A074320 -id:A074320 - cp's OEIS frontend

A046665 Largest prime divisor of n - smallest prime divisor of n (a(1)=0).

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

Offset: 1

N. J. A. Sloane

Even nonzero terms correspond to odd composite numbers that are not powers of primes. Terms of A030173 appear in this sequence infinitely often. - Alonso del Arte, Nov 27 2011

A135093(n) = first occurrence of A030173(n). - Reinhard Zumkeller, Jul 03 2015

Handbook of Number Theory, D. S. Mitrinovic et al., Kluwer, Section IV.1.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A006530, A020639, A074320, A066048, A130064, A130065.

a046665 n = a006530 n - a020639 n  -- Reinhard Zumkeller, Jul 03 2015

a:= n-> `if`(n=1, 0, (s-> max(s)-min(s))(numtheory[factorset](n))):
seq(a(n), n=1..100);  # Alois P. Heinz, Mar 07 2020

f[n_]:=Transpose[FactorInteger[n]][[1]];Table[Last[f[n]-First[f[n]]],{n,200}] (* Vladimir Joseph Stephan Orlovsky, Apr 08 2011 *)
lpd[n_]:=Module[{fi=FactorInteger[n]},fi[[-1,1]]-fi[[1,1]]]; Array[lpd,90] (* Harvey P. Dale, Dec 31 2017 *)

a(n)={if(n==1, 0, my(f=factor(n)[,1]); f[#f]-f[1])} \\ Andrew Howroyd, Mar 07 2020

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

More terms from James Sellers

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).

A199745 Numbers such that the sum of the largest and the smallest prime divisor equals the sum of the other distinct prime divisors.

Original entry on oeis.org

2145, 2730, 4641, 4845, 5005, 5460, 5610, 6435, 7410, 8190, 8778, 9177, 10725, 10920, 11220, 11305, 11730, 13485, 13585, 13650, 13923, 14535, 14820, 16380, 16830, 17017, 17556, 19110, 19305, 20010, 20930, 21489, 21505, 21840, 22230, 22440, 23460, 23529, 23595

Offset: 1

Michel Lagneau, Nov 09 2011

nonn

The definition implies that members of the sequence have at least four distinct prime factors. An even term must have at least five distinct prime factors.

			22440 is in the sequence because the distinct prime divisors are  {2, 3, 5, 11, 17} and 17+2 = 3+5+11.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A020639, A006530, A074320, A008472, A109353.

a199745 n = a199745_list !! (n-1)
a199745_list = filter (\x -> 2 * (a074320 x) == a008472 x) [1..]
-- Reinhard Zumkeller, Nov 10 2011

isA199745 := proc(n)
  local p;
  p := sort(convert(numtheory[factorset](n),list)) ;
  if nops(p) >= 3 then
    return ( op(1,p) + op(-1,p) = add(op(i,p),i=2..nops(p)-1) ) ;
  else
    false;
  end if;
end proc:
for n from 2 to 20000 do
  if isA199745(n) then
    printf("%d,",n) ;
  end if ;
end do: # R. J. Mathar, Nov 10 2011

Select[Range[25000],Plus@@(pl=First/@FactorInteger[#])/2==pl[[1]]+pl[[-1]]&] (* Ray Chandler, Nov 10 2011 *)

def isA199745(n) :
    p = factor(n)
    return len(p) > 2 and p[0][0] + p[-1][0] == add(p[i][0] for i in (1..len(p)-2))
[n for n in (2..20000) if isA199745(n)]  # Peter Luschny, Nov 10 2011

n such that A008472(n)/2 = A074320(n) = A020639(n) + A006530 (n). - Ray Chandler, Nov 10 2011

Sum_{k=2..A001221(a(n))} A027748(a(n),k) = A027748(a(n),1) + A027748(a(n), A001221(a(n))). - Reinhard Zumkeller, Nov 10 2011

A120454 a(n) = ceiling(GPF(n)/LPF(n)) where GPF is greatest prime factor, LPF is least prime factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 4, 2, 1, 1, 2, 1, 3, 3, 6, 1, 2, 1, 7, 1, 4, 1, 3, 1, 1, 4, 9, 2, 2, 1, 10, 5, 3, 1, 4, 1, 6, 2, 12, 1, 2, 1, 3, 6, 7, 1, 2, 3, 4, 7, 15, 1, 3, 1, 16, 3, 1, 3, 6, 1, 9, 8, 4, 1, 2, 1, 19, 2, 10, 2, 7, 1, 3, 1, 21, 1, 4, 4, 22, 10, 6, 1, 3, 2, 12, 11, 24, 4, 2, 1, 4, 4

Offset: 1

Jonathan Vos Post, Aug 16 2006

Given GPF(n) and LPF(n), the sum is A074320, the difference is A046665 and the product is A066048. a(n) = 1 iff n is p^k iff n is in A000961.

			a(26) = ceiling(GPF(26)/LPF(26)) = ceiling(13/2) = 7.

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

Cf. A000961, A006530, A020639, A024619, A046665, A066048, A069897, A074320.

A120454 := proc(n) local ifs ; if n = 1 then RETURN(1) ; else ifs := ifactors(n)[2] ; RETURN( ceil(op(1,op(-1,ifs))/op(1,op(1,ifs))) ) ; fi ; end ; for n from 1 to 100 do printf("%d, ",A120454(n)) ; od ; # R. J. Mathar, Dec 16 2006

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

A120454(n) = if(1==n,1, my(f = factor(n), lpf = f[1, 1], gpf = f[#f~, 1]); ceil(gpf/lpf)); \\ Antti Karttunen, Sep 06 2018

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

a(n) = A069897(n) + 1 if n is not a power of a prime (A024619), and 1 otherwise. - Amiram Eldar, Oct 24 2024

Corrected and extended by R. J. Mathar, Dec 16 2006

A271621 a(1) = 2, a(2) = 3, a(n) = A020639(a(n-2)) + A006530(a(n-1)).

Original entry on oeis.org

2, 3, 5, 8, 7, 9, 10, 8, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Cody M. Haderlie, Apr 10 2016

nonn

Any sequence generated by this formula and any values for a(1) and a(2) will have a finite number of terms not equal to 4; i.e., all such sequences will eventually end up at 4 (and all following terms will be 4; 4 is the only term that can appear more than twice in a row in a sequence because it is the only number equal to the sum of its least and greatest prime factors). Example: a(1) = 77713; a(2) = 16; the sequence is: 77713, 16, 77715, 159, 56, 10, 7, 9, 10, 8, 4, 4, 4, ...

			a(1) = 13; a(2) = 46.
lpf(13) = 13; gpf(46) = 23.
a(3) = 13 + 23 = 36.

Cf. A020639, A006530, A074320.

a[1] = 2; a[2] = 3; a[n_] := a[n] = FactorInteger[a[n - 2]][[1, 1]] +
FactorInteger[a[n - 1]][[-1, 1]]; Array[a, {120}] (* Michael De Vlieger, Apr 12 2016 *)

spf(n) = if (n==1, 1, vecmin(factor(n)[,1]));
gpf(n) = if (n==1, 1, vecmax(factor(n)[,1]));
lista(nn) = {print1(x=2, ", "); print1(y=3, ", "); for (n=1, nn, ny = spf(x) + gpf(y); print1(ny, ", "); x = y; y = ny;);} \\ Michel Marcus, Apr 15 2016

a(n) = lpf(a(n-2)) + gpf(a(n-1)), where lpf(n) is the least prime dividing n and gpf(n) is the greatest prime dividing n.

cp's OEIS Frontend

A046665 Largest prime divisor of n - smallest prime divisor of n (a(1)=0).

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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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A199745 Numbers such that the sum of the largest and the smallest prime divisor equals the sum of the other distinct prime divisors.

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A120454 a(n) = ceiling(GPF(n)/LPF(n)) where GPF is greatest prime factor, LPF is least prime factor.

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A271621 a(1) = 2, a(2) = 3, a(n) = A020639(a(n-2)) + A006530(a(n-1)).

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