A119288 -id:A119288 - cp's OEIS frontend

A076820 Second-largest distinct prime dividing n (or 1 if n is a power of a prime).

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

Offset: 1

R. K. Guy, Nov 19 2002

The 3rd-largest is 1 except 2 at 30, 42, 60, 66, 70, ...

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

Cf. A006530, A119288, A292269.

A076820(n) = { my(o=omega(n)); if(1>=o,1,factor(n)[o-1,1]); }; \\ Antti Karttunen, Sep 25 2018

If p_1 < p_2 < ...

More terms from Jason Earls and John W. Layman, Nov 21 2002

A351563 a(n) is the exponent of the second smallest prime factor of n, or 0 if n is a power of a prime.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 01 2022

nonn

			For n = 4 = 2^2 there is no second smallest prime factor as 4 is a power of prime, therefore a(4) = 0.
For n = 18 = 2^1 * 3^2, the exponent of the second smallest prime factor (3) is 2, therefore a(18) = 2.

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

Cf. A000961 (positions of zeros), A001221, A028234, A067029.

Cf. also A119288, A351567.

Array[If[Length[#] < 2, 0, #[[2, -1]]] &@ FactorInteger[#] &, 108] (* Michael De Vlieger, Apr 01 2022 *)

A351563(n) = if(1>=omega(n), 0, (factor(n))[2,2]);

a(n) = A067029(A028234(n)).

A372864 Numbers k such that A372720(k) = 0.

Original entry on oeis.org

1, 500, 578, 722, 750, 1058, 1500, 1682, 1922, 2646, 2744, 3430, 3645, 4800, 5202, 5346, 5476, 5488, 5625, 6318, 6400, 6724, 7168, 7396, 8000, 8836, 10092, 10976, 11236, 11532, 11979, 12005, 13068, 13924, 14450, 14884, 15309, 16810, 16875, 16896, 18050, 18225

Offset: 1

Michael De Vlieger, Jun 02 2024

nonn

Let tau = A000005, let omega = A001221, let f = A008479, and let g = A372720.
For squarefree k, A372720(k) >= 0, since f(k) = 1 while tau(k) = 2^omega(k).
For prime power p^m, A372720(p^m) = 1, since f(p^m) = m while tau(k) = m+1.
Therefore, apart from a(1) = 1, this sequence is a proper subset of A126706.
In the sequence R = {k = m*s : rad(m) | s, s > 1 in A120944}, there is a smallest term k such that g(k) <= 0 and a largest term k such that g(k) is positive. For instance, in A033845 where s = 6, only {6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864} are such that g(k) > 0.
Apart from terms in this sequence, all the rest of the terms k in R are such that g(k) is negative.
There are no 3-smooth numbers k > 1 in this sequence, however there are 3 terms {500, 6400, 8000} in A033846 (with s = rad(k) = 10). For s = 2*3*23, there are 6 terms {19044, 25392, 38088, 70656, 536544, 953856}.
Conjecture: proper subset of A361098, hence of A360765 and A360768. This is to say that k = a(n) is such that A003557(k) >= A119288(k), i.e., k/rad(k) >= second smallest prime factor of k, and A003557(k) > A053669(k), where A053669(k) is the smallest prime q that does not divide k.

			a(1) = 1 since tau(1) - f(1) = 1 - 1 = 0.
a(2) = 500 = 2^2 * 5*3, since tau(500) - f(500)
     = (2+1)*(3+1) - card({10,20,40,50,80,100,160,200,250,320,400,500})
     = 12 - 12 = 0.
a(3) = 578 = 2*17^2, since tau(578) - f(578)
     = (1+1)*(2+1) - card({34,68,136,272,544,578})
     = 6 - 6 = 0, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..2000

Cf. A000005, A001221, A007947, A008479, A010846, A372720, A372972.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Position[Table[r = rad[n]; DivisorSigma[0, n] - Count[Range[n/r], ?(Divisible[r, rad[#]] &)], {n, 20000}], ?(# == 0 &)][[All, 1]]

A365710 a(n) = second smallest distinct prime factor of A126706(n).

Original entry on oeis.org

3, 3, 5, 3, 7, 3, 5, 11, 5, 3, 5, 13, 3, 7, 3, 7, 17, 3, 5, 19, 5, 3, 11, 3, 23, 3, 7, 11, 5, 13, 3, 7, 29, 13, 3, 31, 3, 3, 5, 17, 5, 3, 7, 37, 3, 19, 17, 3, 5, 3, 41, 3, 19, 43, 7, 11, 3, 23, 47, 7, 3, 7, 3, 5, 3, 23, 13, 53, 3, 5, 7, 5, 3, 29, 3, 59, 3, 11

Offset: 1

Michael De Vlieger, Jan 05 2024

Since omega(A126706(n)) = A001221(A126706(n)) > 1, and since A126706 is infinite, a(n) exists for all n.

			Let b(n) = A126706(n).
a(1) = 3 since b(1) = 12 = 2^2 * 3.
a(2) = 3 since b(2) = 18 = 2 * 3^2.
a(3) = 5 since b(3) = 20 = 2^2 * 5, etc.

Cf. A119288, A126706.

FactorInteger[#][[2, 1]] & /@ Select[Range[250], PrimeOmega[#] > PrimeNu[#] > 1 &]

a(n) = A119288(A126706(n)) > 2.

A376833 Second smallest prime factor of numbers m that are both squarefree and composite.

Original entry on oeis.org

3, 5, 7, 5, 7, 11, 13, 3, 11, 17, 7, 19, 13, 3, 23, 17, 11, 19, 29, 31, 13, 3, 23, 5, 37, 11, 3, 41, 17, 43, 29, 13, 31, 47, 19, 3, 5, 53, 5, 37, 3, 23, 59, 17, 61, 41, 43, 5, 19, 67, 3, 47, 71, 13, 29, 73, 7, 31, 79, 53, 23, 5, 83, 5, 3, 59, 89, 7, 61, 37, 3

Offset: 1

Michael De Vlieger, Oct 05 2024

			Let b(n) = A120944(n).
a(1) = 3 since b(1) = 6, and 3 is the second smallest prime factor.
a(2) = 5 since b(2) = 10, and 5 is the second smallest prime factor.
Table showing select values of a(n):
    n   b(n)          a(n)
  -----------------------
   1    6 = 2*3        3
   2   10 = 2*5        5
   3   14 = 2*7        7
   4   15 = 3*5        5
   5   21 = 3*7        7
   6   22 = 2*11      11
   7   26 = 2*13      13
   8   30 = 2*3*5      3
  14   42 = 2*3*7      3
  22   66 = 2*3*11     3
  24   70 = 2*5*7      5
  82  210 = 2*3*5*7    3

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^20.

Cf. A119288, A120944.

Map[FactorInteger[#][[2, 1]] &, Select[Range[250], And[SquareFreeQ[#], CompositeQ[#]] &]]

from math import isqrt
from sympy import primepi, mobius, primefactors
def A376833(n):
    def f(x): return n+1+primepi(x)+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n+1, f(n+1)
    while m != k:
        m, k = k, f(k)
    return primefactors(m)[1] # Chai Wah Wu, Oct 06 2024

a(n) = A119288(A120944(n)).

For even squarefree semiprime A120944(n) = 2*p with odd prime p, a(n) = p sets a record in this sequence.

A381250 a(n) = least k with n distinct prime factors such that floor(log_q(k)) = floor(log_p(k))-1, where p is the smallest prime factor of k, and q is any other distinct prime factor of k.

Original entry on oeis.org

1, 2, 6, 1001, 81719, 101007559, 84248643949, 78464111896111, 997804397813471821, 1314665322768473913751, 25030469300030639321689313, 93516019518175801382127421211, 1873482639168918364977596279806547, 60958708904928776821774364389940352443, 1089851191947047137351117158610882538395561

Offset: 0

Michael De Vlieger, Apr 21 2025

nonn

Terms are squarefree.

			Let lpf = A020639, slpf = A119288, and gpf = A006530.
Table of a(n), n=0..12, listing the indices of the smallest, second smallest, and greatest prime factors, the latter 2 pertaining to n >= 2 and n >= 3, respectively.
                                           prime indices
 n                                  a(n)   lpf  slpf-gpf  prime factors
-------------------------------------------------------------------------
 0                                    1     0             -
 1                                    2     1             2
 2                                    6     1     2       2*3
 3                                 1001     4     5-6     7*11*13
 4                                81719     5     7-9     11*17*19*23
 5                            101007559     9    13-16    23*41*43*47*53
 6                          84248643949    12    19-23    etc.
 7                       78464111896111    17    25-30
 8                   997804397813471821    26    41-47
 9               1314665322768473913751    32    48-55
10           25030469300030639321689313    47    69-77
11        93516019518175801382127421211    56    83-92
12   1873482639168918364977596279806547    73   108-118
Let f(p,k) = floor(log_p k) and let w be the list of f(p,k) across the sorted list of distinct prime factors of k.
a(0) = 1 since 1 is the only number that does not have prime factors.
a(1) = 2 since prime numbers have just 1 prime factor, and 2 is the smallest prime.
a(2) = 6 since f(2,6) = 2 and f(3,6) = 1; 6 is the smallest squarefree semiprime.
a(3) = 1001 since w(1001) = {3,2,2} and is the smallest sphenic number with this property.
30 is not in the sequence since w(30) = {4,3,2}; 42 is not in since w(42) = {5,3,1}, etc.
a(4) = 81719 since w(81719) = {4,3,3,3} and is the smallest number with 4 distinct prime factors with this property, etc.

Cf. A001221, A005117, A006530, A020639, A119288, A138109, A382022.

f[om_, lm_] := Block[{f, i, j, k, nn, p, q, w, z},
  i = Abs[om]; z = i - 1; j = z; nn = Abs[lm]; w = ConstantArray[1, i];
  Catch@ Do[
    While[Set[{k, p, q}, {Times @@ #, #[[1]], #[[2]]}] &@
      Map[Prime, Accumulate@ w]; k <= nn,
      If[And[q^i > k, p^(i + 1) > k], Throw[k]];
    j = z; w[[-j]]++];
    If[j == i, Break[], j++; w[[-j]]++;
    w = PadRight[w[[;; -j]], i, 1]], {ii, Infinity}] ];
{1, 2}~Join~Table[f[n, 2^(11*n + 2)], {n, 2, 16}]

A384004 a(n) = smallest k such that A010846(k) = n.

Original entry on oeis.org

1, 2, 4, 8, 6, 10, 22, 12, 44, 18, 24, 50, 98, 36, 48, 54, 224, 30, 42, 70, 108, 66, 78, 162, 102, 60, 138, 84, 174, 260, 132, 90, 126, 228, 354, 120, 234, 168, 350, 306, 150, 516, 408, 180, 252, 552, 696, 294, 240, 336, 612, 378, 270, 1416, 300, 702, 1332, 360

Offset: 1

Michael De Vlieger, Jun 10 2025

nonn

For n > 2, a(n) is composite, since A010846(p) = 2 for prime p.
For n <= 3, a(n) = 2^n; for n > 3, a(n) < 2^n, and a(n) is in A024619.
Smallest k with omega(k) = i is A002110(i).
Conjecture: there are only 8 powerful terms (i.e., in A001694) in the sequence.

			Table of n, a(n) for n=1..10, showing row a(n) of A162306, replacing lpf(a(n)) with p, and A119288(a(n)) with q. Note: A010846(n) is the length of row n of A162306.
 n  a(n)  row n of A162306
----------------------------------------------------------
 1:   1   {1}
 2:   2   {1, p}
 3:   4   {1, p, p^2}
 4:   8   {1, p, p^2, p^3}
 5:   6   {1, p, q, p^2, p*q}
 6:  10   {1, p, p^2, q, p^3, p*q}
 7:  22   {1, p, p^2, p^3, q, p^4, p*q}
 8:  12   {1, p, q, p^2, p*q, p^3, q^2, p^2*q}
 9:  44   {1, p, p^2, p^3, q, p^4, p*q, p^5, p^2*q}
10:  18   {1, p, q, p^2, p*q, p^3, q^2, p^2*q, p^4, p*q^2}

Michael De Vlieger, Table of n, a(n) for n = 1..4647
Michael De Vlieger, Log log scatterplot of a(n), n = 1..4647, showing primes in large red, proper prime powers in large gold, composite primorials in large bright green, other squarefree composites in small green, and numbers neither squarefree nor prime powers in blue or magenta, with magenta signifying powerful numbers that are not prime powers.

Cf. A000079, A001222, A010846, A024619, A162306, A244052.

Superset of A002110.

(* First, load the theta program from the algorithms linked in A369609, then: *)
nn = 2310; t[_] := 0; u = 1; Do[(If[t[#] == 0, t[#] = n]; If[# == u, While[t[u] != 0, u++]]) &[theta[n]], {n, nn}]; Array[t, u - 1]

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A380537 Numbers k such that k is a multiple of A351566(k), where A351566 is the radix prime of the second least significant nonzero digit in the primorial base expansion of n, or 1 if there is no such digit.

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A076820 Second-largest distinct prime dividing n (or 1 if n is a power of a prime).

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A351563 a(n) is the exponent of the second smallest prime factor of n, or 0 if n is a power of a prime.

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A372864 Numbers k such that A372720(k) = 0.

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A365710 a(n) = second smallest distinct prime factor of A126706(n).

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A376833 Second smallest prime factor of numbers m that are both squarefree and composite.

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A381250 a(n) = least k with n distinct prime factors such that floor(log_q(k)) = floor(log_p(k))-1, where p is the smallest prime factor of k, and q is any other distinct prime factor of k.

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A384004 a(n) = smallest k such that A010846(k) = n.

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