A161986 -id:A161986 - cp's OEIS frontend

A161850 Subsequence of A161986 consisting of all terms that are prime.

7, 11, 13, 17, 19, 23, 29, 31, 37, 37, 41, 43, 47, 47, 53, 53, 59, 61, 67, 71, 71, 73, 79, 83, 89, 89, 97, 97, 101, 101, 103, 107, 109, 113, 127, 131, 137, 137, 139, 149, 149, 151, 157, 163, 163, 167, 167, 173, 179, 179, 181, 193, 191, 193, 197, 199, 211, 223, 227

Offset: 1

Juri-Stepan Gerasimov, Jun 20 2009

nonn

A161986(n) = k+r where k is n-th composite and r is remainder of (largest prime divisor of k) divided by (smallest prime divisor k).

			A161986(1) to A161986(27) are 4, 7, 8, 9, 11, 13, 15, 17, 16, 19, 21, 22, 23, 25, 25, 27, 27, 29, 31, 32, 35, 35, 37, 37, 39, 40, 41. Hence a(1) to a(11) are the prime terms among them, namely 7, 11, 13, 17, 19, 23, 29, 31 ,37, 37, 41.

Cf. A161986 (A002808(n)+A161849(n)), A002808 (composite numbers), A161849 (A052369(n) mod A056608(n)), A052369 (largest prime factor of n-th composite), A056608 (smallest divisor of n-th composite).

[ p: n in [2..230] | not IsPrime(n) and IsPrime(p) where p is n+D[ #D] mod D[1] where D is PrimeDivisors(n) ];

Edited and corrected (a(19)=57 replaced by 67; a(38)=137, a(49)=179, a(50)=179 inserted) by Klaus Brockhaus, Jun 24 2009

A161849 a(n) = A052369(n) mod A056608(n).

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 3, 1, 1, 2, 1, 1, 1, 2, 1, 4, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 6, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 0, 1, 2, 1, 0, 1, 0, 1, 1, 1, 5, 1, 2, 1, 1, 1

Offset: 1

Juri-Stepan Gerasimov, Jun 20 2009

nonn

(Largest prime divisor) modulo (smallest prime divisor) of n-th composite number.

			a(1) = 0 = 2 mod 2;
a(2) = 1 = 3 mod 2;
a(3) = 0 = 2 mod 2;
a(4) = 0 = 3 mod 3;
a(5) = 1 = 5 mod 2.

Cf. A052369, A056608, A002808.

Cf. A161850, A161986. - Klaus Brockhaus, Jun 24 2009

[ D[ #D] mod D[1]: n in [2..140] | not IsPrime(n) where D is PrimeDivisors(n) ]; // Klaus Brockhaus, Jun 24 2009

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) : fi; od: fi; end:
A006530 := proc(n) local u: if n=1 then 1 else u:= numtheory[factorset](n): max(seq(u[j], j=1..nops(u))) end if end:
A020639 := proc(n) local u: if n=1 then 1 else u:= numtheory[factorset](n): min(seq(u[j], j=1..nops(u))) end if end:
A052369 := proc(n) A006530(A002808(n)) ; end:
A056608 := proc(n) A020639(A002808(n)) ; end:
A161849 := proc(n) A052369(n) mod A056608(n) ; end: seq(A161849(n),n=1..120) ; # R. J. Mathar, Jun 23 2009

Composite[n_] := FixedPoint[n + PrimePi[#] + 1&, n + PrimePi[n] + 1];
a[n_] := With[{f = FactorInteger[Composite[n]]}, f[[-1, 1]]~Mod~f[[1, 1]]];
Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Oct 15 2023 *)

a(102) corrected by R. J. Mathar, Jun 23 2009

cp's OEIS Frontend

A161850 Subsequence of A161986 consisting of all terms that are prime.

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