A253567 -id:A253567 - cp's OEIS frontend

A253784 Numbers which have no two successive prime factors (when sorted into monotonic order) where the latter prime factor would be greater than the square of the former.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 32, 35, 36, 37, 41, 42, 43, 45, 47, 48, 49, 53, 54, 55, 59, 60, 61, 63, 64, 65, 67, 71, 72, 73, 75, 77, 79, 81, 83, 84, 85, 89, 90, 91, 95, 96, 97, 101, 103, 105, 107, 108, 109, 113, 115, 119, 120, 121, 125, 126, 127, 128, 131, 133, 135, 137, 139, 143, 144

Offset: 1

Antti Karttunen, Jan 16 2015

nonn

In other words, {1} together with primes and such composite numbers n = p_i * p_j * p_k * ... * p_u, p_i <= p_j <= p_k <= ... <= p_u, where each successive prime factor (when sorted into a nondecreasing order) is less than the square of the previous: (p_i)^2 > p_j, (p_j)^2 > p_k, etc.

Whenever gcd(a(i),a(j)) > 1, then a(i)*a(j) and lcm(a(i),a(j)) are also members of this sequence.

			1 is present as it has an empty prime factorization.
2 like all primes is present.
4 = 2*2 is present as 2^2 > 2.
9 = 3*3 is present as 3^2 > 3.
10 = 2*5 is NOT present, as 2^2 < 5.
30 = 2*3*5 is present, as 2^2 > 3 and 3^2 > 5.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A253785.
Subsequences: A251726 (a(n+1) differs from A251726(n) for the first time at n=23, where a(24) = 30, while A251726(23) = 31), A251728 (semiprimes only).
Subsequence of A253567.
Cf. A000290.

A253569 Composite numbers n = p_i * p_j * p_k * ... * p_u, p_i <= p_j <= p_k <= ... <= p_u, where each successive prime factor (when sorted into a nondecreasing order) is greater than the square of the previous: (p_i)^2 < p_j, (p_j)^2 < p_k, etc.

Original entry on oeis.org

10, 14, 22, 26, 33, 34, 38, 39, 46, 51, 57, 58, 62, 69, 74, 82, 86, 87, 93, 94, 106, 111, 118, 122, 123, 129, 134, 141, 142, 145, 146, 155, 158, 159, 166, 177, 178, 183, 185, 194, 201, 202, 205, 206, 213, 214, 215, 218, 219, 226, 235, 237, 249, 254, 262, 265, 267, 274, 278, 290

Offset: 1

Antti Karttunen & Reinhard Zumkeller, Jan 09 2015

nonn

Numbers n = A020639(n) * A014673(n) * A054576(n), for which A020639(n)^2 < A014673(n) and either A054576(n) = 1 or A032742(n) satisfies the same condition (is the term of this sequence).

			290 = 2*5*29 is a member, because 2^2 < 5 and 5^2 < 29.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Complement: A253567.
Subsequence of A002808, A005117, A088381, A251727, A245729 and A253785.
A138511 is a subsequence, from which this sequence differs for the first time at n=60, where A138511(60) = 291, while here a(60) = 290.
Cf. A000290, A001222, A020639, A032742, A014673, A054576.

a253569 n = a253569_list !! (n-1)
a253569_list = filter f [1..] where
                    f x = (p ^ 2 < a020639 q) && (a010051' q == 1 || f q)
                          where q = div x p; p = a020639 x
-- Antti Karttunen after Reinhard Zumkeller's code for A138511, Jan 09 2015
a253569 n = a253569_list !! (n-1)
a253569_list = filter (not . f''') a002808_list where
   f''' x = p ^ 2 > a020639 q || (a010051 q == 0 && f''' q)
            where q = div x p; p = a020639 x
-- Reinhard Zumkeller, Jan 12 2015
(Scheme, with Antti Karttunen's IntSeq-library)
(define A253569 (MATCHING-POS 1 1 (lambda (n) (and (> (A001222 n) 1) (numbers-sparsely-distributed? (ifactor n))))))
(define (numbers-sparsely-distributed? lista) (cond ((null? lista) #t) ((null? (cdr lista)) #t) ((> (A000290 (car lista)) (cadr lista)) #f) (else (numbers-sparsely-distributed? (cdr lista)))))
;; Antti Karttunen, Jan 16 2015

cnQ[n_]:=CompositeQ[n]&&Union[Boole[#[[2]]>#[[1]]^2&/@Partition[Flatten[Table[ #[[1]], #[[2]]]&/@FactorInteger[n]],2,1]]]=={1}; Select[Range[300],cnQ] (* Harvey P. Dale, Jul 10 2023 *)

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

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.

Original entry on oeis.org

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

A269131 Composite numbers whose largest prime factor is less than its second-largest prime factor's square, counting with multiplicity so that the factors of 18 are 2, 3, 3.

Original entry on oeis.org

4, 6, 8, 9, 12, 15, 16, 18, 21, 24, 25, 27, 30, 32, 35, 36, 42, 45, 48, 49, 50, 54, 55, 60, 63, 64, 65, 70, 72, 75, 77, 81, 84, 85, 90, 91, 95, 96, 98, 100, 105, 108, 110, 115, 119, 120, 121, 125, 126, 128, 130, 133, 135, 140, 143, 144, 147, 150, 154, 161, 162, 165

Offset: 1

Marc Moskowitz, Feb 19 2016

These are numbers that a naïve factoring algorithm can declare done at the penultimate prime factor.

Marc Moskowitz, Table of n, a(n) for n = 1..10000

Cf. A002808, A251728 (semiprime terms).

Subsequence of A253567.

cnQ[n_]:=Module[{pfs=Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[n]]}, CompositeQ[ n]&&Last[pfs]Harvey P. Dale, Nov 05 2017 *)

is(n)=my(f=factor(n),e=#f~); e && (f[e,2]>1 || (e>1 && f[e-1,1]^2>f[e,1])) \\ Charles R Greathouse IV, Feb 19 2016

seq = []
for n in range(2, 1000):
    temp = n
    factor = 2
    while temp > 1:
        if temp % factor == 0:
            temp //= factor
            if temp == 1:
                continue
            if temp < factor * factor:
                seq.append(n)
        else:
            factor += 1
print(seq)

cp's OEIS Frontend

A253784 Numbers which have no two successive prime factors (when sorted into monotonic order) where the latter prime factor would be greater than the square of the former.

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A253569 Composite numbers n = p_i * p_j * p_k * ... * p_u, p_i <= p_j <= p_k <= ... <= p_u, where each successive prime factor (when sorted into a nondecreasing order) is greater than the square of the previous: (p_i)^2 < p_j, (p_j)^2 < p_k, etc.

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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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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.

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Mathematica

PARI

Python

A269131 Composite numbers whose largest prime factor is less than its second-largest prime factor's square, counting with multiplicity so that the factors of 18 are 2, 3, 3.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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.

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.