A046386 -id:A046386 - cp's OEIS frontend

A376864 4-brilliant numbers with distinct prime factors.

210, 46189, 55913, 62491, 70499, 75361, 78793, 81719, 84227, 89947, 95381, 96577, 99671, 100529, 101959, 103037, 104533, 110143, 111397, 114257, 116831, 121693, 121771, 124729, 127699, 128557, 128843, 130169, 131461, 133331, 134849, 139403, 141427, 143429

Offset: 1

Paul Duckett, Oct 07 2024

			210 = 2*3*5*7 is a term.
130169 = 13*17*19*31 is a term.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Intersection of A046386 and A376704.

from sympy import factorint
def ok(n):
    f = factorint(n)
    return len(f) == sum(f.values()) == 4 and len(set([len(str(p)) for p in f])) == 1
print([k for k in range(144000) if ok(k)]) # Michael S. Branicky, Oct 08 2024

from math import prod
from sympy import primerange
from itertools import count, combinations, islice
def bgen(d): # generator of terms that are products of d-digit primes
    primes, out = list(primerange(10**(d-1), 10**d)), set()
    for t in combinations(primes, 4): out.add(prod(t))
    yield from sorted(out)
def agen(): # generator of terms
    for d in count(1): yield from bgen(d)
print(list(islice(agen(), 34))) # Michael S. Branicky, Oct 08 2024

Terms corrected by Michael S. Branicky, Oct 08 2024

A379762 Products of 4 distinct prime numbers (or tetraprimes) that are abundant.

Original entry on oeis.org

210, 330, 390, 462, 510, 546, 570, 690, 714, 770, 798, 858, 870, 910, 930, 966, 1110, 1122, 1190, 1218, 1230, 1254, 1290, 1302, 1326, 1330, 1410, 1430, 1482, 1518, 1554, 1590, 1610, 1722, 1770, 1794, 1806, 1830, 1870, 1914, 1938, 1974, 2002, 2010, 2030, 2046, 2090, 2130, 2170, 2190, 2210, 2226, 2262, 2346, 2370, 2418, 2442, 2470, 2478, 2490, 2530, 2562, 2590, 2622

Offset: 1

Massimo Kofler, Jan 09 2025

nonn

This sequence is not 2*{A046389}. 2618  = 2*1309 is not in this sequence, while 1309 is in A046389.
Contains 6*p*q if p and q are distinct primes > 3. The first term not of this form is 770. - Robert Israel, Jan 09 2025
a(43) = 2002 is the only term coprime to 15. - Charles R Greathouse IV, Jan 13 2025

			210 is a term because 210=2*3*5*7 is the product of four distinct primes and it is smaller than the sum of its proper divisors 366.
1155 is not a term because 1155=3*5*7*11 is the product of four distinct primes and it is larger than the sum of its proper divisors 1149.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A005101 and A046386.

Cf. A001065, A378717, A379917.

filter:= proc(n) local F,t;
   F:= ifactors(n)[2];
   F[..,2] = [1,1,1,1] and mul(t[1]+1, t = F) > 2*n
end proc:
select(filter, [seq(i,i=2..3000, 4)]); # Robert Israel, Jan 09 2025

q[n_] := Module[{f = FactorInteger[n]}, f[[;; , 2]] == {1, 1, 1, 1} && Times @@ (1 + 1/f[[;; , 1]]) > 2]; Select[Range[3000], q] (* Amiram Eldar, Jan 09 2025 *)

list(lim)=my(v=List(select(k->k<=lim, [1430, 1870, 2002, 2090, 2210, 2470, 2530, 2990, 3190, 3230, 3410, 3770, 4030, 4070, 4510, 4730, 5170, 5830]))); forprime(p=5,sqrtint(lim\6), my(t=6*p); forprime(q=p+2,lim\t, listput(v,t*q))); forprime(p=11,lim\70,listput(v,70*p)); Set(v) \\ Charles R Greathouse IV, Jan 13 2025

a(n) == 2 (mod 4).

a(n) ~ (1/6)*n log n/log log n. - Charles R Greathouse IV, Jan 13 2025

A382208 Numbers k for which pi(bigomega(k)) = omega(k).

Original entry on oeis.org

1, 4, 9, 12, 18, 20, 24, 25, 28, 36, 40, 44, 45, 49, 50, 52, 54, 56, 63, 68, 75, 76, 88, 92, 98, 99, 100, 104, 116, 117, 120, 121, 124, 135, 136, 147, 148, 152, 153, 164, 168, 169, 171, 172, 175, 180, 184, 188, 189, 196, 207, 212, 225, 232, 236, 240, 242, 244, 245

Offset: 1

Felix Huber, Mar 30 2025

nonn

Numbers k for which A000720(A001222(k)) = A001221(k).
Numbers k = p_1^e_1 * ... * p_j^e_j for which pi(Sum_{i=1..j} e_i) = j where pi = A000720.
Supersequence of A001248, A054753, A065036, A085986, A162143, A179643, A179644, A179693, A179700, A179704.

			240 = 2^4*3*5 is in the sequence because pi(Omega(240)) = pi(6) = 3 = omega(240).

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A000720, A001221, A001222, A001248, A046386, A054753, A065036, A085986, A162143, A179644, A179693, A179700, A179704.

with(NumberTheory):
A382208:=proc(n)
    option remember;
    local k;
    if n=1 then
        1
    else
        for k from procname(n-1)+1 do
            if pi(Omega(k))=Omega(k,distinct) then
                return k
            fi
        od
    fi;
end proc;
seq(A382208(n),n=1..59);
# second Maple program:
q:= n-> (l-> is(numtheory[pi](add(i[2], i=l))=nops(l)))(ifactors(n)[2]):
select(q, [$1..245])[];  # Alois P. Heinz, Apr 05 2025

Select[Range[250], PrimePi[PrimeOmega[#]] == PrimeNu[#] &] (* Amiram Eldar, Apr 05 2025 *)

isok(k) = primepi(bigomega(k)) == omega(k); \\ Michel Marcus, Apr 05 2025

a(1) inserted by Michel Marcus, Apr 05 2025

A331669 List of distinct numbers that occur in A318366 (the Dirichlet convolution square of bigomega).

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 12, 20, 24, 34, 35, 40, 48, 52, 56, 70, 72, 84, 95, 104, 112, 116, 120, 130, 156, 160, 164, 165, 168, 180, 189, 212, 220, 224, 238, 240, 258, 280, 284, 286, 300, 304, 322, 330, 344, 348, 352, 364, 380, 420, 438, 440, 455, 460, 464, 472, 477, 480

Offset: 1

Torlach Rush, Jan 23 2020

nonn

There is a strong correlation between values of this function and values of other arithmetic functions. In other words, a(n) correlates to a single distinct value from one or more of the arithmetic functions.
Terms of this sequence select from the positive integers as follows:
A318366(k) = a(1), 1 followed by the primes (A008578).
A318366(k) = A008836(k) = A001221(k) = a(2), primes squared (A001248).
A318366(k) = A001221(k) = a(3), squarefree semiprimes (A006881).
A318366(k) = A000005(k) = a(4), primes cubed (A030078).
A318366(k) = a(5), a prime squared times a prime (A054753).
A318366(k) = a(6), primes to the fourth power (A030514).
A318366(k) = a(7), sphenic numbers (A007304).
A318366(k) = a(8), union of A050997 and A065036.
A318366(k) = a(9), squarefree semiprimes squared (A085986).
A318366(k) = a(10), product of four primes, three distinct (A085987).
A318366(k) = a(11), primes to the sixth power (A030516).
A318366(k) = a(12), product of prime to fourth power and a different prime (A178739).
A318366(k) = a(13), product of four distinct primes (A046386).
...

			0 is a term because the only divisors of a prime (p) are 1 and a prime itself and bigomega(1) * bigomega(p) + bigomega(p) * bigomega(1) = 0 * 1 + 1 * 0 = 0.
1 is a term because a prime squared gives bigomega(1) * bigomega(p^2) + bigomega(p) * bigomega(p) + bigomega(p^2) * bigomega(1) = 0 * 2 + 1 * 1 + 2 * 0 = 1.

Cf. A001222, A318366.

Cf. also A101296.

More terms, using A318366 extended b-file, from Michel Marcus, Jan 24 2020

A351867 Heptagonal numbers which are products of four distinct primes.

Original entry on oeis.org

3010, 4774, 10465, 14326, 20566, 28462, 54538, 59059, 59830, 66178, 66994, 87142, 104755, 112042, 120670, 121771, 131905, 137710, 138886, 168610, 179158, 201214, 212722, 223054, 249166, 273406, 288490, 290191, 314530, 343546, 358534, 375778, 401401, 405418, 419635, 461605

Offset: 1

Massimo Kofler, Apr 12 2022

nonn

A squarefree subsequence of heptagonal numbers.

			3010 = 2*5*7*43 = 35(5*35-3)/2.
4774 = 2*7*11*31 = 44(5*44-3)/2.
10465 = 5*7*13*23 = 65(5*65-3)/2.
14326 = 2*13*19*29 = 76(5*76-3)/2.

Intersection of A000566 and A046386.

from itertools import count, islice
from sympy import factorint
def A351867_gen(): return filter(lambda k:sum((f := factorint(k)).values()) == 4 == len(f), (n*(5*n-3)//2 for n in count(1)))
A351867_list = list(islice(A351867_gen(),20)) # Chai Wah Wu, Apr 14 2022

A351878 Products of four distinct primes between products of two primes.

Original entry on oeis.org

870, 1190, 1254, 1590, 1794, 1938, 2046, 2190, 2470, 2490, 2562, 2814, 2982, 3030, 3094, 3102, 3198, 3666, 3810, 4170, 4182, 4386, 4470, 4530, 4578, 4686, 4710, 5330, 5358, 5434, 5730, 5754, 5910, 5934, 5970, 6018, 6118, 6402, 6438, 6486, 6594

Offset: 1

Massimo Kofler, Mar 31 2022

nonn

			   870 = 2*3*5*29   (between  869 = 11 * 79  and   871 = 13 * 67);
  1190 = 2*5*7*17   (between 1189 = 29 * 41  and  1191 =  3 * 397);
  5330 = 2*5*13*41  (between 5329 = 73 * 73  and  5331 =  3 * 1777).

Subsequence of A046386.

Cf. A001358.

A354086 11-gonal numbers which are products of four distinct primes.

Original entry on oeis.org

4785, 8170, 11526, 14421, 27105, 30710, 38595, 59110, 60146, 77946, 94105, 107570, 118990, 120458, 121935, 132526, 140361, 141955, 156706, 158390, 161785, 181101, 199606, 203415, 213095, 215058, 217030, 221001, 243485, 249806, 267058, 287155, 298635, 303290

Offset: 1

Massimo Kofler, Jun 08 2022

nonn

A squarefree subsequence of 11-gonal numbers, i.e., numbers of the form k*(9*k-7)/2.

			4785 = 33*(9*33-7)/2 = 3*5*11*29.
30710 = 83*(9*83-7)/2 = 2*5*37*83.
140361 = 177*(9*177-7)/2 = 3*13*59*61.
303290 = 260*(9*260-7)/2 = 2*5*13*2333.

Intersection of A051682 and A046386.

q:= n-> is(map(x-> x[2], ifactors(n)[2])=[1$4]):
select(q, [n*(9*n-7)/2$n=1..300])[];  # Alois P. Heinz, Jun 15 2022

Select[Table[n*(9*n - 7)/2, {n, 1, 300}], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &] (* Amiram Eldar, Jun 08 2022 *)

A354447 Taxicab numbers (sums of 2 cubes in more than 1 way) which are products of four distinct primes.

Original entry on oeis.org

684019, 704977, 2691451, 3242197, 3375001, 4931101, 5318677, 5772403, 8872487, 10702783, 16983854, 20616463, 24897817, 41258737, 46343059, 60698521, 66469429, 69625969, 79692193, 89576767, 95731489, 96753187, 97867441, 116773741, 119793457, 126516061, 147187369

Offset: 1

Massimo Kofler, May 30 2022

nonn

A squarefree subsequence of taxicab numbers.

			684019 =  51^3 +  82^3 =  64^3 +  75^3 =  7 * 19 *  37 *  139.
8872487 =  14^3 + 207^3 = 140^3 + 183^3 = 13 * 17 *  19 * 2113.
96753187 = 147^3 + 454^3 = 283^3 + 420^3 = 19 * 37 * 229 *  601.

Eric Weisstein's World of Mathematics, Taxicab Number

Intersection of A001235 and A046386.

Select[Range[10^6], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} && Length @ PowersRepresentations[#, 2, 3] > 1 &] (* Amiram Eldar, May 30 2022 *)

A364766 Products k of 4 distinct primes (or tetraprimes) such that none of k-2, k-1, k+1 and k+2 is squarefree.

Original entry on oeis.org

3774, 6726, 10934, 11726, 12426, 13674, 15042, 16226, 17630, 17974, 18278, 18998, 21574, 23374, 23426, 24038, 27710, 27874, 28826, 32390, 34390, 35074, 35126, 37630, 37774, 38170, 38626, 41210, 41426, 46342, 46774, 46990, 47874, 50518, 50806, 51794, 53074, 53846

Offset: 1

Massimo Kofler, Aug 06 2023

nonn

			3772 = 2^2 * 23 * 41, 3773 = 7^3 * 11, 3774 = 2 * 3 * 17 * 37, 3775 = 5^2 * 151, 3776 = 2^6 * 59, so 3774 is a term.
6724 = 2^2 * 41^2, 6725 = 5^2 * 269, 6726 = 2 * 3 * 19 * 59, 6727 = 7 * 31^2, 6728 = 2^3 * 29^2, so 6726 is a term.

Cf. A013929, A046386. Subsequence of A364141.

Select[Range[55000], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} && !AnyTrue[# + {-2, -1, 1, 2}, SquareFreeQ] &] (* Amiram Eldar, Aug 06 2023 *)

A367791 Lesser of 2 successive tetraprimes (k, k+4) sandwiching three consecutive not squarefree numbers.

Original entry on oeis.org

3770, 12122, 12426, 17574, 19158, 22074, 28574, 31506, 40922, 46322, 47382, 50930, 52854, 57174, 60378, 61586, 66174, 72474, 74222, 77231, 78774, 85074, 85526, 87954, 89090, 91322, 91374, 95226, 97622, 99582, 104210, 106674, 113734, 118374, 120786, 122822, 124674, 126870, 127673

Offset: 1

Massimo Kofler, Nov 30 2023

nonn

Tetraprimes are the product of four distinct prime numbers (cf. A046386).

			3770 = 2*5*13*29, 3771 = 3^2*419, 3772 = 2^2*23*41, 3773 = 7^3*11, 3774 = 2*3*17*37, so 3770 is a term.
12122 = 2*11*19*29, 12123 = 3^3*449, 12124 = 2^2*7*433, 12125 = 5^3*97, 12126 = 2*3*43*47, so 12122 is a term.

Cf. A046386, A013929.

f[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, If[e == {1, 1, 1, 1}, 1, If[AnyTrue[e, # > 1 &], 2, 0]]]; Position[Partition[Array[f, 130000], 5, 1], {1, 2, 2, 2, 1}][[;; , 1]] (* Amiram Eldar, Nov 30 2023 *)

cp's OEIS Frontend

A376864 4-brilliant numbers with distinct prime factors.

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A379762 Products of 4 distinct prime numbers (or tetraprimes) that are abundant.

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A382208 Numbers k for which pi(bigomega(k)) = omega(k).

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A331669 List of distinct numbers that occur in A318366 (the Dirichlet convolution square of bigomega).

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A351867 Heptagonal numbers which are products of four distinct primes.

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A351878 Products of four distinct primes between products of two primes.

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A354086 11-gonal numbers which are products of four distinct primes.

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A354447 Taxicab numbers (sums of 2 cubes in more than 1 way) which are products of four distinct primes.

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