A382022 -id:A382022 - cp's OEIS frontend

A383177 Sphenic numbers k such that floor(log(k)/log(lpf(k))) = 1+floor(log(k)/log(p)) for all primes p | k such that p > lpf(k), where lpf = A020639(k).

1001, 1309, 1547, 1729, 2093, 2261, 3553, 4199, 4301, 4807, 5681, 6061, 6479, 7337, 7843, 8671, 9269, 9361, 9889, 10373, 10879, 11063, 11339, 11687, 11803, 11891, 12121, 12617, 13079, 13717, 13949, 13981, 14911, 15283, 15457, 16211, 16523, 17081, 17329, 17719

Offset: 1

Michael De Vlieger, Apr 21 2025

nonn

Subset of A382022, a subset of A007304.
Let primes p, q, r, p < q < r divide k.
Then floor(log(k)/log(p)) = 3 and floor(log(k)/log(q)) = floor(log(k)/log(r)) = 2.
Row a(n) of A162306 is the set {1, p, p^2, p^3, q, p*q, p^2*q, q^2, p*q^2, r, p*r, p^2*r, q*r, p*q*r, r^2}.

			Let s(n) = A010846(a(n)).
Table of a(n) for n = 1..12, showing prime factors of a(n) and
 n   a(n)  facs(a(n))  s(n)
---------------------------
 1   1001    7*11*13    15
 2   1309    7*11*17    15
 3   1547    7*13*17    15
 4   1729    7*13*19    15
 5   2093    7*13*23    15
 6   2261    7*17*19    15
 7   3553   11*17*19    15
 8   4199   13*17*19    15
 9   4301   11*17*23    15
10   4807   11*19*23    15
11   5681   13*19*23    15
12   6061   11*19*29    15
Let f(p,k) = floor(log(k)/log(p)) and let w be the list of f(p,k) across the sorted list of distinct prime factors of k.
30 = 2*3*5 is not in the sequence since f(30,2) = 4, f(30,3) = 3, f(30,5) = 2.
a(1) = 1001 = 7*11*13; f(7,1001) = 3, f(11,1001) = 2, f(13,1001) = 2.
a(2) = 1309 = 7*11*17; w(1309) = {3,2,2}, etc.
Pattern of numbers in row a(n) of A275280:
  Level r^0                    Level r^1               Level r^2
  1,   p,     p^2,  p^3   |    r,   p*r,   p^2*r   |   r^2
  q,   p*q,   p^2*q       |    q*r, p*q*r          |
  q^2, p*q^2;             |
Example: k = 1001 = 7*11*13
    1,   7,  49, 343   |    13,   91, 637   |   169
   11,  77, 539        |   143, 1001        |
  121, 847             |

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Hasse diagram of R(1001) with logarithmic vertical scale. Gray represents the empty product, red represents primes, gold represents proper prime powers, green squarefree composites, and blue numbers that are neither squarefree nor prime powers.
Michael De Vlieger, Three dimensional diagram of R(a(n)), labeling exponents along axes, showing p^3, q^2, and r^2, and using the color scheme above.
Michael De Vlieger, Plot prime(i) | a(n) at (x,y) = (n,i) for n = 1..2048, 8X vertical exaggeration. The green bar at the bottom of the graph emphasizes the x axis that rides on the top edge of the bar.

Cf. A005117, A007304, A010846, A162306, A381250.

Intersection of A380995 and A382022.

f[om_, lm_ : 0] := Block[{f, i, j, k, nn, w}, i = Abs[om]; j = 1;
  If[lm == 0, nn = Times @@ Prime@ Range[i], nn = Abs[lm]]; w = ConstantArray[1, i];
  Union@ Reap[Do[
    While[Set[k, Times @@ Map[Prime, Accumulate@w]]; k <= nn,
      If[Or[k == 1, Union[#2] == #1 - 1 & @@
        TakeDrop[Map[Floor@Log[#, k] &, FactorInteger[k][[All, 1]] ], 1] ],
        Sow[k]];
      j = 1; w[[-j]]++];
      If[j == i, Break[], j++; w[[-j]]++;
        w = PadRight[w[[;; -j]], i, 1]], {n, Infinity}] ][[-1, 1]] ];
f[3, 20000]

A010846(a(n)) = 15.

A381736 Integers k = pqr, where p < q < r are distinct primes and p*q > r.

Original entry on oeis.org

30, 70, 105, 154, 165, 182, 195, 231, 273, 286, 357, 374, 385, 399, 418, 429, 442, 455, 494, 561, 595, 598, 627, 646, 663, 665, 715, 741, 759, 782, 805, 874, 897, 935, 957, 969, 986, 1001, 1015, 1023, 1045, 1054, 1085, 1102, 1105, 1131, 1173, 1178, 1209

Offset: 1

Matthew Goers, Mar 05 2025

nonn

These are squarefree 3-almost-primes, called sphenic numbers, that are greater than the square of the largest of its prime factors. As all sphenic numbers are, by definition, less than the cube of their largest prime factor, numbers in this sequence satisfy r^2 < k < r^3, where k = p*q*r, p < q < r.

			30 = 2*3*5 and 2*3 > 5, so 30 is in the sequence.
70 = 2*5*7 and 2*5 > 7, so 70 is in the sequence.
110 = 2*5*11 but 2*5 < 11, so 110 is not in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000
Matthew Goers, Factors of Terms

Intersection of A007304 (sphenic numbers) and A164596.

Cf. A382022.

N:= 2000: # for terms < N
P:= select(isprime, [2,seq(i,i=3..isqrt(N),2)]):
R:= NULL:
for k from 1 to nops(P) do
  for i from 1 to k-2 while P[i]*P[i+1]*P[k] < N do
     jmin:= max(i+1,ListTools:-BinaryPlace(P,P[k]/P[i])+1);
     jmax:= min(k-1,ListTools:-BinaryPlace(P,N/(P[i]*P[k])));
     R:= R, seq(P[i]*P[j]*P[k],j=jmin .. jmax);
od od:
sort([R]); # Robert Israel, Mar 28 2025

q[n_] := Module[{f = FactorInteger[n]}, f[[;; , 2]] == {1, 1, 1} && f[[1, 1]]*f[[2, 1]] > f[[3, 1]]]; Select[Range[1500], q] (* Amiram Eldar, Mar 20 2025 *)

is_a381736(n) = my(F=factor(n)); omega(F)==3 && bigomega(F)==3 && F[1,1]*F[2,1]>F[3,1] \\ Hugo Pfoertner, Mar 08 2025

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A381736(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(primepi(min(x//(p*q),p*q-1))-b for a,p in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,q in enumerate(primerange(p+1,isqrt(x//p)+1),a+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 28 2025

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}]

cp's OEIS Frontend

A383177 Sphenic numbers k such that floor(log(k)/log(lpf(k))) = 1+floor(log(k)/log(p)) for all primes p | k such that p > lpf(k), where lpf = A020639(k).

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A381736 Integers k = pqr, where p < q < r are distinct primes and p*q > r.

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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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cp's OEIS Frontend

A383177 Sphenic numbers k such that floor(log(k)/log(lpf(k))) = 1+floor(log(k)/log(p)) for all primes p | k such that p > lpf(k), where lpf = A020639(k).

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A381736 Integers k = p*q*r, where p < q < r are distinct primes and p*q > r.

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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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A381736 Integers k = pqr, where p < q < r are distinct primes and p*q > r.