of A024619.
For prime-powers (A246655) we have the latter terms of A057820.
Essentially the same as the exclusive version, A375708.
Positions of 1's are A375713(n) - 1.
For runs of non-prime-powers:
- length: A110969
- first: A373676
- last: A373677
- sum: A373678
A000040 lists all of the primes, first differences A001223.
A000961 lists prime-powers (inclusive).
A007916 lists non-perfect-powers, first differences A375706.
A013929 lists the nonsquarefree numbers, first differences A078147.
A246655 lists prime-powers (exclusive).
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Prime-power anti-runs: A373576, min A120430, max A006549, length A373671.
Non-prime-power anti-runs: A373679, min A373575, max A255346, len A373672.
Cf. A046933, A061399, A093555, A176246, A251092, A323054, A375709, A375736.

Differences[Select[Range[2,100],!PrimePowerQ[#]&]]

from itertools import count
from sympy import primepi, integer_nthroot, primefactors
def A375735(n):
    def f(x): return int(n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return next(i for i in count(m+1) if len(primefactors(i))>1)-m # Chai Wah Wu, Sep 10 2024

A065310 Number of occurrences of n-th prime in A065308, where A065308(j) = prime(j - pi(j)).

Original entry on oeis.org

3, 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, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2

Offset: 1

Labos Elemer, Oct 29 2001

nonn

Seems identical to A054546. Each odd prime arises once or twice!?

First differences of A018252 (positive nonprime numbers). Including 0 gives A054546. Removing 1 gives A073783. - Gus Wiseman, Sep 15 2024

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A000720, A054576, A065308-A065309.
For twin 2's see A169643.
Positions of 1's are A375926, complement A014689 (except first term).
Other families of numbers and their first-differences:
For prime numbers (A000040) we have A001223.
For composite numbers (A002808) we have A073783.
For nonprime numbers (A018252) we have A065310 (this).
For perfect powers (A001597) we have A053289.
For non-perfect-powers (A007916) we have A375706.
For squarefree numbers (A005117) we have A076259.
For nonsquarefree numbers (A013929) we have A078147.
For prime-powers inclusive (A000961) we have A057820.
For prime-powers exclusive (A246655) we have A057820(>1).
For non-prime-powers inclusive (A024619) we have A375735.
For non-prime-powers exclusive (A361102) we have A375708.
Cf. A046933, A053707, A054546, A110969, A176246, A251092, A373403, A375929.

t=Table[Prime[w-PrimePi[w]], {w, a, b}] Table[Count[t, Prime[n]], {n, c, d}]
Differences[Select[Range[100],!PrimeQ[#]&]] (* Gus Wiseman, Sep 15 2024 *)

{ p=1; f=2; m=1; for (n=1, 1000, a=0; p=nextprime(p + 1); while (p==f, a++; m++; f=prime(m - primepi(m))); write("b065310.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 16 2009

A073051 Least k such that Sum_{i=1..k} (prime(i) + prime(i+2) - 2*prime(i+1)) = 2n + 1.

Original entry on oeis.org

1, 3, 8, 23, 33, 45, 29, 281, 98, 153, 188, 262, 366, 428, 589, 737, 216, 1182, 3301, 2190, 1878, 1830, 7969, 3076, 3426, 2224, 3792, 8027, 4611, 4521, 3643, 8687, 14861, 12541, 15782, 3384, 34201, 19025, 17005, 44772, 23282, 38589, 14356

Offset: 1

Robert G. Wilson v, Aug 15 2002

nonn

Also, least k such that 2n = A001223(k-1) = prime(k+1) - prime(k), where prime(k) = A001223(n). - Alexander Adamchuk, Jul 30 2006

Also the least number k>0 such that the k-th maximal run of composite numbers has length 2n-1. For example, the 8th such run (24,25,26,27,28) is the first of length 2(3)-1, so a(3) = 8. Also positions of first appearances in A176246 (A046933 without first term). - Gus Wiseman, Jun 12 2024

			a(3) = 8 because 1+0+2-2+2-2+2+2 = 5 and (5+1)/2 = 3.

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Cf. A000230, A001223.
Position of first appearance of 2n+1 in A176246.
For nonsquarefree runs we have a bisection of A373199.
A000040 lists the primes, first differences A001223.
A002808 lists the composite numbers, differences A073783, sums A053767.
A046933 counts composite numbers between primes.
A065855 counts composite numbers up to n.
Cf. A005381, A027833, A038664, A045881, A068780, A174965, A371201, A373403.

NextPrim[n_Integer] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; a = Table[0, {50}]; s = 0; k = 1; p = 0; q = 2; r = 3; While[k < 10^6, p = q; q = r; r = NextPrim[q]; s = s + p + r - 2q; If[s < 101 && a[[(s + 1)/2]] == 0, a[[(s + 1)/2]] = k]; k++ ]; a

a001223(n) = prime(n+1) - prime(n);
a(n) = {my(k = 1); while(2*n != A001223(k+1), k++); k;} \\ Michel Marcus, Nov 20 2016

a(n) = A038664(n) - 1. - Filip Zaludek, Nov 19 2016

A377033 Array read by antidiagonals downward where A(n,k) is the n-th term of the k-th differences of the composite numbers (A002808).

Original entry on oeis.org

4, 6, 2, 8, 2, 0, 9, 1, -1, -1, 10, 1, 0, 1, 2, 12, 2, 1, 1, 0, -2, 14, 2, 0, -1, -2, -2, 0, 15, 1, -1, -1, 0, 2, 4, 4, 16, 1, 0, 1, 2, 2, 0, -4, -8, 18, 2, 1, 1, 0, -2, -4, -4, 0, 8, 20, 2, 0, -1, -2, -2, 0, 4, 8, 8, 0, 21, 1, -1, -1, 0, 2, 4, 4, 0, -8, -16, -16

Offset: 0

Gus Wiseman, Oct 17 2024

Row n is the k-th differences of A002808 = the composite numbers.

			Array begins:
        n=1:  n=2:  n=3:  n=4:  n=5:  n=6:  n=7:  n=8:  n=9:
  ----------------------------------------------------------
  k=0:   4     6     8     9    10    12    14    15    16
  k=1:   2     2     1     1     2     2     1     1     2
  k=2:   0    -1     0     1     0    -1     0     1     0
  k=3:  -1     1     1    -1    -1     1     1    -1    -1
  k=4:   2     0    -2     0     2     0    -2     0     2
  k=5:  -2    -2     2     2    -2    -2     2     2    -2
  k=6:   0     4     0    -4     0     4     0    -4    -1
  k=7:   4    -4    -4     4     4    -4    -4     3    10
  k=8:  -8     0     8     0    -8     0     7     7   -29
  k=9:   8     8    -8    -8     8     7     0   -36    63
Triangle begins:
    4
    6    2
    8    2    0
    9    1   -1   -1
   10    1    0    1    2
   12    2    1    1    0   -2
   14    2    0   -1   -2   -2    0
   15    1   -1   -1    0    2    4    4
   16    1    0    1    2    2    0   -4   -8
   18    2    1    1    0   -2   -4   -4    0    8
   20    2    0   -1   -2   -2    0    4    8    8    0
   21    1   -1   -1    0    2    4    4    0   -8  -16  -16

Initial rows: A002808, A073783, A073445.
The version for primes is A095195 or A376682.
A version for partitions is A175804, cf. A053445, A281425, A320590.
Triangle row-sums are A377034, absolute version A377035.
Column n = 1 is A377036, for primes A007442 or A030016.
First position of 0 in each row is A377037.
Other arrays of differences: A095195 (prime), A376682 (noncomposite), A377033 (composite), A377038 (squarefree), A377046 (nonsquarefree), A377051 (prime-power).
A000040 lists the primes, differences A001223, seconds A036263.
A008578 lists the noncomposites, differences A075526.
Cf. A065310, A065890, A084758, A173390, A350004, A376602 (zero), A376603 (nonzero), A376651 (positive), A376652 (negative), A376680.

nn=9;
t=Table[Take[Differences[NestList[NestWhile[#+1&, #+1,PrimeQ]&,4,2*nn],k],nn],{k,0,nn}]

A(i,j) = Sum_{k=0..j} (-1)^(j-k) binomial(j,k) A002808(i+k).

cp's OEIS Frontend

A002808 The composite numbers: numbers n of the form x*y for x > 1 and y > 1.

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A375706 First differences of non-perfect-powers.

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A027833 Distances between successive 2's in sequence A001223 of differences between consecutive primes.

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A375708 First differences of non-prime-powers (exclusive, so 1 is not a prime-power).

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A073445 Second differences of A002808, the sequence of composites.

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A095195 T(n,0) = prime(n), T(n,k) = T(n,k-1)-T(n-1,k-1), 0<=k

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