A087040 -id:A087040 - cp's OEIS frontend

A378680 a(n) = numerator(Sum_{k=1..n} 1/P_2(k)), where P_2(k) = A087040(k) is the second largest prime dividing the k-th composite number.

1, 1, 3, 11, 7, 17, 10, 11, 25, 9, 5, 16, 35, 19, 98, 211, 221, 118, 41, 87, 271, 143, 146, 151, 317, 109, 57, 176, 367, 377, 196, 407, 2879, 2921, 997, 516, 1583, 1604, 3313, 3383, 1744, 593, 1221, 3733, 1919, 388, 395, 811, 275, 1389, 4237, 2171, 2192, 4489

Offset: 1

Amiram Eldar, Dec 03 2024

			Fractions begin: 1/2, 1, 3/2, 11/6, 7/3, 17/6, 10/3, 11/3, 25/6, 9/2, 5, 16/3, ...

Amiram Eldar, Table of n, a(n) for n = 1..1000
Jean-Marie De Koninck, Sur les plus grands facteurs premiers d'un entier, Monatshefte für Mathematik, Vol. 116, No. 1 (1993), pp. 13-37; alternative link; author's copy.

Cf. A006530, A087039, A087040, A378681 (denominators).

p2[c_] := Module[{f = FactorInteger[c]}, If[f[[-1, 2]] > 1, f[[-1, 1]], f[[-2, 1]]]]; Numerator@ Accumulate[Table[1/p2[c], {c, Select[Range[100], CompositeQ]}]]

lista(nmax) = {my(s = 0); forcomposite(n = 1, nmax, f = factor(n); s += if(f[#f~, 2] > 1, 1/f[#f~, 1], 1/f[#f~ - 1, 1]); print1(numerator(s), ", "));}

a(n)/A378681(n) = Sum_{k=1..m} c_k * n/log(n)^k + O(n/log(n)^(m+1)) for any integer m >= 1, where c_k are constants. c_1 = Sum_{k>=1} (1/k)*Sum_{p prime > P(k)} 1/p^2 = Sum_{p prime} (1/p^2)*Product_{primes q < p} (1/(1-1/q)) = 1.254435359..., where P(k) = A006530(k) is the greatest prime dividing k for k >= 2, and P(1) = 1.

A378681 a(n) = denominator(Sum_{k=1..n} 1/P_2(k)), where P_2(k) = A087040(k) is the second largest prime dividing the k-th composite number.

Original entry on oeis.org

2, 1, 2, 6, 3, 6, 3, 3, 6, 2, 1, 3, 6, 3, 15, 30, 30, 15, 5, 10, 30, 15, 15, 15, 30, 10, 5, 15, 30, 30, 15, 30, 210, 210, 70, 35, 105, 105, 210, 210, 105, 35, 70, 210, 105, 21, 21, 42, 14, 70, 210, 105, 105, 210, 210, 210, 105, 35, 70, 210, 210, 105, 105, 210

Offset: 1

Amiram Eldar, Dec 03 2024

See A378680 for more details.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A087039, A087040, A378680 (numerators).

p2[c_] := Module[{f = FactorInteger[c]}, If[f[[-1, 2]] > 1, f[[-1, 1]], f[[-2, 1]]]]; Denominator@ Accumulate[Table[1/p2[c], {c, Select[Range[50], CompositeQ]}]]

lista(nmax) = {my(s = 0); forcomposite(n = 1, nmax, f = factor(n); s += if(f[#f~, 2] > 1, 1/f[#f~, 1], 1/f[#f~ - 1, 1]); print1(denominator(s), ", "));}

A087039 If n is prime then 1 else 2nd largest prime factor of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 5, 2, 3, 2, 1, 3, 1, 2, 3, 2, 5, 3, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 7, 5, 3, 2, 1, 3, 5, 2, 3, 2, 1, 3, 1, 2, 3, 2, 5, 3, 1, 2, 3, 5, 1, 3, 1, 2, 5, 2, 7, 3, 1, 2, 3, 2, 1, 3, 5, 2, 3, 2, 1, 3, 7, 2, 3, 2, 5, 2, 1, 7, 3, 5, 1, 3

Offset: 1

Reinhard Zumkeller, Aug 01 2003

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Greatest Prime Factor

Cf. A002808, A006530, A085392, A087040.

Cf. A027746, A052126.

a087039 n | null ps   = 1
          | otherwise = head ps
          where ps = tail $ reverse $ a027746_row n
-- Reinhard Zumkeller, Oct 03 2012

A087039 := proc(n)
    local pset ,t;
    if isprime(n) or n= 1 then
        1;
    else
        pset := [] ;
        for p in ifactors(n)[2] do
            pset := [op(pset),seq(op(1,p),t=1..op(2,p))] ;
        end do:
        op(-2,sort(pset)) ;
    end if;
end proc: # R. J. Mathar, Sep 14 2012

gpf[n_] := FactorInteger[n][[-1, 1]];
a[n_] := If[PrimeQ[n], 1, gpf[n/gpf[n]]];
Array[a, 105] (* Jean-François Alcover, Dec 16 2021 *)

from sympy import factorint
def a(n):
    pf = factorint(n, multiple=True)
    return 1 if len(pf) < 2 else pf[-2]
print([a(n) for n in range(1, 103)]) # Michael S. Branicky, Dec 16 2021

a(n) = A006530(A052126(n)) = A006530(n/A006530(n));

A087040(n) = a(A002808(n)).

A179938 Third largest prime factor of numbers that are divisible by at least three different primes (A000977).

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 3, 3, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 5, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2, 5, 3, 3, 3, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 5, 2, 2, 3, 3, 3, 2, 2, 2, 2, 3, 5, 2, 2, 3, 2, 3

Offset: 1

Jonathan Vos Post, Jan 12 2011

Third largest prime factor of numbers k such that omega(k) = A001221(k) > 2. The 3rd largest prime factor may equal the second largest. This is not identical to third largest distinct prime factor of numbers that are divisible by at least three different primes. Indices n where a(n) equals 2, 3, 5, 7, 11, 13, 17, 19, 23, ... for the first time are 1, 8, 72, 299, 905, 1718, 3302, 6020, 10330, ... the corresponding numbers from A000977 are 30, 90, 350, 1001, 2431, 4199, 7429, 12673, 20677, ...

			a(1) = 2 because 30 = 2 * 3 * 5 has third largest prime factor 2.
a(2) = 2 because 42 = 2 * 3 * 7 has third largest prime factor 2.
a(3) = 2 because 60 = 2 * 2 * 3 * 5 has both third and fourth largest prime factor 2.
a(8) = 3 because 90 = 2 * 3 * 3 * 5 has both second and third largest prime factor 3.

Cf. A000040, A000977, A001221, A002808, A033992, A007774, A000961, A033993, A051270, A087040, A088739, A179312.

b:= proc(n) option remember; local k;
      if n=1 then 30
      else for k from b(n-1)+1 while
              nops(ifactors(k)[2])<3 do od;
           k
      fi
    end:
a:= n-> sort(map(x-> x[1]$x[2], ifactors(b(n))[2]))[-3]:
seq(a(n), n=1..120);

b[n_] := b[n] = Module[{k}, If[n==1, 30, For[k = b[n-1]+1, PrimeNu[k] < 3, k++]; k]];
a[n_] := (Table[#[[1]], {#[[2]]}]& /@ FactorInteger[b[n]] // Flatten // Sort)[[-3]];
Array[a, 120] (* Jean-François Alcover, Nov 28 2020, after Alois P. Heinz *)

Edited by Alois P. Heinz, Jan 14 2011

cp's OEIS Frontend

A378680 a(n) = numerator(Sum_{k=1..n} 1/P_2(k)), where P_2(k) = A087040(k) is the second largest prime dividing the k-th composite number.

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A378681 a(n) = denominator(Sum_{k=1..n} 1/P_2(k)), where P_2(k) = A087040(k) is the second largest prime dividing the k-th composite number.

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A087039 If n is prime then 1 else 2nd largest prime factor of n.

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A179938 Third largest prime factor of numbers that are divisible by at least three different primes (A000977).

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