A162361 -id:A162361 - cp's OEIS frontend

A162908 Pairs of central divisors of A089229(n).

2, 3, 2, 4, 2, 5, 3, 4, 2, 7, 3, 5, 3, 6, 4, 5, 3, 7, 2, 11, 4, 6, 2, 13, 3, 9, 4, 7, 5, 6, 4, 8, 3, 11, 2, 17, 5, 7, 2, 19, 3, 13, 5, 8, 6, 7, 4, 11, 5, 9, 2, 23, 6, 8, 5, 10, 3, 17, 4, 13, 6, 9, 5, 11, 7, 8, 3, 19, 2, 29, 6, 10, 2, 31, 7, 9, 5, 13, 6, 11, 4, 17, 3, 23, 7, 10, 8, 9, 2, 37, 5, 15, 4, 19

Offset: 1

Juri-Stepan Gerasimov, Jul 17 2009

The central divisors are the two complementary divisors closest to the square

root of a number, as tabulated in A162348.

			a(1)=2 and a(2)=3 since A089229(1)=6 with divisors 1, 2, 3, 4.
a(3)=2 and a(4)=4 since A089229(2)=8 with divisors 1, 2, 4, 8.
a(5)=2 and a(6)=5 since A089229(3)=10 with divisors 1, 2, 5, 10.

Cf. A089229, A162361.

a(2n-1)= A033676(A089229(n)). a(2n)=A033677(A089229(n)). - R. J. Mathar, Jul 19 2009

Missing pairs inserted by R. J. Mathar, Jul 19 2009

A163173 The n-th product of three primes divided by its central prime factor.

Original entry on oeis.org

4, 6, 6, 10, 9, 14, 10, 14, 22, 15, 10, 26, 21, 22, 34, 14, 15, 38, 26, 46, 14, 33, 34, 21, 22, 38, 58, 39, 62, 25, 26, 46, 21, 74, 51, 22, 82, 33, 34, 57, 86, 58, 35, 26, 62, 94, 38, 39, 69, 106, 74, 46, 33, 118, 34, 22, 122, 35, 82, 51, 86, 87, 38, 134, 39, 55, 93, 94, 142, 57, 26

Offset: 1

Juri-Stepan Gerasimov, Jul 22 2009

All entries are in A001358.

			a(1)=2*2*2/2=4. a(2)=2*2*3/2=6. a(3)=2*3*3/3=6. a(4)=2*2*5/2=10. a(5)=3*3*3/3 = 9.

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

Cf. A014612, A162361.

f:= proc(n) local P;
   if numtheory:-bigomega(n) <> 3 then return NULL fi;
   P:= numtheory:-factorset(n);
   min(P)*max(P)
end proc:
map(f, [$1..500]); # Robert Israel, Jun 05 2023

a(n) = A014612(n)/A162361(n).

a(57) corrected by R. J. Mathar, Jul 25 2009

A163178 a(n) is the central prime factor of A046339(n).

Original entry on oeis.org

2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 5, 2, 3, 3, 2, 5, 2, 5, 2, 3, 2, 2, 7, 3, 3, 5, 3, 5, 2, 3, 2, 3, 2, 2, 5, 2, 5, 3, 7, 2, 3, 7, 3, 2, 5, 2, 5, 3, 2, 3, 5, 2, 3, 7, 3, 2, 5, 2, 5, 2, 3, 2, 2, 3, 5, 7, 2, 7, 11, 3, 2, 7, 3, 3, 5, 3, 3, 2, 7, 2, 3, 2, 7, 5, 3, 2, 3, 2, 5, 11, 2, 5, 2, 3, 2, 5, 2, 2, 3, 2, 7, 5

Offset: 1

Juri-Stepan Gerasimov, Jul 22 2009

nonn

			a(8)=2 since A046339(8) = 32 = 2*2*2*2*2.
a(9)=3 since A046339(9) = 42 = 2*3*7.
a(10)=2 since A046339(10) = 44 = 2*2*11.
a(11)=3 since A046339(11) = 45 = 3*3*5.
a(12)=2 since A046339(12) = 48 = 2*2*2*2*3.

Cf. A046339, A162361.

cpr := proc(n) nred := n ; while numtheory[bigomega](nred) > 1 do pf := sort(convert(numtheory[factorset](nred), list)) ; nred := nred/op(1, pf)/op(-1, pf) ; od: nred ; end:
A046339 := proc(n) option remember ; if n = 1 then 8 ; else for a from procname(n-1)+1 do if not isprime(a) and numtheory[bigomega](a) mod 2 = 1 then RETURN(a) ; fi; od: fi; end:
A163178 := proc(n) cpr(A046339(n)) ; end; seq(A163178(n), n=1..120) ; # R. J. Mathar, Jul 25 2009

More terms from R. J. Mathar, Jul 25 2009

A162731 a(n) is the n-th triprime (A014612) minus its central prime factor.

Original entry on oeis.org

6, 10, 15, 18, 24, 26, 27, 39, 42, 42, 45, 50, 60, 63, 66, 65, 70, 74, 75, 90, 91, 96, 99, 100, 105, 111, 114, 114, 122, 120, 125, 135, 140, 146, 150, 147, 162, 160, 165, 168, 170, 171, 170, 175, 183, 186, 185, 190, 204, 210, 219, 225, 224, 234, 231, 231, 242, 238, 243, 250

Offset: 1

Juri-Stepan Gerasimov, Jul 12 2009

			a(1) =  6 = 2*2*2 - 2;
a(2) = 10 = 2*2*3 - 2;
a(3) = 15 = 2*3*3 - 3;
a(4) = 18 = 2*2*5 - 2;
a(5) = 24 = 3*3*3 - 3.

Cf. A014612, A162361.

a(n) = A014612(n) - A162361(n).

Definition reformulated, a(11), a(38) etc. corrected by R. J. Mathar, Jul 16 2009

A162915 Absolute values of the first differences of A162908.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 1, 2, 5, 4, 2, 2, 3, 2, 1, 2, 4, 5, 9, 7, 2, 4, 11, 10, 6, 5, 3, 2, 1, 2, 4, 5, 8, 9, 15, 12, 2, 5, 17, 16, 10, 8, 3, 2, 1, 3, 7, 6, 4, 7, 21, 17, 2, 3, 5, 7, 14, 13, 9, 7, 3, 4, 6, 4, 1, 5, 16, 17, 27, 23, 4, 8, 29, 24, 2, 4, 8, 7, 5, 7, 13, 14, 20, 16, 3, 2, 1, 7, 35, 32, 10, 11, 15

Offset: 1

Juri-Stepan Gerasimov, Jul 17 2009

Cf. A089229, A162361, A162908.

Corrected as a followup to correcting A162908. - R. J. Mathar, Jul 19 2009

cp's OEIS Frontend

A162908 Pairs of central divisors of A089229(n).

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A163173 The n-th product of three primes divided by its central prime factor.

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A163178 a(n) is the central prime factor of A046339(n).

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Examples

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Programs

Maple

Extensions

A162731 a(n) is the n-th triprime (A014612) minus its central prime factor.

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Keywords

Examples

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Formula

Extensions

A162915 Absolute values of the first differences of A162908.

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