A046665 -id:A046665 - cp's OEIS frontend

A231813 Number of iterations of A046665(n) = (greatest prime divisor of n) - (least prime divisor of n) [with A046665(1) = 0] required to reach zero.

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 2, 1, 2, 3, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 2, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 3, 3, 2, 1, 2, 1, 3, 2, 2, 2, 2, 1, 2, 1, 4, 1, 2, 3, 2, 3, 2, 1, 2, 3, 3, 3, 3, 3, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2

Offset: 0

Clark Kimberling, Dec 11 2013

nonn

			A046665(6) = 3 - 2, and A046665(1) = 0, so a(6) = 2.

Antti Karttunen, Table of n, a(n) for n = 0..16384 (terms 1..1000 from Clark Kimberling)
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..100000

Cf. A006530, A020639, A046665, A233510.

z = 400; h[n_] := h[n] = FactorInteger[n][[-1, 1]] - FactorInteger[n][[1, 1]]; t[n_] := Drop[FixedPointList[h, n], -2]; Table[t[n], {n, 1, z}]; a = Table[Length[t[n]], {n, 1, z}]

A046665(n) = if(1==n,0, my(f = factor(n), lpf = f[1, 1], gpf = f[#f~, 1]); (gpf-lpf));
A231813(n) = if(0==n,0, 1+A231813(A046665(n))); \\ Antti Karttunen, Jan 03 2019

a(0) = 0; for n > 0, a(n) = 1 + a(A046665(n)). - Antti Karttunen, Jan 03 2019

Name edited, term a(0)=0 prepended and more terms added by Antti Karttunen, Jan 03 2019

A074320 a(n) = sum of smallest and largest prime factors of n, for n>1; a(1)=2.

Original entry on oeis.org

2, 4, 6, 4, 10, 5, 14, 4, 6, 7, 22, 5, 26, 9, 8, 4, 34, 5, 38, 7, 10, 13, 46, 5, 10, 15, 6, 9, 58, 7, 62, 4, 14, 19, 12, 5, 74, 21, 16, 7, 82, 9, 86, 13, 8, 25, 94, 5, 14, 7, 20, 15, 106, 5, 16, 9, 22, 31, 118, 7, 122, 33, 10, 4, 18, 13, 134, 19, 26, 9, 142, 5, 146, 39, 8, 21, 18, 15

Offset: 1

Jason Earls, Sep 26 2002

If n is prime, a(n) = 2n; the only prime factor, n itself, is taken as both the "smallest" and "largest" prime factor of n. - Jon E. Schoenfield, Jan 14 2015

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

Cf. A046665, A066048, A130064, A130065.

f[n_]:=Transpose[FactorInteger[n]][[1]];Table[First[f[n]]+Last[f[n]],{n,200}] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)

a(n) = if (n==1, 2, my(f=factor(n)); f[1,1] + f[#f~,1]); \\ Michel Marcus, Jan 14 2015

a(n) = A020639(n) + A006530(n).

A130064 a(n) = (n / SmallestPrimeFactor(n)) * GreatestPrimeFactor(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 8, 9, 25, 11, 18, 13, 49, 25, 16, 17, 27, 19, 50, 49, 121, 23, 36, 25, 169, 27, 98, 29, 75, 31, 32, 121, 289, 49, 54, 37, 361, 169, 100, 41, 147, 43, 242, 75, 529, 47, 72, 49, 125, 289, 338, 53, 81, 121, 196, 361, 841, 59, 150, 61, 961, 147, 64, 169, 363

Offset: 1

Reinhard Zumkeller, May 05 2007

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Paul Erdős and J. H. van Lint, On the average ratio of the smallest and largest prime divisor of n, Indagationes Mathematicae (Proceedings), Vol. 85, No. 2 (1982), pp. 127-132.

Cf. A006530, A020639, A032742, A000961.
Cf. A001221, A001222.
Cf. A000290, A046665, A074320, A066048, A130065

a[n_] := With[{pp = FactorInteger[n][[All, 1]]}, n*pp[[-1]]/pp[[1]]];
Array[a, 100] (* Jean-François Alcover, Nov 18 2021 *)

a(n) = if (n==1, 1, my(f=factor(n)[, 1]~); n*vecmax(f)/vecmin(f)); \\ Michel Marcus, Sep 24 2022

from sympy import factorint
def a(n): f = factorint(n); return 1 if n == 1 else n//min(f)*max(f)
print([a(n) for n in range(1, 67)]) # Michael S. Branicky, Sep 24 2022

a(n) = n*A006530(n)/A020639(n) = A032742(n)*A006530(n);
a(n) >= n.
a(n) = n iff n is a prime power: a(A000961(n)) = A000961(n);
a(A001221(n)) <= A001221(n); a(A001222(n)) = A001222(n);
a(n) = A130065(n)+n*A046665(n)*A074320(n)/A066048(n) = A000290(n)/A130065(n).
Sum_{k=1..n} k/a(k) ~ n/log(n) + 3*n/log(n)^2 + o(n/log(n)^2) (Erdős and van Lint, 1982). - Amiram Eldar, Oct 14 2022

A130065 a(n) = (n / GreatestPrimeFactor(n)) * SmallestPrimeFactor(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 8, 9, 4, 11, 8, 13, 4, 9, 16, 17, 12, 19, 8, 9, 4, 23, 16, 25, 4, 27, 8, 29, 12, 31, 32, 9, 4, 25, 24, 37, 4, 9, 16, 41, 12, 43, 8, 27, 4, 47, 32, 49, 20, 9, 8, 53, 36, 25, 16, 9, 4, 59, 24, 61, 4, 27, 64, 25, 12, 67, 8, 9, 20, 71, 48, 73, 4, 45, 8, 49, 12, 79, 32, 81

Offset: 1

Reinhard Zumkeller, May 05 2007

nonn

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

Cf. A000290, A001221, A001222, A006530, A000961, A020639, A046665, A052126, A066048, A074320, A130064.

a[n_] := Module[{ps = First /@ FactorInteger[n]}, n * First[ps] / Last[ps]]; Array[a, 100] (* Amiram Eldar, Nov 27 2020 *)

a(n) = if (n==1, 1, my(f=factor(n)[,1]~); n*vecmin(f)/vecmax(f)); \\ Michel Marcus, Nov 27 2020

a(n) = n*A020639(n)/A006530(n) = A052126(n)*A020639(n);
a(n) <= n; a(n) = n iff n is a prime power: a(A000961(n)) = A000961(n);
a(A001221(n)) <= A001221(n); a(A001222(n)) = A001222(n);
a(n) = A130064(n) - n*A046665(n)*A074320(n)/A066048(n) = A000290(n)/A130064(n).

A069897 Integer quotient of the largest and the smallest prime factors of n, with a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 2, 5, 1, 1, 1, 6, 1, 3, 1, 2, 1, 1, 3, 8, 1, 1, 1, 9, 4, 2, 1, 3, 1, 5, 1, 11, 1, 1, 1, 2, 5, 6, 1, 1, 2, 3, 6, 14, 1, 2, 1, 15, 2, 1, 2, 5, 1, 8, 7, 3, 1, 1, 1, 18, 1, 9, 1, 6, 1, 2, 1, 20, 1, 3, 3, 21, 9, 5, 1, 2, 1, 11, 10, 23, 3, 1, 1, 3, 3, 2, 1

Offset: 1

Labos Elemer, Apr 10 2002

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A006530, A020639, A046665, A120454.

a[n_] := Module[{p = FactorInteger[n][[;;, 1]]}, Floor[p[[-1]] / p[[1]]]]; Array[a, 100] (* Amiram Eldar, Oct 24 2024 *)

A069897(n) = if(1==n,1, my(f = factor(n), lpf = f[1, 1], gpf = f[#f~, 1]); (gpf\lpf)); \\ Antti Karttunen, Sep 07 2018

a(n) = floor(A006530(n)/A020639(n)).

Term a(1) = 1 prepended by Antti Karttunen, Sep 07 2018

A081303 gpf(m) - 2*spf(m), where gpf(m) is the greatest and spf(m) is the smallest prime factor of m (A006530, A020639).

Original entry on oeis.org

-1, -2, -3, -2, -5, -1, -7, -2, -3, 1, -11, -1, -13, 3, -1, -2, -17, -1, -19, 1, 1, 7, -23, -1, -5, 9, -3, 3, -29, 1, -31, -2, 5, 13, -3, -1, -37, 15, 7, 1, -41, 3, -43, 7, -1, 19, -47, -1, -7, 1, 11, 9, -53, -1, 1, 3, 13, 25, -59, 1, -61, 27, 1, -2, 3, 7, -67, 13, 17, 3, -71, -1, -73, 33, -1, 15

Offset: 1

Reinhard Zumkeller, Mar 17 2003

sign

Cf. A046665, A081304, A081305, A081306, A069900.

A111426 Difference between largest and smallest prime factor of the n-th composite number.

Original entry on oeis.org

0, 1, 0, 0, 3, 1, 5, 2, 0, 1, 3, 4, 9, 1, 0, 11, 0, 5, 3, 0, 8, 15, 2, 1, 17, 10, 3, 5, 9, 2, 21, 1, 0, 3, 14, 11, 1, 6, 5, 16, 27, 3, 29, 4, 0, 8, 9, 15, 20, 5, 1, 35, 2, 17, 4, 11, 3, 0, 39, 5, 12, 41, 26, 9, 3, 6, 21, 28, 45, 14, 1, 5, 8, 3, 15, 11, 4, 51, 1, 9, 34, 5, 17, 18, 27, 10, 57, 10, 3, 0

Offset: 1

Giovanni Teofilatto, Nov 13 2005

nonn

a(n) = 0 iff the n-th composite number is a perfect power.

First occurrence of k or 0 if impossible: 2,8,5,12,7,38,0,21,13,26,16,61,0,35,22,40,25,84,0,49,31,156,0,111,0...,.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Composite[n_] := FixedPoint[n + 1 + PrimePi[ # ] &, n]; f[n_] := Block[{a = First@Transpose@FactorInteger@n}, a[[ -1]] - a[[1]]]; f[n_] := Block[{a = First@Transpose@FactorInteger@n}, a[[ -1]] - a[[1]]] (* Robert G. Wilson v *)
dif[n_]:=Module[{f=Transpose[FactorInteger[n]][[1]]},If[PrimeQ[n],{},Last[ f]- First[f]]]; Flatten[Table[dif[n],{n,4,200}]] (* Harvey P. Dale, May 12 2014 *)

a(n) = A046665(A002808(n)). - R. J. Mathar, Feb 19 2008

More terms from Robert G. Wilson v, Nov 17 2005
Corrected a(19). - Juri-Stepan Gerasimov, Jun 16 2009
a(19)=3 inserted by Klaus Brockhaus, Jun 25 2009

A069859 (Largest prime factor of n) modulo (smallest prime factor of n).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Apr 23 2002

a(n) = A006530(n) - A069897(n)*A020639(n).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A006530, A020639, A046665, A069897.

lpfmspf[n_]:=Module[{fs=Transpose[FactorInteger[n]][[1]]},Mod[Last[fs], First[fs]]]; Array[lpfmspf,100] (* Harvey P. Dale, Jul 10 2015 *)

A069859(n) = if(1==n,0, my(f = factor(n), lpf = f[1, 1], gpf = f[#f~, 1]); (gpf%lpf)); \\ Antti Karttunen, Sep 25 2018

a(n) = A006530(n) mod A020639(n).

More terms from Antti Karttunen, Sep 25 2018

A115090 a(n) = A115074(n) - A117183(n).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 3, 1, 0, 0, 5, 0, 1, 3, 9, 2, 1, 0, 3, 11, 1, 5, 3, 4, 0, 15, 1, 2, 17, 3, 0, 5, 9, 3, 21, 1, 5, 8, 3, 11, 1, 5, 27, 10, 3, 0, 29, 0, 5, 0, 9, 2, 15, 5, 1, 4, 35, 3, 17, 14, 11, 3, 1, 39, 5, 0, 16, 41, 2, 9, 3, 21, 45, 4, 1, 5, 9, 3, 15, 20, 11, 51, 1, 9, 5, 2, 17, 27, 11, 57, 3, 9, 0

Offset: 1

Leroy Quet, Mar 01 2006

nonn

The union of A115090 is the complement of A007921: numbers that are not the difference of two primes. - Robert G. Wilson v, Mar 09 2006

			12, the 4th nonsquarefree positive integer, is 2^2 * 3. 3 is the largest prime dividing 12. 2 is the smallest prime dividing 12. So a(4) = 3 - 2 = 1.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A007921, A013929, A046665, A115074, A115090, A117183.

(Max@# - Min@#) & /@ (First /@ FactorInteger@# & /@ Select[ Range@243, !SquareFreeQ@# &]) (* Robert G. Wilson v, Mar 09 2006 *)

a(n) = A046665(A013929(n)). - Amiram Eldar, Jan 06 2024

More terms from Robert G. Wilson v, Mar 09 2006

A135093 Least composite number k for each possible difference gpf(k)-lpf(k).

Original entry on oeis.org

4, 6, 15, 10, 21, 14, 55, 33, 22, 39, 26, 85, 51, 34, 57, 38, 115, 69, 46, 203, 145, 87, 58, 93, 62, 259, 185, 111, 74, 205, 123, 82, 129, 86, 235, 141, 94, 371, 265, 159, 106, 413, 295, 177, 118, 183, 122, 469, 335, 201, 134, 355, 213, 142, 219, 146, 553, 395, 237

Offset: 0

Rick L. Shepherd, Nov 18 2007

nonn

Clearly all terms are semiprimes. a(0)=prime(1)^2=4. For n>=1, a(n)=k, a squarefree semiprime, where gpf(k)-lpf(k)=A006530(k)-A020639(k)=A030173(k).

For n > 0: first occurrences of A030173(n) in A046665. - Reinhard Zumkeller, Jul 03 2015

			a(3)=2*5=10 because 5-2=3=A030173(3), where the latter terms are ordered by the increasing possible differences between two distinct primes and no smaller composite number has a difference of 3 between its least and greatest prime factors.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A001358, A006881, A030173, A020639, A006530.

Cf. A046665.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a135093 0 = 4
a135093 n = (+ 1) $ fromJust $ (`elemIndex` a046665_list) $ a030173 n
-- Reinhard Zumkeller, Jul 03 2015

cp's OEIS Frontend

A231813 Number of iterations of A046665(n) = (greatest prime divisor of n) - (least prime divisor of n) [with A046665(1) = 0] required to reach zero.

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A074320 a(n) = sum of smallest and largest prime factors of n, for n>1; a(1)=2.

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A130064 a(n) = (n / SmallestPrimeFactor(n)) * GreatestPrimeFactor(n).

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A130065 a(n) = (n / GreatestPrimeFactor(n)) * SmallestPrimeFactor(n).

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A069897 Integer quotient of the largest and the smallest prime factors of n, with a(1) = 1.

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A081303 gpf(m) - 2*spf(m), where gpf(m) is the greatest and spf(m) is the smallest prime factor of m (A006530, A020639).

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A111426 Difference between largest and smallest prime factor of the n-th composite number.

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A069859 (Largest prime factor of n) modulo (smallest prime factor of n).

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