A380995 -id:A380995 - 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.

A380438 Integers k that are the product of 3 distinct primes, the smallest of which is larger than the 5th root of k: k = pqr, where p, q, r are primes and k^(1/5) < p < q < r.

Original entry on oeis.org

30, 105, 165, 195, 231, 385, 455, 595, 665, 715, 805, 935, 1001, 1015, 1045, 1085, 1105, 1235, 1265, 1295, 1309, 1435, 1463, 1495, 1505, 1547, 1595, 1615, 1645, 1705, 1729, 1771, 1855, 1885, 1955, 2015, 2035, 2065, 2093, 2135, 2185, 2233, 2255, 2261, 2345, 2365, 2387, 2405, 2431, 2465, 2485

Offset: 1

Matthew Goers, Jan 24 2025

nonn

This subsequence of the sphenics (A007304) is similar to A362910 or A138109 for semiprimes. Ishmukhametov and Sharifullina defined semiprimes n = p*q where each prime is greater than n^(1/4) as strongly semiprime. This sequence defines sphenic numbers with an analogous 'strength' as a product of 3 distinct primes k = p*q*r where each prime is greater than k^(1/5), or, alternately, k < p^5.
The only even term is 30 = 2*3*5.
As there are many equivalent ways of expressing Ishmukhametov and Sharifullina's "strongly semiprime" criterion, it is not obvious how it should most appropriately be extended to measure an equivalent "strength" of numbers with more prime factors. Here we follow a comparison of the least prime factor, p, to the factored number, k; but we could instead compare the greatest prime factor, r, to k; or p to r; or measure the variance/standard deviation of the prime factors (more precisely, after twice taking the logarithm of each factor as is done in A379271). Furthermore, it looks clear that the comparison used here (p against k^(1/5)) could be shown to give a substantially lower density asymptotically within the sphenics than Ishmukhametov and Sharifullina's equivalent for semiprimes. - Peter Munn, Feb 18 2025 and May 13 2025

			231 = 3*7*11 and 231^(1/5) < 3, so 231 is in the sequence.
255 = 3*5*17 but 255^(1/5) > 3, so 255 is not in the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..420 from Matthew Goers).
Sh. T. Ishmukhametov and F. F. Sharifullina, On distribution of semiprime numbers, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 53-59. On distribution of semiprime numbers, English translation, Russian Mathematics, Vol. 58, No. 8 (2014), pp. 43-48, ResearchGate link.

Subsequence of A253567, A290965, A379271, and A007304.
A046301 is a subsequence (product of 3 successive primes).
Cf. A115957, A138109, A251728, A362910 (strong semiprimes), A380995.

q[k_] := Module[{f = FactorInteger[k]}, f[[;; , 2]] == {1, 1, 1} && f[[1, 1]]^5 > k]; Select[Range[2500], q] (* Amiram Eldar, Feb 14 2025 *)

isok(k) = my(f=factor(k)); (bigomega(f)==3) && (omega(f)==3) && (k < vecmin(f[,1])^5); \\ Michel Marcus, Jan 27 2025

list(lim)=my(v=List()); forprime(p=2,sqrtnint(lim\=1,3), forprime(q=p+1,min(sqrtint(lim\p),p^2), forprime(r=q+2,min(lim\(p*q),p^4\q), listput(v,p*q*r)))); Set(v) \\ Charles R Greathouse IV, May 20 2025

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A380438(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(max(0,primepi(min(x//(p*q),p**4//q))-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

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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A380438 Integers k that are the product of 3 distinct primes, the smallest of which is larger than the 5th root of k: k = pqr, where p, q, r are primes and k^(1/5) < p < q < r.

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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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Mathematica

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A380438 Integers k that are the product of 3 distinct primes, the smallest of which is larger than the 5th root of k: k = p*q*r, where p, q, r are primes and k^(1/5) < p < q < r.

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Mathematica

PARI

PARI

Python

A380438 Integers k that are the product of 3 distinct primes, the smallest of which is larger than the 5th root of k: k = pqr, where p, q, r are primes and k^(1/5) < p < q < r.