A141468 -id:A141468 - cp's OEIS frontend

A018252 The nonprime numbers: 1 together with the composite numbers, A002808.

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

d(a(n)) != 2 (cf. A000005). - Juri-Stepan Gerasimov, Oct 17 2009
Number of prime divisors of a(n) (counted with multiplicity) != 1. - Juri-Stepan Gerasimov, Oct 30 2009
Largest nonprime < n-th composite. - Juri-Stepan Gerasimov, Oct 29 2009
The nonnegative nonprimes A141468 without zero; the natural nonprimes; the whole nonprimes; the counting nonprimes. If the nonprime numbers A141468 which are also the nonnegative integers A001477, then the nonprimes A141468 also called the nonnegative nonprimes. If the nonprime numbers A018252 which are also the natural (or whole or counting) numbers A000027, then the nonprimes A018252 also called the natural nonprimes, the whole nonprimes and the counting nonprimes. - Juri-Stepan Gerasimov, Nov 22 2009
Smallest nonprime > n-th nonnegative nonprime. - Juri-Stepan Gerasimov, Dec 04 2009
a(n) = A175944(A014284(n)) = A175944(A175965(n)). - Reinhard Zumkeller, Mar 18 2011

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.

N. J. A. Sloane, List of nonprimes up to 20000: Table of n, a(n) for n = 1..17738
Eric Weisstein's World of Mathematics, Monica Set
Eric Weisstein's World of Mathematics, Suzanne Set
Index entries for "core" sequences

Cf. A000040 (complement), A002808.
Cf. A000005, A001222, A141468.
Boustrophedon transforms: A230955, A230954.
Cf. A010051, A239968.

A018252 := Difference([1..10^5], Filtered([1..10^5], IsPrime)); # Muniru A Asiru, Oct 21 2017

a018252 n = a018252_list !! (n-1)
a018252_list = filter ((== 0) . a010051) [1..]
-- Reinhard Zumkeller, Mar 31 2014

Magma
```
[n : n in [1..100] | not IsPrime(n) ];
```

with(numtheory); sort(convert(convert([ seq(i,i=1..541) ],set) minus convert([ seq(ithprime(i),i=1..100) ],set),list));
seq(`if`(not isprime(n),n,NULL),n=1..88); # Peter Luschny, Jul 29 2009
A018252 := proc(n) option remember; if n = 1 then 1; 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 22 2010

nonPrime[n_Integer] := FixedPoint[n + PrimePi@# &, n + PrimePi@ n]; Array[ nonPrime, 75] (* Robert G. Wilson v, Jan 29 2015, based on the algorithm by Labos Elemer in A006508 *)
max = 90; Complement[Range[max], Prime[Range[PrimePi[max]]]] (* Harvey P. Dale, Aug 12 2011 *)
Join[{1}, Select[Range[100], CompositeQ]] (* Jean-François Alcover, Nov 07 2021 *)

isA018252(n) = !isprime(n)
A018252(n) = {local(a,b);b=n;a=1;while(a!=b,a=b;b=n+primepi(a));b} \\ Michael B. Porter, Nov 06 2009

a(n) = my(k=0); while(-n+n-=k-k=primepi(n), ); n; \\ Ruud H.G. van Tol, Jul 15 2024 (after code in A002808)

from sympy import isprime
def ok(n): return not isprime(n)
print([k for k in range(1, 89) if ok(k)]) # Michael S. Branicky, Nov 10 2022

from sympy import composite
def A018252(n): return 1 if n == 1 else composite(n-1) # Chai Wah Wu, Nov 15 2022

def A018252_list(n) :
    return [k for k in (1..n) if not k.is_prime()]
A018252_list(88)  # Peter Luschny, Feb 03 2012

Let b(0) = n + pi(n) and b(n+1) = n + pi(b(n)), with pi(n) = A000720(n); then a(n) is the limit value of b(n). - Floor van Lamoen, Oct 08 2001
a(n) = A137621(A137624(n)). - Reinhard Zumkeller, Jan 30 2008
A010051(a(n)) = 0. - Reinhard Zumkeller, Mar 31 2014
A239968(a(n)) = n. - Reinhard Zumkeller, Dec 02 2014

A008864 a(n) = prime(n) + 1.

Original entry on oeis.org

3, 4, 6, 8, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 54, 60, 62, 68, 72, 74, 80, 84, 90, 98, 102, 104, 108, 110, 114, 128, 132, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 258, 264, 270, 272, 278, 282, 284

Offset: 1

N. J. A. Sloane, R. K. Guy

Sum of divisors of prime(n). - Labos Elemer, May 24 2001
For n > 1, there are a(n) more nonnegative Hurwitz quaternions than nonnegative Lipschitz quaternions, which are counted in A239396 and A239394, respectively. - T. D. Noe, Mar 31 2014
These are the numbers which are in A239708 or in A187813, but excluding the first 3 terms of A187813, i.e., a number m is a term if and only if m is a term > 2 of A187813, or m is the sum of two distinct powers of 2 such that m - 1 is prime. This means that a number m is a term if and only if m is a term > 2 such that there is no base b with a base-b digital sum of b, or b = 2 is the only base for which the base-b digital sum of m is b. a(6) is the only term such that a(n) = A187813(n); for n < 6, we have a(n) > A187813(n), and for n > 6, we have a(n) < A187813(n). - Hieronymus Fischer, Apr 10 2014
Does not contain any number of the format 1 + q + ... + q^e, q prime, e >= 2, i.e., no terms of A060800,  A131991, A131992, A131993 etc. [Proof: that requires 1 + p = 1 + q + ... + q^e, or p = q*(1 + ... + q^(e-1)). This is not solvable because the left hand side is prime, the right hand side composite.] - R. J. Mathar, Mar 15 2018
1/a(n) is the asymptotic density of numbers whose prime(n)-adic valuation is odd. - Amiram Eldar, Jan 23 2021

C. W. Trigg, Problem #1210, Series Formation, J. Rec. Math., 15 (1982), 221-222.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
R. P. Boas and N. J. A. Sloane, Correspondence, 1974
N. J. A. Sloane and Brady Haran, Eureka Sequences, Numberphile video (2021).

Cf. A000040, A060800, A131991, A131992, A131993, A141468.
Cf. A007953, A079696, A187813, A239703, A239708.
Column 1 of A341605, column 2 of A286623 and of A328464.
Partial sums of A125266.

a008864 = (+ 1) . a000040
-- Reinhard Zumkeller, Sep 04 2012, Oct 08 2012

[NthPrime(n)+1: n in [1..70]]; // Vincenzo Librandi, Jul 30 2016

A008864:=n->ithprime(n)+1; seq(A008864(n), n=1..50); # Wesley Ivan Hurt, Apr 11 2014

Prime[Range[70]]+1 (* Vladimir Joseph Stephan Orlovsky, Apr 27 2008 *)

forprime(p=2,1e3,print1(p+1", ")) \\ Charles R Greathouse IV, Jun 16 2011

A008864(n) = (1+prime(n)); \\ Antti Karttunen, Mar 14 2021

[nth_prime(n) +1 for n in (1..70)] # G. C. Greubel, May 20 2019

a(n) = prime(n) + 1 = A000040(n) + 1.
a(n) = A000005(A034785(n)) = A000203(A000040(n)). - Labos Elemer, May 24 2001
a(n) = A084920(n) / A006093(n). - Reinhard Zumkeller, Aug 06 2007
A239703(a(n)) <= 1. - Hieronymus Fischer, Apr 10 2014
From Ilya Gutkovskiy, Jul 30 2016: (Start)
a(n) ~ n*log(n).
Product_{n>=1} (1 + 2/(a(n)*(a(n) - 2))) = 5/2. (End)

A158611 0, 1 and the primes.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

Jaroslav Krizek, Mar 22 2009

Essentially a duplicate of A008578. - N. J. A. Sloane, Mar 27 2009
Or, nonnegative noncomposite numbers. - Juri-Stepan Gerasimov, Nov 22 2009, Dec 12 2009
Numbers with all divisors in arithmetic progression. - Charles R Greathouse IV, Oct 19 2015

A158611 U A002808 = A141468 U A000040 = A001477. - Juri-Stepan Gerasimov, Sep 27 2009

Cf. A008578.

A158611:=Concatenation([0,1],Filtered([1..10^5],IsPrime)); # Muniru A Asiru, Sep 05 2017

[0,1] cat [NthPrime(n): n in [1..100]]; // Vincenzo Librandi, Dec 08 2015

Join[{0, 1}, Table[Prime[n], {n, 400}]] (* Vladimir Joseph Stephan Orlovsky, Feb 04 2012 *)

a(n)=if(n<3,n-1,prime(n-2)) \\ Charles R Greathouse IV, Aug 26 2011

a(n) = A008578(n-1), n > 1.

Edited by R. J. Mathar, Aug 21 2009

Further edited by R. J. Mathar, Klaus Brockhaus and N. J. A. Sloane, Sep 13 2009

A051349 Sum of first n nonprimes.

Original entry on oeis.org

0, 1, 5, 11, 19, 28, 38, 50, 64, 79, 95, 113, 133, 154, 176, 200, 225, 251, 278, 306, 336, 368, 401, 435, 470, 506, 544, 583, 623, 665, 709, 754, 800, 848, 897, 947, 998, 1050, 1104, 1159, 1215, 1272, 1330, 1390, 1452, 1515, 1579, 1644, 1710, 1778, 1847, 1917

Offset: 0

Armand Turpel (armandt(AT)unforgettable.com)

Partial sums of A141468 or A018252. - R. J. Mathar, Mar 01 2009

The lexicographically earliest sequence with first differences as increasing sequence of composites A002808. Complement of A175970. See A175965, A175966, A175967, A014284, A175969, A175970. - Jaroslav Krizek, Oct 31 2010

Nathaniel Johnston, Table of n, a(n) for n = 0..10000

Cf. A014284, A018252, A122998, A141468, A175965, A175966, A175967, A175969, A175970.

ithnonprime := proc(n)local k: option remember: if(n=1)then return 1: fi: for k from procname(n-1)+1 do if(not isprime(k))then return k fi: od: end: A051349 := proc(n) option remember: local k: if(n<=1)then return n: fi: return ithnonprime(n)+procname(n-1): end: seq(A051349(n),n=0..51); # Nathaniel Johnston, May 25 2011

lst={};s=0;Do[If[ !PrimeQ[n], s=s+n;AppendTo[lst, s]], {n, 0, 10^2}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 14 2008 *)

Sum_{n>=1} 1/a(n) = A122998. - Amiram Eldar, Nov 17 2020

A163300 Even numbers without 2.

Original entry on oeis.org

0, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132

Offset: 1

Juri-Stepan Gerasimov, Jul 26 2009

Zero together with the even nonprimes. - Omar E. Pol, Aug 04 2009
A163300 U A014076 = A141468.
The nonprime numbers (A018252) begin: 1,4,6,8,9,10,12,14,15,... So the words "prime" and "nonprime" normally refer to the natural numbers or positive integers: 1,2,3,4,5,6,... (Zero is not a member of A018252. See also the definition of A141468). - Omar E. Pol, Aug 04 2009

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A087156, A014076, A141468, A018252.

Drop[Range[0,132,2],{2}] (* Harvey P. Dale, May 05 2012 *)

a(n)=if(n>1,2*n,0) \\ Charles R Greathouse IV, Apr 17 2024

def A163300(n): return n<<1 if n>1 else 0 # Chai Wah Wu, Jul 31 2024

a(n) = 2*A087156(n).

New definition from Charles R Greathouse IV, Jun 23 2024

A014692 a(n) = prime(n) - (n-1).

Original entry on oeis.org

2, 2, 3, 4, 7, 8, 11, 12, 15, 20, 21, 26, 29, 30, 33, 38, 43, 44, 49, 52, 53, 58, 61, 66, 73, 76, 77, 80, 81, 84, 97, 100, 105, 106, 115, 116, 121, 126, 129, 134, 139, 140, 149, 150, 153, 154, 165, 176, 179, 180, 183, 188, 189, 198, 203, 208, 213, 214, 219, 222, 223

Offset: 1

Mohammad K. Azarian

Also, number of primes between prime(n) and prime(prime(n)) inclusive. For example, prime(1) = 2, prime(prime(1)) = prime(2) = 3 and there are two primes between 2 and 3 inclusive. - Zak Seidov, Sep 22 2003; N. J. A. Sloane, Mar 07 2007
Since a(n+1) - a(n) = prime(n+1) - prime(n) - 1 >= 1 for n > 1, the sequence is monotonic for n > 1. - N. J. A. Sloane, Mar 07 2007
a(n) = number of terms < prime(n) in A141468. - David James Sycamore, Oct 14 2017

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000040, A141468.

Equals A014689 + 1.

A014692:=n->ithprime(n)-(n-1): seq(A014692(n), n=1..100); # Wesley Ivan Hurt, Oct 15 2017

Table[Prime[n] - n + 1, {n, 61}]  (* Geoffrey Critzer, May 02 2013 *)

first(n) = {my(t, res = vector(n)); forprime(p=2, , t++; res[t] = p - t + 1; if(t>=n, return(res)))} \\ David A. Corneth, Oct 04 2017

a(n) = prime(n)-n+1; \\ Altug Alkan, Oct 05 2017

from sympy import prime
def A014692(n): return prime(n)-n+1 # Chai Wah Wu, Oct 11 2024

More terms from Andrew J. Gacek (andrew(AT)dgi.net)

A164276 The non-isolated nonprimes.

Original entry on oeis.org

0, 1, 8, 9, 10, 14, 15, 16, 20, 21, 22, 24, 25, 26, 27, 28, 32, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 64, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104, 105, 106, 110

Offset: 1

Juri-Stepan Gerasimov, Aug 11 2009

Non-isolated nonprimes in the sense that at least one of the two adjacent integers is also a nonprime.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A014574, A141468.

Select[Range[0,100], !PrimeQ[#] && (!PrimeQ[# - 1] || !PrimeQ[# + 1]) & ] (* G. C. Greubel, Jul 07 2016 *)

isA164276(n) = !isprime(n)&&(!isprime(n-1)||!isprime(n+1)) \\ Michael B. Porter, Feb 02 2010

A141468 \ A014574.

a(n) = n + n / log n + O(n / (log n)^2) by Brun's theorem. - Charles R Greathouse IV, Mar 15 2011

40 added by R. J. Mathar, Aug 27 2009

A054546 First differences of nonprimes (including 0 and 1, A002808).

Original entry on oeis.org

1, 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

Offset: 1

G. L. Honaker, Jr., Apr 09 2000

Sum of first n terms equals n-th nonprime number.

First differences of A141468. - Omar E. Pol, Oct 21 2011

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A141468, A066110, A001223

t=Flatten[Position[Table[PrimeQ[w], {w, 2, 256}], False]]+1 Delete[t-RotateRight[t], 1]
Differences[Select[Range[0,200],!PrimeQ[#]&]] (* Harvey P. Dale, May 27 2018 *)

a(n) = A018252(n) - A141468(n). - Omar E. Pol, Oct 21 2011

More terms from James Sellers, Apr 11 2000

A182986 Zero together with the prime numbers (A000040).

Original entry on oeis.org

0, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269

Offset: 1

Charles R Greathouse IV, Feb 05 2011

These numbers are the possible characteristics of a field.
First differences are in A054541. - Omar E. Pol, Oct 31 2013
Also A158611 without its second term. - Omar E. Pol, Nov 01 2013
The ideals generated by a(n) form Spec(Z), the set of prime ideals of the ring of integers. Due to its importance in algebraic geometry, algebraic geometers often consider 0 to be an honorary prime. - Keith J. Bauer, Jan 09 2024

Cf. A141468.

Complement of A018252. - Arkadiusz Wesolowski, Sep 15 2011

Prepend[Table[Prime[n], {n, 60}], 0] (* Arkadiusz Wesolowski, Sep 15 2011 *)
NestList[ NextPrime, 0, 57] (* Robert G. Wilson v, Jul 21 2015 *)

a(n)=if(n>1,prime(n-1),0) \\ Charles R Greathouse IV, Jul 15 2013

A144291 Triangular numbers n*(n-1)/2 with n and n -1 nonprime.

Original entry on oeis.org

0, 36, 45, 105, 120, 210, 231, 300, 325, 351, 378, 528, 561, 595, 630, 741, 780, 990, 1035, 1176, 1225, 1275, 1326, 1485, 1540, 1596, 1653, 1953, 2016, 2080, 2145, 2346, 2415, 2775, 2850, 2926, 3003, 3240, 3321, 3570, 3655, 3741, 3828, 4095, 4186, 4278

Offset: 1

Juri-Stepan Gerasimov, Dec 01 2008

nonn

			If n=1, then 1*(1-1)/2=0=a(1).
If n=9, then 9*(9-1)/2=36=a(2).
etc.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A000217, A068780, A141468.

p:= -1: Res:= NULL: count:= 0:
while count < 100 do
  q:= p; p:= nextprime(p);
  if p - q > 2 then
    count:= count + p-q-2;
    Res:= Res, seq(k*(k+1)/2, k=q+1..p-2);
  fi
od:
Res; # Robert Israel, Jul 03 2018

Reap[For[n = 1, n <= 100, n++, If[!PrimeQ[n] && !PrimeQ[n-1], Sow[n(n-1)/2] ] ] ][[2, 1]] (* Jean-François Alcover, Mar 27 2019 *)

a(n) = A000217(A068780(n-1)), n>1. - R. J. Mathar, Dec 10 2008

3570 inserted by R. J. Mathar, Dec 10 2008

cp's OEIS Frontend

A018252 The nonprime numbers: 1 together with the composite numbers, A002808.

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A008864 a(n) = prime(n) + 1.

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A158611 0, 1 and the primes.

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A051349 Sum of first n nonprimes.

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A163300 Even numbers without 2.

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A014692 a(n) = prime(n) - (n-1).

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