A065858 -id:A065858 - cp's OEIS frontend

A065134 Remainder when n is divided by the number of primes not exceeding n.

0, 1, 0, 2, 0, 3, 0, 1, 2, 1, 2, 1, 2, 3, 4, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 0, 1, 9, 0, 9, 10, 0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 6, 7, 5, 6, 7, 8, 9, 10, 8, 9, 7, 8, 9, 10, 11, 12, 10, 11, 12, 13, 11, 12, 10, 11, 12, 13, 14, 15, 13, 14, 15, 16, 14, 15, 16, 17, 18, 19, 17, 18, 19

Offset: 2

Labos Elemer, Oct 15 2001

nonn

Also remainder when the number of nonprimes is divided by the number of primes (not exceeding n).

			n = 2: Pi[2] = 1,Mod[1,1] = 0, the first term = a(2) = 0; n = 100: Pi[100] = 25, Mod[100,25] = 0 = a(100); n = 20: Pi[20] = 8, Mod[20,8] = 4 = a(20).

Harry J. Smith, Table of n, a(n) for n = 2..1000

Cf. A000720, A057809, A057810, A000040, A065134, A004648, A062298, A065858-A065864, A065133.

Table[Last@ QuotientRemainder[n, PrimePi[n]], {n, 2, 91}] (* Michael De Vlieger, Jul 04 2016 *)

{ for (n=2, 1000, write("b065134.txt", n, " ", n%primepi(n)) ) } \\ Harry J. Smith, Oct 11 2009

a(n) = n (mod pi(n)).

Term a(1) removed so OFFSET changed from 1,5 to 2,4 by Harry J. Smith, Oct 11 2009

Since OFFSET is 2,4; Term a(1) removed and a(91) added by Harry J. Smith, Oct 11 2009

A022449 c(p(n)) where p(k) is k-th prime including p(1)=1 and c(k) is k-th composite number.

Original entry on oeis.org

4, 6, 8, 10, 14, 20, 22, 27, 30, 35, 44, 46, 54, 58, 62, 66, 75, 82, 85, 92, 96, 99, 108, 114, 120, 129, 134, 136, 142, 144, 148, 166, 171, 178, 182, 194, 196, 204, 210, 215, 221, 230, 232, 245, 247, 252, 254, 268, 285, 289, 291, 296, 302, 304, 318

Offset: 1

Clark Kimberling

nonn

a(n) U A050435(n) = A002808(n), a(n+1) U A175251(n) = A002808(n) for n >= 1. a(n) = A065858(n-1) = composites (A002808) with prime (A000040) subscripts for n >=2. [From Jaroslav Krizek, Mar 13 2010]

			a(5) = 14 because a(5) = composite(noncomposite(5)) = composite(7) =14. _Jaroslav Krizek_, Mar 13 2010

C. Kimberling, Fractal sequences and interspersions, Ars Combinatoria, vol. 45 p 157 1997.

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

A065858 with a leading 4.

a022449 = a002808 . a008578
a022449_list = map a002808 a008578_list
-- Reinhard Zumkeller, Jan 12 2013

A022449 := proc(n)
    A002808(A008578(n)) ;
end proc:
seq(A022449(n),n=1..40) ; # R. J. Mathar, Jan 28 2014

p[1] = 1; p[n_] := Prime[n - 1];
Composite[n_] := FixedPoint[n + PrimePi[#] + 1 & , n + PrimePi[n] + 1];
a[n_] := Composite[p[n]];
Array[a, 100] (* Jean-François Alcover, Jan 26 2018, after Robert G. Wilson v *)

a(n) = A002808(A008578(n)). - Jaroslav Krizek, Mar 13 2010

Definition corrected by Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Mar 30 2005

A065864 Remainder when n is divided by the number of nonprimes not exceeding n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 21

Offset: 1

Labos Elemer, Nov 26 2001

nonn

			For n=100, pi(100)=25, so a(100) = 100 mod (100-25) = 25.

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

Cf. A000040, A004648, A062298, A065133, A065134, A065858-A065864.

Table[Mod[n, n - PrimePi@ n], {n, 78}] (* or *)
Table[Mod[n, Count[Range@ n, k_ /; ! PrimeQ@ k]], {n, 78}] (* Michael De Vlieger, Jan 01 2017 *)

{ for (n = 1, 1000, a=n%(n - primepi(n)); write("b065864.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 02 2009

a(n) = n mod (n-pi(n)) = n mod (n-A000720(n)) = n mod A062298(n).

A175251 Composites (A002808) with nonprime (A018252) subscripts.

Original entry on oeis.org

4, 9, 12, 15, 16, 18, 21, 24, 25, 26, 28, 32, 33, 34, 36, 38, 39, 40, 42, 45, 48, 49, 50, 51, 52, 55, 56, 57, 60, 63, 64, 65, 68, 69, 70, 72, 74, 76, 77, 78, 80, 81, 84, 86, 87, 88, 90, 91, 93, 94, 95, 98, 100

Offset: 1

Jaroslav Krizek, Mar 13 2010

nonn

a(n) = composite(nonprime(n)) = A002808(A018252(n)). a(n) U A065858(n) = A002808(n), a(n+1) U A022449(n) = A002808(n) for n >= 1. a(1) = 4, a(n) = A050435(n-1) = composites (A002808) with composite (A002808) subscripts for n >=2.

			a(5) = 16 because a(5) = c(b(5)) = c(9) = 16, c = composite, b = nonprime.

A065863 Remainder when n-th prime is divided by the number of nonprimes not exceeding n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 3, 3, 5, 1, 2, 6, 3, 2, 3, 9, 6, 1, 11, 8, 9, 13, 14, 1, 16, 13, 12, 14, 13, 7, 5, 5, 1, 5, 1, 7, 7, 5, 5, 11, 7, 17, 13, 11, 7, 19, 25, 23, 19, 17, 17, 19, 23, 23, 23, 23, 19, 25, 23, 25, 29, 37, 35, 31, 29, 43, 43, 47, 43, 47, 47, 3, 2, 1, 53, 53, 55, 2, 3, 6, 1, 11, 6

Offset: 1

Labos Elemer, Nov 26 2001

nonn

			For n=25, prime(25)=97, n - pi(n) = 25 - 9 = 16, a(25)=1 because 97 = 6*16 + 1.

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

Cf. A000040, A062298, A065858-A065864, A065134, A004648, A065133, A065134.

Table[Mod[Prime[n],n-PrimePi[n]],{n,90}] (* Harvey P. Dale, Aug 04 2015 *)

a(n) = { prime(n)%(n - primepi(n)) } \\ Harry J. Smith, Nov 02 2009

a(n) = prime(n) mod (n - pi(n)) = A000040(n) mod A062298(n).

A065862 Remainder when n-th composite number is divided by the number of nonprimes not exceeding n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 3, 1, 0, 2, 0, 1, 0, 7, 6, 7, 6, 8, 8, 7, 6, 7, 6, 6, 5, 4, 4, 6, 5, 6, 6, 5, 4, 3, 2, 4, 3, 2, 1, 2, 2, 4, 3, 2, 1, 2, 2, 1, 0, 0, 0, 1, 0, 38, 38, 39, 39, 40, 41, 42, 42, 42, 42, 43, 43, 44, 44, 44, 44, 45, 46, 47, 47, 48, 49, 49, 49, 51, 52, 52, 52, 54, 54, 54, 54, 54

Offset: 1

Labos Elemer, Nov 26 2001

nonn

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

Cf. A002808, A000720, A062298, A065858-A065864, A065134.

Module[{nn=150,cmps,len},cmps=Select[Range[nn],CompositeQ];len=Length[ cmps];Mod[#[[1]],#[[2]]-PrimePi[#[[2]]]]&/@Thread[{cmps,Range[len]}]] (* Harvey P. Dale, Feb 21 2020 *)

Composite(n) = { local(k); k=n + primepi(n) + 1; while (k != n + primepi(k) + 1, k = n + primepi(k) + 1); return(k) } { for (n = 1, 1000, a=Composite(n)%(n - primepi(n)); write("b065862.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 02 2009

a(n) = c(n) mod (n - pi(n)) = A002808(n) mod (n - A000720(n)) = A002808(n) mod A062298(n).

A145351 Prime-indexed composites k such that lpf(k) + gpf(k) is a prime.

Original entry on oeis.org

6, 10, 20, 22, 30, 44, 54, 58, 66, 82, 96, 108, 120, 136, 142, 144, 204, 232, 324, 330, 340, 352, 384, 464, 492, 544, 596, 616, 704, 738, 750, 792, 870, 894, 918, 960, 990, 1062, 1234, 1312, 1318, 1326, 1498, 1534, 1540, 1566, 1576, 1632, 1694, 1700, 1722

Offset: 1

Juri-Stepan Gerasimov, Jan 04 2009

nonn

			6 is a term because it is the 2nd composite number, 6=2*3, and 2+3=5 is prime;
10 is a term because it is the 5th composite number, 10=2*5, and 2+5=7 is prime;
22 is a term because it is the 13th composite number, 22=2*11, and 2+11=13 is prime;
44 is a term because it is the 29th composite number, 44=2*2*11, and 2+11=13 is prime.

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

Cf. A000040, A002808, A020639 (lpf), A006530 (gpf).

A020639 := proc(n) numtheory[factorset](n) ; min(op(%)) ; end proc:
A006530 := proc(n) numtheory[factorset](n) ; max(op(%)) ; end proc:
A002808 := proc(n) 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:
A065858 := proc(n) A002808(ithprime(n)) ; end proc:
A145351 := proc(n) c := A065858(n) ; if isprime(A020639(c) + A006530(c)) then printf("%d,",c) ; end if; end proc:
seq(A145351(n),n=1..400) ; # R. J. Mathar, May 01 2010

pfiQ[n_]:=Module[{f=FactorInteger[n]},PrimeQ[f[[1,1]]+f[[-1,1]]]]; Module[ {nn=2000,c},c=Select[ Range[nn],CompositeQ];Select[ Table[ Take[c,{n}][[1]],{n,Prime[Range[PrimePi[Length[c]]]]}],pfiQ]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 18 2019 *)

Corrected (inserted 20 from n=5, 30 from n=8, removed 200) and extended beyond 204 by R. J. Mathar, May 01 2010

Edited by Jon E. Schoenfield, Feb 07 2019

A177997 p and A002808(p)/2 are both prime.

Original entry on oeis.org

2, 5, 7, 13, 31, 41, 43, 59, 101, 107, 127, 137, 149, 199, 239, 277, 359, 389, 479, 613, 743, 757, 809, 829, 937, 991, 1031, 1103, 1439, 1487, 1499, 1847, 1877, 2011, 2083, 2179, 2609, 2663, 2711, 2741, 2749, 2857, 2909, 3329, 3559, 3623, 3643, 3697, 3823

Offset: 1

Juri-Stepan Gerasimov, May 17 2010, May 23 2010

nonn

Primes p such that composite(p) is an even semiprime.

			a(1)=2 because 2=prime and composite(2)/2=6/2=3=prime.

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

Cf. A002808, A065858, A065897.

P,C:= selectremove(isprime, [$2..10000]):
select(t -> t <= nops(C) and C[t]::even and isprime(C[t]/2), P); # Robert Israel, Mar 20 2018

c=[]; for(n=2, 10000, if(!isprime(n), c=concat(c, n))); c; \\ The composites
s=[]; forprime(p=2, #c, if(c[p]%2==0 && isprime(c[p]\2), s=concat(s, p))); s \\ Colin Barker, Jun 28 2014

Corrected by D. S. McNeil and R. J. Mathar, May 23 2010

A260621 Let b(k, n) = number obtained when the map x->A002808(x) is applied k times to n; a(n) is the smallest k such that b(k, n) + 1 is prime.

Original entry on oeis.org

1, 1, 12, 2, 1, 1, 3, 11, 1, 1, 7, 9, 1, 2, 10, 4, 2, 1, 1, 6, 8, 3, 3, 1, 9, 3, 1, 1, 18, 3, 1, 5, 7, 2, 2, 1, 4, 8, 2, 14, 1, 1, 6, 17, 2, 6, 1, 4, 6, 1, 1, 2, 2, 3, 7, 1, 13, 6, 1, 4, 16, 5, 16, 1, 5, 31, 35, 3, 5, 2, 1, 2, 3, 1, 1, 2, 6, 1, 1, 12, 5, 1, 2

Offset: 1

Matthew Campbell, Sep 25 2015

nonn

a(n) is also the smallest value of k at which b(k, n+1) - b(k, n) > 1.

			When n = 3, writing Composite(x) for A002808(x):
1. Composite(3) = 8. 8 + 1 = 9 = 3^2. 9 is not prime.
2. Composite(8) = 15. 15 + 1 = 16 = 2^4. 16 is not prime.
3. Composite(15) = 25. 25 + 1 = 26 = 2*13. 26 is not prime.
4. Composite(25) = 38. 38 + 1 = 39 = 3*13. 39 is not prime.
5. Composite(38) = 55. 55 + 1 = 56 = 2^3*7. 56 is not prime.
6. Composite(55) = 77. 77 + 1 = 78 = 2*3*13. 78 is not prime.
7. Composite(77) = 105. 105 + 1 = 106 = 2*53. 106 is not prime.
8. Composite(105) = 140. 140 + 1 = 141 = 3*47. 141 is not prime.
9. Composite(140) = 183. 183 + 1 = 184 = 2^3*23. 184 is not prime.
10. Composite(183) = 235. 235 + 1 = 236 = 2^2*59. 236 is not prime.
11. Composite(235) = 298. 298 + 1 = 299 = 13*23. 299 is not prime.
12. Composite(298) = 372. 372 + 1 = 373. 373 is prime.
--------------------------------------------------------------
Since the composite function was applied 12 times, a(3)=12.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Primes and nonprimes: A000040, A002808, A008578, A018252.
a(1) = p, a(n+1) = a(n)-th composite number: A006508, A022450, A022451, A025010, A025011, A059407, A059408.
Composites with order n > 1: A050435, A050436, A050438, A050439, A050440.
Composites with order n = b, n >= 1: A022449.
Composites with prime subscripts: A065858.
Composites without prime subscripts: A175251.
Order of compositeness: A059981, A236536.
Prime(n)-1: A006093.

c = Select[Range[10^5], CompositeQ]; Table[k = 1; While[! PrimeQ[Nest[c[[#]] &, n, k] + 1], k++]; k, {n, 120}] (* Michael De Vlieger, Jul 15 2016 *)

Terms from a(12) onward from Jon E. Schoenfield, Sep 27 2015

cp's OEIS Frontend

A065134 Remainder when n is divided by the number of primes not exceeding n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A022449 c(p(n)) where p(k) is k-th prime including p(1)=1 and c(k) is k-th composite number.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

Extensions

A065864 Remainder when n is divided by the number of nonprimes not exceeding n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A175251 Composites (A002808) with nonprime (A018252) subscripts.

Views

Author

Keywords

Comments

Examples

A065863 Remainder when n-th prime is divided by the number of nonprimes not exceeding n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A065862 Remainder when n-th composite number is divided by the number of nonprimes not exceeding n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A145351 Prime-indexed composites k such that lpf(k) + gpf(k) is a prime.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A177997 p and A002808(p)/2 are both prime.

Views