A087039 -id:A087039 - cp's OEIS frontend

A143766 a(n+1) = a(n)^2 + 3na(n) + n^2, a(1) = 1.

1, 5, 59, 4021, 16216709, 262981894041341, 69159476593575838635509822455, 4783033202697364284917104840982811414253511628131328498629, 22877406618105405861781317490149379589769149890660405723416585348109182559037843469373513563751798569651299138846801

Offset: 1

Reinhard Zumkeller, Sep 01 2008

nonn

Let f(n+1,k) = f(n,k)^2 + k*n*f(n,k) + n^2, f(1, k) = 1:
f(n,0)=A143760(n), f(n,-1)=A143761(n), f(n,+1)=A143762(n),
f(n,-2)=A143763(n), f(n,+2)=A143764(n), f(n,-3)=A143765(n), f(n,+3)=a(n).

			Contribution from _Reinhard Zumkeller_, Sep 11 2008: (Start)
a(4)=A000040(556);
a(5)=19*199*4289;
a(6)=3686299*71340359;
a(7)=5*89*23581*36190079671*182112572569;
A055642(a(8))=58; A001221(a(8))=A001222(a(8))=3;
A055642(A020639(a(8)))=4, A020639(a(8))=2459;
A055642(A006530(a(8)))=43, A006530(a(8))=1145781805709434583439407716589323093429591;
A055642(a(9))=116; A001221(a(9))=A001222(a(9))=3;
A055642(A020639(a(9)))=5, A020639(a(9))=52291;
A055642(A087039(a(9)))=39, A087039(a(9))=823717865733493312451872329574156137131;
A055642(A006530(a(9)))=72, A006530(a(9))=531130643259166452223939782963931943654770628199012648274446497807560081;
factorizations made with Dario Alpern's ECM applet. (End)

Index entries for sequences of form a(n+1)=a(n)^2 + ... [From _Reinhard Zumkeller_, Sep 11 2008]
Dario A. Alpern, Factorization using the Elliptic Curve Method [From _Reinhard Zumkeller_, Sep 11 2008]

RecurrenceTable[{a[n+1] == a[n]^2 + 3*n*a[n] + n^2, a[1] == 1}, a, {n, 1, 10}] (* Vaclav Kotesovec, Dec 18 2014 *)

a(n) ~ c^(2^n), where c = 1.68000796750332615134775696497253700657744224375254906378714756508286... . - Vaclav Kotesovec, Dec 18 2014

A087040 2nd largest prime factor of n-th composite number.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Aug 01 2003

nonn

a(n) = A087039(A002808(n));

a(n) = A006530(A002808(n)/A052369(n)).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002808, A006530, A052369, A087039.

f:= proc(n) local F;
  if isprime(n) then return NULL fi;
  F:= sort(ifactors(n)[2],(a,b) -> a[1]>b[1]);
  if F[1][2] >= 2 then F[1][1] else F[2][1] fi
end proc:
map(f, [$2..200]); # Robert Israel, Feb 20 2024

Table[l=FactorInteger[ResourceFunction["Composite"][n]];If[Last[l][[2]]>1,Last[l][[1]],First[Part[l,-2]]],{n,102}] (* James C. McMahon, Feb 20 2024 *)

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.

Original entry on oeis.org

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), ", "));}

A189172 Largest prime number tried when factoring n using trial division.

Original entry on oeis.org

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

Offset: 1

Dan Uznanski, May 02 2011

nonn

When factoring a number via trial division, one generally continues trying primes until it is certain that the remaining portion of n is prime. Sometimes, it is already clear that the remaining portion is prime before that portion is found; in this case, the last prime tried is the second to last prime factor.

			A(22) is 3, because after 3 is tried, it is clear that 11 is prime and no more factorization can be done.
A(18) is 3, because despite the largest prime factor (3) being obviously prime, it is not obviously the last factor until the first 3 is factored out.

T. D. Noe, Table of n, a(n) for n = 1..10000

Like A059396 but also works on composites; uses A006530, A087039, A000040.

prime(k), not shown, gives A000040[k].
function a(n) {
  var k = 1;
  while (Math.pow(prime(k),2) <= n) {
    var p = prime(k);
    if (n % p == 0) {
      n /= p;
    } else {
      k += 1;
    }
  }
  return p;
}

a[n_] := Module[{m = n, k = 1, p = 1, q}, While[q = Prime[k]; q^2 <= m, p = q; m = m/p^IntegerExponent[m, p]; k++]; p]; Array[a,100] (* T. D. Noe, May 04 2011 *)

a(n) = max(A087039(n), A007917(A000196(A006530(n)))).

A215776 Second-largest prime factor of the n-th number that is a product of exactly n primes.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 3, 3, 3, 2, 3, 5, 2, 3, 3, 3, 3, 2, 5, 5, 2, 3, 3, 2, 3, 7, 3, 3, 3, 5, 5, 5, 3, 2, 3, 2, 5, 5, 3, 3, 3, 7, 2, 3, 3, 3, 7, 5, 2, 5, 5, 5, 3, 2, 3, 5, 3, 7, 3, 5, 2, 5, 5, 3, 3, 2, 3, 7, 3, 3, 3, 3, 5, 7, 2, 5, 7, 11, 2, 7, 3, 5, 5, 5, 3, 3, 3

Offset: 1

Jonathan Vos Post, Aug 23 2012

nonn

This is to A215405 as 2nd largest prime factor is to largest (greatest) prime factor. Technically, the prime numbers are "1-almost prime."

			a(2) = 2 because the 2nd number that is a product of exactly 2 primes
(semiprime) is 6 = 2*3, so 2 is the 2nd largest of those two prime factors.
a(4) = 2 because the 4th number that is a product of exactly 4 primes is 40 = 2*2*2*5, so 2 is the 2nd largest of those two distinct prime factors {2,5}. This requires clarity in "distinct prime factors" versus merely "prime factors."
a(87) = 3 because the 87th number that is a product of 87 primes is 5048474222710691433572990976 = 2^84 3^2 29, and 3 is the 2nd largest prime factor.

Cf. A087039, A101695, A215405.

A215776 := proc(n)
    A087039(A101695(n)) ;
end proc: # R. J. Mathar, Sep 14 2012

a(n) = A087039(A101695(n)).

Corrected by R. J. Mathar, Sep 14 2012

More terms from Lars Blomberg, Mar 02 2016

cp's OEIS Frontend

A143766 a(n+1) = a(n)^2 + 3*n*a(n) + n^2, a(1) = 1.

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A087040 2nd largest prime factor of n-th composite number.

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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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A189172 Largest prime number tried when factoring n using trial division.

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A215776 Second-largest prime factor of the n-th number that is a product of exactly n primes.

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Extensions

A143766 a(n+1) = a(n)^2 + 3na(n) + n^2, a(1) = 1.