A066246 -id:A066246 - cp's OEIS frontend

A169643 Numbers k such that neither composite(k)+-1 is composite.

1, 2, 6, 10, 19, 28, 42, 51, 75, 79, 104, 114, 138, 148, 152, 178, 187, 212, 221, 247, 278, 338, 348, 372, 423, 465, 490, 504, 525, 539, 669, 679, 683, 709, 729, 848, 858, 873, 883, 909, 961, 1028, 1071, 1080, 1089, 1104, 1202, 1221, 1247, 1251, 1354, 1363

Offset: 1

Juri-Stepan Gerasimov, Apr 04 2010

nonn

a(n) = A066246(A014574(n)). - Reinhard Zumkeller, Apr 06 2010

			a(1)=1 because composite(1)-1=3=prime and composite(1)+1=5=prime.

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

Cf. A002808, A014574, A066246.

Comp:= remove(isprime,[$4..2000]): nC:= nops(Comp):
select(t -> isprime(Comp[t]-1) and isprime(Comp[t]+1), [$1..nC]); # Robert Israel, Oct 21 2024

Position[Select[Range[2000],CompositeQ],?(AllTrue[#+{1,-1},PrimeQ]&),1,Heads->False]// Flatten (* _Harvey P. Dale, Jan 16 2024 *)

Corrected by Ray Chandler, Apr 05 2010

A139636 If n = the k-th prime, then a(n) = the (k+1)th prime. If n = the k-th composite, then a(n) = the (k+1)th composite.

Original entry on oeis.org

3, 5, 6, 7, 8, 11, 9, 10, 12, 13, 14, 17, 15, 16, 18, 19, 20, 23, 21, 22, 24, 29, 25, 26, 27, 28, 30, 31, 32, 37, 33, 34, 35, 36, 38, 41, 39, 40, 42, 43, 44, 47, 45, 46, 48, 53, 49, 50, 51, 52, 54, 59, 55, 56, 57, 58, 60, 61, 62, 67, 63, 64, 65, 66, 68, 71, 69, 70, 72, 73, 74

Offset: 2

Leroy Quet, Apr 28 2008

nonn

This is a permutation of the positive integers sans 1,2,4.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A139637.

A000040 := proc(n) ithprime(n) ; end: A002808 := proc(n) local a; if n = 1 then 4; else for a from A002808(n-1)+1 do if not isprime(a) then RETURN(a) ; fi ; od: fi ; end: A066246 := proc(n) local k ; if isprime(n) then 0 ; else for k from 1 do if A002808(k) = n then RETURN(k) ; fi ; od: fi ; end: A049084 := proc(n) if not isprime(n) then 0; else numtheory[pi](n) ; fi ; end: A139636 := proc(n) local k; if isprime(n) then k := A049084(n) ; RETURN(A000040(k+1)) ; else k := A066246(n) ; RETURN(A002808(k+1)) ; fi ; end: seq(A139636(n),n=2..160) ; # R. J. Mathar, May 12 2008

npc[n_]:=Module[{k=1},If[PrimeQ[n],NextPrime[n],While[PrimeQ[n+k],k++];n+k]]; Array[npc,80,2] (* Harvey P. Dale, Jun 26 2022 *)

More terms from R. J. Mathar, May 12 2008

A139637 If n = the k-th prime, then a(n) = the (k-1)th prime. If n = the k-th composite, then a(n) = the (k-1)th composite. a(2) = 1. a(4) = 0.

Original entry on oeis.org

1, 2, 0, 3, 4, 5, 6, 8, 9, 7, 10, 11, 12, 14, 15, 13, 16, 17, 18, 20, 21, 19, 22, 24, 25, 26, 27, 23, 28, 29, 30, 32, 33, 34, 35, 31, 36, 38, 39, 37, 40, 41, 42, 44, 45, 43, 46, 48, 49, 50, 51, 47, 52, 54, 55, 56, 57, 53, 58, 59, 60, 62, 63, 64, 65, 61, 66, 68, 69, 67, 70, 71, 72

Offset: 2

Leroy Quet, Apr 28 2008

nonn

This is a permutation of the nonnegative integers.

Cf. A000040, A002808, A139636.

A000040 := proc(n) ithprime(n) ; end: A002808 := proc(n) local a; if n = 1 then 4; else for a from A002808(n-1)+1 do if not isprime(a) then RETURN(a) ; fi ; od: fi ; end: A066246 := proc(n) local k ; if isprime(n) then 0 ; else for k from 1 do if A002808(k) = n then RETURN(k) ; fi ; od: fi ; end: A049084 := proc(n) if not isprime(n) then 0; else numtheory[pi](n) ; fi ; end: A139637 := proc(n) local k; if n = 2 then 1; elif n = 4 then 0 ; else if isprime(n) then k := A049084(n) ; A000040(k-1) ; else k := A066246(n) ; A002808(k-1) ; fi ; fi ; end: seq(A139637(n),n=2..160) ; # R. J. Mathar, May 12 2008

HC[k_] := If[k <= 4, None, Max[Select[Range[4, k-1], !PrimeQ[#]&]]];a[2]=1;a[4]=0;a[n_]:=If[PrimeQ[n],Prime[PrimePi[n-1]],HC[n]];Array[a,73,2] (* James C. McMahon, Jul 01 2025 *)

More terms from R. J. Mathar, May 12 2008

A133555 Order of A113709(n) among composite positive integers.

Original entry on oeis.org

1, 2, 3, 6, 9, 10, 11, 14, 19, 24, 27, 28, 29, 32, 37, 42, 47, 48, 51, 56, 57, 60, 71, 74, 75, 76, 79, 82, 95, 96, 99, 104, 105, 114, 119, 124, 125, 128, 133, 138, 147, 148, 151, 152, 157, 168, 175, 178, 181, 182, 187, 196, 197, 202, 207, 212, 217, 220, 221, 228, 231

Offset: 2

Leroy Quet, Dec 25 2007

nonn

			The 10th prime - the 9th prime = 29-23 = 6. The integer between 23 and 29 that is divisible by 6 is 24. 24 is the 14th composite, so a(9) = 14.

Cf. A066246, A113709.

A113709 := proc(n) local d,a ; d := ithprime(n+1)-ithprime(n) ; for a from ithprime(n)+1 do if a mod d = 0 then RETURN(a) ; fi ; od: end: A066246 := proc(n) local a,i; if n = 1 or isprime(n) then 0 ; else a := 0 ; for i from 4 to n do if not isprime(i) then a := a+1 ; fi ; od: RETURN(a) ; fi ; end: A133555 := proc(n) A066246(A113709(n)) ; end: seq(A133555(n),n=2..80) ; # R. J. Mathar, Jan 12 2008

compositePi[n_] := n - PrimePi[n] - 1;
a[n_] := Module[{p1 = Prime[n], p2 = Prime[n+1], c}, c = SelectFirst[ Range[p1+1, p2-1], Divisible[#, p2-p1]&]; compositePi[c]];
Table[a[n], {n, 2, 62}] (* Jean-François Alcover, Apr 02 2024 *)

a(n) = A066246(A113709(n)). - R. J. Mathar, Jan 12 2008

More terms from R. J. Mathar, Jan 12 2008

A161003 A list of the composite numbers divided by their largest prime factors.

Original entry on oeis.org

2, 2, 4, 3, 2, 4, 2, 3, 8, 6, 4, 3, 2, 8, 5, 2, 9, 4, 6, 16, 3, 2, 5, 12, 2, 3, 8, 6, 4, 9, 2, 16, 7, 10, 3, 4, 18, 5, 8, 3, 2, 12, 2, 9, 32, 5, 6, 4, 3, 10, 24, 2, 15, 4, 7, 6, 16, 27, 2, 12, 5, 2, 3, 8, 18, 7, 4, 3, 2, 5, 32, 14, 9, 20, 6, 8, 15, 2, 36, 10, 3, 16, 6, 5, 4, 9, 2, 7, 24, 11, 2, 3, 4

Offset: 1

Trevor Cassiliano (casstjc(AT)gmail.com), Jun 01 2009

nonn

a(A120389(n)) = A000040(n). - Gionata Neri, May 07 2015

For n >= 2, a(x) = n where x = A066246(n*A006530(n)). - Robert Israel, May 07 2015

			n=1 4/2; n=2 6/3; n=3 8/2.

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

Cf. A000040, A002808, A006530, A052369, A066246, A120389, A161004, A160180.

with(numtheory): a := proc (n) if isprime(n) = false then n/factorset(n)[nops(factorset(n))] else end if end proc: seq(a(n), n = 2 .. 130); # Emeric Deutsch, Jun 27 2009

With[{cmps=Select[Range[200],CompositeQ]},#/FactorInteger[#][[-1,1]]&/@ cmps] (* Harvey P. Dale, Mar 29 2017 *)

a(n) = A002808(n)/A052369(n). - Robert Israel, May 07 2015

Extended by Emeric Deutsch, Jun 27 2009

cp's OEIS Frontend

A169643 Numbers k such that neither composite(k)+-1 is composite.

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A139636 If n = the k-th prime, then a(n) = the (k+1)th prime. If n = the k-th composite, then a(n) = the (k+1)th composite.

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A139637 If n = the k-th prime, then a(n) = the (k-1)th prime. If n = the k-th composite, then a(n) = the (k-1)th composite. a(2) = 1. a(4) = 0.

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A133555 Order of A113709(n) among composite positive integers.

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A161003 A list of the composite numbers divided by their largest prime factors.

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Maple

Mathematica

Formula

Extensions