Matthew Goers - cp's OEIS frontend

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

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

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

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

A366867 Products of sexy prime triples: sphenic numbers with prime factorization (p - 6)p(p + 6).

Original entry on oeis.org

935, 1729, 4301, 11339, 49321, 102131, 146969, 298351, 386389, 1089019, 1221191, 3864241, 5171489, 12640949, 16965341, 18181979, 21243961, 43974269, 51881689, 178433279, 208506509, 223626691, 230324329, 270816731, 278421569, 393806449, 849244031, 932539661

Offset: 1

Matthew Goers, Oct 25 2023

nonn

			5, 11, and 17 are primes p, p+6, p+12, called a sexy prime triple. 5*11*17 = 935, so 935 is a term.
7, 13, and 19 are the second set of sexy prime triples. 7*13*19=1729, so 1729 is the second term.

Matthew Goers, Table of n, a(n) for n = 1..1000 [a(535) corrected by _Georg Fischer_, Sep 21 2024]

Cf. A006489, A111192. Subsequence of A007304.

(#*(#^2 - 36)) & /@ Select[Prime[Range[180]], PrimeQ[# - 6] && PrimeQ[# + 6] &] (* Amiram Eldar, Oct 27 2023 *)

apply(x->x*(x-6)*(x+6), select(x->(isprime(x-6) && isprime(x) && isprime(x+6)), [1..1000])) \\ Michel Marcus, Oct 27 2023

a(n) = (A006489(n) - 6)*A006489(n)*(A006489(n) + 6).

A272899 Product of next n prime numbers greater than n.

Original entry on oeis.org

1, 2, 15, 385, 5005, 323323, 7436429, 955049953, 35336848261, 1448810778701, 62298863484143, 14107860812636383, 832363787945546597, 261682369333342226303, 18579448222667298067513, 1356299720254712758928449, 107147677900122307955347471, 46558817449894322874479515781

Offset: 0

Matthew Goers, May 09 2016

a(n) is of comparable size to n^n. - Charles R Greathouse IV, May 09 2016

a(n) is the product of the terms of the n-th row of A084754. - Michel Marcus, May 09 2016

			a(0) = 1 (the empty product).
a(1) = 2 = 2.
a(2) = 3 * 5 = 15.
a(3) = 5 * 7 * 11 = 385.
a(4) = 5 * 7 * 11 * 13 = 5005.

Alois P. Heinz, Table of n, a(n) for n = 0..322

Cf. A000720, A002110, A007918, A034386, A084754.

a:= n-> mul((nextprime@@i)(n), i=1..n):
seq(a(n), n=0..17);  # Alois P. Heinz, Jun 24 2024

Table[Times@@Prime[Range[PrimePi[n] + 1, PrimePi[n] + n]], {n, 25}] (* Alonso del Arte, May 09 2016 *)

a(n)=my(v=primes(primepi(n)+n)); prod(i=0,n-1,v[#v-i]) \\ Charles R Greathouse IV, May 09 2016

from math import prod
from sympy import prime, primepi
def a(n): r = primepi(n); return prod(prime(i) for i in range(r+1, r+n+1))
print([a(n) for n in range(1, 17)]) # Michael S. Branicky, Feb 15 2021

a(n) = A002110(n + A000720(n))/A034386(n), where A002110(n) are the primorials, A000720(n) is the pi(n) prime counting function, and A034386(n) is the primorial of primes less than or equal to n. E.g., a(7) = 955049953 = A002110(11) / A034386(7).

a(0)=1 prepended by Alois P. Heinz, Jun 24 2024

A229467 Related to Pisano periods: numbers n such that there are n+10 distinct Fibonacci numbers mod n.

Original entry on oeis.org

170, 230, 530, 830, 1370, 1670, 1730, 1970, 2270, 2570, 2930, 3170, 3830, 4430, 4670, 5030, 5870, 5930, 6170, 6470, 6530, 6830, 7430, 7730, 8270, 8570, 8630, 8870, 9470, 9770, 9830, 10130, 11630, 11870, 11930, 12170, 12830, 13070, 13670, 13730, 14330

Offset: 1

Matthew Goers, Sep 24 2013

nonn

These are a subset of the terms of A071774 multiplied by 10, where A071774 are numbers m such that Fibonacci numbers mod m = 2*(m+1). All A071774 terms multiplied by 10 have Pisano periods 3*(n+10) or (n+10). This sequence is the (n+10) subset.

			The Pisano period of the Fibonacci numbers mod 170 = 180, which is 170+10.
The Pisano period of the Fibonacci numbers mod 1670 = 1680, which is 1670+10.

Matthew Goers, Table of n, a(n) for n = 1..60

Cf. A000045, A001175, A071774, A229466.

Added 3 terms - Matthew Goers, Oct 14 2013

A229466 Numbers k such that the period of Fibonacci numbers mod k is 3*(k+10).

Original entry on oeis.org

10, 30, 70, 130, 370, 430, 670, 730, 970, 1030, 1270, 1570, 1630, 1930, 2230, 2770, 2830, 3130, 3370, 3670, 3730, 3970, 4330, 4570, 4630, 4870, 5230, 5470, 5770, 6070, 6130, 6430, 6730, 7270, 7330, 7570, 7870, 8230, 8530, 8770, 8830, 9070, 9370, 9970

Offset: 1

Matthew Goers, Sep 24 2013

nonn

Related to Pisano periods. Other than the initial term 10, these are a subset of the terms of A071774 multiplied by 10, where A071774 are numbers m such that Fibonacci numbers mod m = 2*(m+1). All A071774 terms multiplied by 10 have Pisano periods 3*(n+10) or (n+10). This sequence is the 3*(n+10) subset. A229467 is the n+10 subset.

			The Pisano period of the Fibonacci numbers mod 30 = 120, which is 3*(30+10).
The Pisano period of the Fibonacci numbers mod 1570 = 4740, which is 3*(1570+10).

Matthew Goers, Table of n, a(n) for n = 1..74

Cf. A000045, A001175, A071774, A229467.

t = {}; Do[a = {1, 0}; a0 = a; k = 0; While[k++; s = Mod[Plus @@ a, n]; a = RotateLeft[a]; a[[2]] = s; k <= 3*(n + 10) && a != a0]; If[k == 3*(n + 10), AppendTo[t, n]], {n, 2, 10000}]; t (* T. D. Noe, Oct 02 2013 *)

A227397 Related to Pisano periods: Numbers k such that the period of Fibonacci numbers mod k equals k+2.

Original entry on oeis.org

4, 34, 46, 94, 106, 166, 226, 274, 334, 346, 394, 454, 514, 526, 586, 634, 694, 706, 766, 886, 934, 1006, 1126, 1174, 1186, 1234, 1294, 1306, 1354, 1366, 1486, 1546, 1654, 1714, 1726, 1774, 1894, 1954, 1966, 2026, 2326, 2374, 2386, 2434, 2566, 2614, 2734, 2746

Offset: 1

Matthew Goers, Sep 20 2013

nonn

This sequence is a subsequence of A220168, where k divides the Fibonacci number F(k+2). There is no discernible pattern among the terms of A220168 terms that are not in this sequence.

All terms are 2 less than a multiple of 6, and all except the first term (4) are 2 less than a multiple of 12.

			The Pisano period (A001175) for dividing the Fibonacci numbers (A000045) by 4 is 6; 6 = 4 + 2, so 4 is a term.
The Pisano period for the Fibonacci numbers mod 34 is 36; 36 = 34 + 2, so 34 is a term.

Matthew Goers, Table of n, a(n) for n = 1..125

Cf. A000045, A001175, A220168, A071776.

A224401 a(n) is the row number of triangle A085612 in which n appears.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 4, 6, 3, 5, 4, 7, 4, 5, 5, 8, 9, 7, 9, 7, 5, 10, 9, 11, 3, 10, 6, 7, 9, 12, 9, 13, 10, 10, 10, 14, 9, 10, 10, 11, 9, 12, 9, 7, 7, 10, 9, 15, 16, 7, 10, 17, 18, 11, 10, 11, 19, 19, 18, 20, 18, 19, 17, 21, 19, 12, 18, 17, 19, 12, 18, 22, 18, 19

Offset: 1

Matthew Goers, Apr 05 2013

nonn

Row number in A085612 triangle of prime signatures.

a(n) is not the same for all numbers n with the same prime signature. For such a sequence, see A101296. - Peter Munn, Oct 23 2023

			a(9) = 3, because 9 is in the 3rd row (1st 3 prime^2) of A085612.
a(10) = 5, because 10 is in the 5th row (1st 5 semiprimes) of A085612.
a(11) = 4, because 11 is in the 4th row (4 primes, prime(3)..prime(6)) of A085612.

Rémy Sigrist, Table of n, a(n) for n = 1..10000 (first 150 terms from Matthew Goers)
Rémy Sigrist, PARI program for A224401

Cf. A085612, A101296.

First and last positions of each number: A085834, A085836.

import Data.List (findIndex); import Data.Maybe (fromJust)
a224401 = (+ 1) . fromJust . (`findIndex` a085612_tabf) . elem
-- Reinhard Zumkeller, Jun 05 2013

PARI
```
See Links section.
```

cp's OEIS Frontend

User: Matthew Goers

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

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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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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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A366867 Products of sexy prime triples: sphenic numbers with prime factorization (p - 6)*p*(p + 6).

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A272899 Product of next n prime numbers greater than n.

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A229467 Related to Pisano periods: numbers n such that there are n+10 distinct Fibonacci numbers mod n.

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

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

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.

A366867 Products of sexy prime triples: sphenic numbers with prime factorization (p - 6)p(p + 6).