A073432 -id:A073432 - cp's OEIS frontend

A071259 Integer part of the arithmetic mean of the n-th prime p(n) and the n-th composite number C(n).

3, 4, 6, 8, 10, 12, 15, 17, 19, 23, 25, 29, 31, 33, 36, 39, 43, 44, 48, 51, 53, 56, 59, 62, 67, 70, 71, 74, 76, 79, 86, 89, 93, 94, 100, 101, 105, 109, 111, 115, 118, 120, 126, 128, 130, 132, 138, 145, 148, 149, 152, 156, 158, 163, 167, 170, 174, 176, 179, 182, 184

Offset: 1

Amarnath Murthy, May 30 2002

nonn

			a(7) = {p(7) + C(7)}/2 = (17 + 14)/2 = 15.
p(9) = 23, C(9) = 16, average = 19.5, a(9) = 19.

Harvey P. Dale, Table of n, a(n) for n = 1..2500

Cf. A000040, A002808, A073432, A073433, A071260.

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] Do[s=(Prime[n]+c[n])/2; If[IntegerQ[s], Print[s]], {n, 1, 256}]
IntegerPart[Mean[#]]&/@Module[{nn=100,cmps},cmps=Select[Range[nn], CompositeQ]; Thread[{cmps,Prime[Range[Length[cmps]]]}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 24 2018 *)

Floor [ (p(n) + C(n))/2]

a(n)=Floor[[A000040(n)+A002808(n)]/2]; Prime[n]>a(n)>Composite[n]

More terms from Labos Elemer, Jul 31 2002

A073433 Value of [p(n)+c(n)]/2 when it is an integer; p(n) is n-th prime, c(n) is n-th composite.

Original entry on oeis.org

3, 8, 17, 29, 36, 43, 53, 59, 70, 79, 93, 100, 109, 115, 128, 132, 148, 158, 167, 176, 184, 197, 204, 215, 222, 233, 247, 265, 274, 280, 288, 292, 301, 308, 319, 337, 346, 359, 368, 373, 380, 397, 401, 410, 419, 433, 448, 455, 466, 473, 496, 503, 507, 514

Offset: 1

Labos Elemer, Jul 31 2002

nonn

			p(8)=19, c(8)=15, average=17, corresponds to 3rd term.

Harvey P. Dale, Table of n, a(n) for n = 1..2000

Cf. A000040, A002808, A073432.

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] Do[s=(Prime[n]+c[n])/2; If[IntegerQ[s], Print[s]], {n, 1, 256}]
Module[{nn=200,pr,cm,len},pr=Prime[Range[nn]];cm=Select[Range[ nn], CompositeQ]; len=Min[Length[pr],Length[cm]];Select[Total[#]/2&/@ Thread[ {Take[pr,len],Take[cm,len]}],IntegerQ]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Oct 29 2019 *)

[A000040(n)+A002808(n)]/2 when it is an integer.

cp's OEIS Frontend

A071259 Integer part of the arithmetic mean of the n-th prime p(n) and the n-th composite number C(n).

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