A380438 -id:A380438 - cp's OEIS frontend

A380995 Integers k that are the product of 3 distinct primes, the smallest of which is larger than the 4th root of k: k = pqr, where p, q, r are primes and k^(1/4) < p < q < r.

385, 455, 595, 1001, 1309, 1463, 1547, 1729, 1771, 2093, 2233, 2261, 2387, 2431, 2717, 3289, 3553, 4147, 4199, 4301, 4433, 4807, 5083, 5291, 5423, 5681, 5797, 5863, 6061, 6149, 6409, 6479, 6721, 6851, 6919, 7163, 7337, 7429, 7579, 7657, 7667, 7733, 7843, 8041, 8177, 8437, 8569, 8671, 8723, 8789, 8987, 9061

Offset: 1

Matthew Goers, Feb 12 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 lists sphenic numbers that are a product of 3 distinct primes k = p*q*r where each prime is greater than k^(1/4).
Sequence is intersection of A007304 (sphenics) and A088382 (numbers not exceeding the 4th power of their smallest prime factor).
No terms have 2 or 3 as a prime factor, as all sphenic numbers are greater than 2^4 = 16 and all odd sphenic numbers are greater than 3^4 = 81.
A380438 is the 'less strong' sequence of sphenic numbers k = p*q*r, where k^(1/5) < p < q < r.

			595 = 5*7*17 and 595^(1/4) < 5, so 595 is in the sequence.
665 = 5*7*19 but 665^(1/4) > 5, so 665 is not in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 231 terms 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.
Matthew Goers, Factors of Terms

Cf. A007304 (sphenics), A088382, A380438, A115957, A362910 (strong semiprimes), A251728, A138109.

Subsequence of A253567, A290965.

q[k_] := Module[{f = FactorInteger[k]}, f[[;; , 2]] == {1, 1, 1} && f[[1, 1]]^4 > k]; Select[Range[10^4], q] (* Amiram Eldar, Feb 14 2025 *)

is(n) = my(f = factor(n)); f[,2] == [1,1,1]~ && f[1,1]^4 > n \\ David A. Corneth, Apr 24 2025

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A380995(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**3//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

A379271 Composite numbers, k, whose prime factors, viewed on a log log scale, have a small standard deviation defined with respect to bigomega(k), as specified in the comments.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 66, 68, 70, 72, 75, 76, 77, 78, 80, 81, 84, 85, 88, 90, 91, 92, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 115, 116, 117, 119, 120, 121, 124, 125, 126, 128

Offset: 1

Peter Munn, Feb 18 2025

nonn

Composite numbers k (written as a product of primes p_1 * p_2 * ... * p_m) such that s( {log(log(p_i)) : 1 <= i <= m} ) < s( {i : 1 <= i <= m} ), where s is standard deviation and m = bigomega(k).
Loosely described, these are numbers whose prime factors, including repetitions, are relatively close together. (Note we get the same criterion irrespective of whether s is sample standard deviation or population standard deviation.)
The author's intent is to divide the set of composite numbers into 2 parts whose asymptotic densities differ at most by a small factor. So his choice of criterion was guided by particular information relating to the statistics of prime factors of large numbers.
From Charles R Greathouse IV, May 19 2025: (Start)
For example, semiprimes p*q with p <= q are in this sequence if (and only if) q < p^e where e = 2.71... is the base of the natural logarithm.
For any m, there are finitely many primes p (perhaps none) such that p*m is in the sequence. (End)

Subsequences: A251728, the composites in A253784, A380438.

Select[Select[Range[128], CompositeQ], Less @@ Map[StandardDeviation, Transpose@ MapIndexed[{Log@ Log[#1], First[#2]} &, Flatten[ConstantArray[#1, #2] & @@@ FactorInteger[#] ] ] ] &] (* Michael De Vlieger, May 04 2025 *)

cp's OEIS Frontend

A380995 Integers k that are the product of 3 distinct primes, the smallest of which is larger than the 4th root of k: k = pqr, where p, q, r are primes and k^(1/4) < p < q < r.

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A379271 Composite numbers, k, whose prime factors, viewed on a log log scale, have a small standard deviation defined with respect to bigomega(k), as specified in the comments.

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Author

Keywords

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Mathematica

cp's OEIS Frontend

A380995 Integers k that are the product of 3 distinct primes, the smallest of which is larger than the 4th root of k: k = p*q*r, where p, q, r are primes and k^(1/4) < p < q < r.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

A379271 Composite numbers, k, whose prime factors, viewed on a log log scale, have a small standard deviation defined with respect to bigomega(k), as specified in the comments.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

A380995 Integers k that are the product of 3 distinct primes, the smallest of which is larger than the 4th root of k: k = pqr, where p, q, r are primes and k^(1/4) < p < q < r.