A375008 -id:A375008 - cp's OEIS frontend

A372209 Primes p_1 where products m of k = 3 consecutive primes p_1..p_k are such that only p_1 < m^(1/k).

3, 7, 13, 23, 31, 37, 47, 53, 61, 67, 73, 89, 97, 103, 113, 131, 139, 151, 157, 167, 173, 181, 193, 199, 211, 223, 233, 241, 251, 257, 263, 271, 277, 293, 307, 317, 337, 359, 367, 373, 389, 409, 421, 433, 449, 457, 467, 479, 491, 509, 523, 547, 557, 563, 577, 587

Offset: 1

Michael De Vlieger, Sep 11 2024

Primes p such that the second differences of p and the next 2 primes is never positive.
Superset of A022005.
Does not intersect A022004.

			3 is in the sequence since m = 3*5*7 = 105 is such that 3 is less than the cube root of 105, but both 5 and 7 exceed it.
5 is not in the sequence because m = 5*7*11 = 385 is such that both 5 and 7 are less than the cube root.
7 is in the sequence since m = 7*11*13 = 1001 is such that 7 < 1001^(1/3), but both 11 and 13 are larger than 1001^(1/3), etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A022004, A022005, A375008.

k = 3; s = {1}~Join~Prime[Range[k - 1]]; Reap[Do[s = Append[Rest[s], Prime[i + k - 1]]; r = Surd[Times @@ s, k]; If[Count[s, _?(# < r &)] == 1, Sow[Prime[i]] ], {i, 600}] ][[-1, 1]]

A382022 Composite integers k = pqr where p < q < r are distinct primes such that p*r < q^2.

Original entry on oeis.org

70, 105, 110, 154, 182, 231, 238, 266, 273, 286, 322, 374, 418, 429, 442, 494, 506, 561, 598, 627, 638, 646, 663, 682, 715, 741, 754, 759, 782, 806, 814, 874, 897, 902, 935, 946, 957, 962, 969, 986, 1001, 1023, 1034, 1045, 1054, 1066, 1102, 1105, 1118

Offset: 1

Matthew Goers, Mar 12 2025

nonn

These are squarefree, 3-almost primes, called sphenic numbers, that are less than the cube of the middle prime factor. If k = p*q*r and p < q < r, it is always true that p^3 < k < r^3. This sequence includes the terms where k < q^3.

			70 = 2*5*7 and 2*7 < 5^2, so 70 is in the sequence.
105 = 3*5*7 and 3*7 < 5^2, so 105 is in the sequence.
165 = 3*5*11 but 3*11 > 5^2, so 165 is not in the sequence.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1776 from Matthew Goers)
Matthew Goers, Factors of Terms

Subsequence of A007304 (sphenic numbers).
Supersequence of A375008 (consecutive primes p, q, r).
Cf. A381736.

q[n_] := Module[{f = FactorInteger[n]}, f[[;; , 2]] == {1, 1, 1} && f[[1, 1]] * f[[3, 1]] < f[[2, 1]]^2]; Select[Range[1200], q] (* Amiram Eldar, Mar 12 2025 *)

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A382022(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),q**2//p))-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

A376331 a(n) is the smallest product of n consecutive primes p_1..p_k, where only p_1 < m^(1/n).

Original entry on oeis.org

6, 105, 257557397, 362469273063260281, 15119658537284521518782249, 117383204057701408834470517376101793436427, 23238824136447515117641387686174787861885627837847997511, 139957288120766060385660710153537529132218663535147563443966068820553

Offset: 2

Michael De Vlieger, Sep 20 2024

Proper subset of A120944, since squarefree m is the smallest number in the sequence of numbers that have m as squarefree kernel, and since more than 1 prime is a factor.

			a(2) = 6 since m = 2*3 = 6 and 3 > sqrt(6).
a(3) = 105 since m = 3*5*7 = 105 and 5 > 105^(1/3).
a(4) = 257557397 since m = 113 * 127 * 131 * 137 = 257557397 and 127 > 257557397^(1/4), etc.

Cf. A120944, A374873, A375008, A375975, A376261.

k = 1; Table[r = Range[0, n - 1]; While[(Set[{p, q, m}, {#[[1]], #[[2]], Times @@ #}]; q < Surd[m, n]) &[Prime[k + r]], k++]; m, {n, 2, 6}]

A374873(n) = lpf(a(n)) = A020639(a(n)).

a(n) = Product_{k=0..n-1} k + pi(A374873(n)), where pi = A000720.

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A372209 Primes p_1 where products m of k = 3 consecutive primes p_1..p_k are such that only p_1 < m^(1/k).

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A382022 Composite integers k = pqr where p < q < r are distinct primes such that p*r < q^2.

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A376331 a(n) is the smallest product of n consecutive primes p_1..p_k, where only p_1 < m^(1/n).

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cp's OEIS Frontend

A372209 Primes p_1 where products m of k = 3 consecutive primes p_1..p_k are such that only p_1 < m^(1/k).

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A382022 Composite integers k = p*q*r where p < q < r are distinct primes such that p*r < q^2.

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A376331 a(n) is the smallest product of n consecutive primes p_1..p_k, where only p_1 < m^(1/n).

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A382022 Composite integers k = pqr where p < q < r are distinct primes such that p*r < q^2.