A120389 -id:A120389 - cp's OEIS frontend

A056608 Least prime factor of the n-th composite number.

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

Offset: 1

Odimar Fabeny, Aug 07 2000

Record values are seen when n = A120389(m). Conjecture: at each new record the count of the prior record follows A247509. Records seen are 2, 3, 5, 7, 11, ... and when 3, 5, 7, 11 are first seen, there have been 3, 3, 2, and 4 occurrences of 2, 3, 5, and 7. These are A247509(1) through A247509(4). Thus, the count for prime(60) would be A247509(60). - Bill McEachen, Jun 17 2024

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

Cf. A052369 (largest prime factor of n, where n runs through composite numbers). - Klaus Brockhaus, Jun 23 2009

Cf. A160180.

a056608 = a020639 . a002808  -- Reinhard Zumkeller, Mar 29 2014

[ PrimeDivisors(n)[1]: n in [2..140] | not IsPrime(n) ]; // Klaus Brockhaus, Jun 23 2009

DeleteCases[Table[FactorInteger[n][[1, 1]] * Boole[Not[PrimeQ[n]]], {n, 2, 100}], 0] (* Alonso del Arte, Aug 21 2014 *)
FactorInteger[#][[1,1]]&/@Select[Range[200],CompositeQ] (* Harvey P. Dale, Mar 16 2023 *)

forcomposite(n=1, 1e2, p=factor(n)[1, 1]; print1(p, ", ")) \\ Felix Fröhlich, Aug 03 2014

from sympy import composite, factorint
def A056608(n):
    return min(factorint(composite(n))) # Chai Wah Wu, Jul 22 2019

a(n) = A020639(A002808(n)) = A000040(A118663(n)). - M. F. Hasler, Apr 03 2012

More terms from James Sellers, Aug 25 2000
Definition corrected by Reinhard Zumkeller, Mar 29 2014
Name changed by Alonso del Arte, Aug 21 2014

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

A073814 a(n) is the smallest number k such that A073813(k) = prime(n).

Original entry on oeis.org

2, 4, 15, 33, 90, 129, 227, 288, 429, 694, 798, 1149, 1417, 1565, 1879, 2399, 2993, 3201, 3879, 4365, 4623, 5429, 6002, 6920, 8245, 8948, 9314, 10067, 10457, 11245, 14251, 15184, 16627, 17130, 19711, 20253, 21919, 23653, 24845, 26687, 28604, 29248, 32612, 33303, 34719, 35436, 39893, 44622, 46254

Offset: 1

Labos Elemer, Aug 15 2002

nonn

			a(6)=129 means that c[129]-Max[URS[c[129]]=Prime[6]: c[129]=169, Max[URS[169]]=Max{26,39,...,143,156}=156; difference=169-156=13=6th prime. Suspicion: A073813(n) is always prime number!

Cf. A045763, A073759, A002808, A073813.

Cf. A120389. [From R. J. Mathar, Aug 07 2008]

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x]; tn[x_] := Table[j, {j, 1, x}]; di[x_] := Divisors[x]; rrs[x_] := Flatten[Position[GCD[tn[x], x], 1]]; rs[x_] := Union[rrs[x], di[x]]; urs[x_] := Complement[tn[x], rs[x]]; Do[s=c[n]-Max[urs[c[n]]]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 10}]; t
nn = 6 * 10^4; s = Function[k, k - SelectFirst[Range[k - 2, 2, -1], 1 < GCD[#, k] < # &]] /@ Select[Range[6, nn], ! PrimeQ@ # &]; Table[SelectFirst[Range@ Length@ s, s[[# - 1]] == Prime@ n &], {n, 49}] (* Michael De Vlieger, Mar 28 2016, Version 10 *)

Min{x; c[x]-Max[URS[c[x]]]=p(n)}, p(n)=n-th prime. See program.

Definition corrected by Gionata Neri, Mar 28 2016

More terms from Michael De Vlieger, Mar 28 2016

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A056608 Least prime factor of the n-th composite number.

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

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A073814 a(n) is the smallest number k such that A073813(k) = prime(n).

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