A085986 -id:A085986 - cp's OEIS frontend

A307342 Products of four primes, except fourth powers of primes.

24, 36, 40, 54, 56, 60, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 184, 189, 196, 198, 204, 210, 220, 225, 228, 232, 234, 248, 250, 260, 276, 294, 296, 297, 306, 308, 315, 328, 330, 340, 342, 344, 348, 350, 351, 364, 372, 375, 376, 380, 390

Offset: 1

Kalle Siukola, Apr 02 2019

Numbers with exactly four prime factors (counted with multiplicity) and more than one distinct prime factor.

Numbers n such that bigomega(n) = 4 and omega(n) > 1.

Setwise difference of A014613 and A030514.
Union of A046386, A065036, A085986 and A085987.
Cf. A307682.

Select[Range@ 400, And[! PrimePowerQ@ #, PrimeOmega@ # == 4] &] (* Michael De Vlieger, Apr 21 2019 *)
Select[Range[400],PrimeOmega[#]==4&&PrimeNu[#]>1&] (* Harvey P. Dale, Aug 27 2021 *)

isok(n) = (bigomega(n)==4) && (omega(n) > 1); \\ Michel Marcus, Apr 03 2019

import sympy
def bigomega(n): return sympy.primeomega(n)
def omega(n): return len(sympy.primefactors(n))
print([n for n in range(1, 1000) if bigomega(n) == 4 and omega(n) > 1])

A307682 Products of four primes, two of which are distinct.

Original entry on oeis.org

24, 36, 40, 54, 56, 88, 100, 104, 135, 136, 152, 184, 189, 196, 225, 232, 248, 250, 296, 297, 328, 344, 351, 375, 376, 424, 441, 459, 472, 484, 488, 513, 536, 568, 584, 621, 632, 664, 676, 686, 712, 776, 783, 808, 824, 837, 856, 872, 875, 904, 999, 1016, 1029

Offset: 1

Kalle Siukola, Apr 21 2019

nonn

Numbers with exactly four prime factors (counted with multiplicity) and exactly two distinct prime factors.
Numbers n such that bigomega(n) = 4 and omega(n) = 2.
Products of a prime and the cube of a different prime (pq^3) together with squares of squarefree semiprimes (p^2*q^2).

Union of A065036 and A085986.
Intersection of A007774 and A067801.
Intersection of A007774 and A195086.
Intersection of A014613 and A067801.
Intersection of A014613 and A195086.
Cf. A307342.

Select[Range@ 1050, And[PrimeNu@ # == 2, PrimeOmega@ # == 4] &] (* Michael De Vlieger, Apr 21 2019 *)

isok(n) = (bigomega(n) == 4) && (omega(n) == 2); \\ Michel Marcus, Apr 22 2019

import sympy
def bigomega(n): return sympy.primeomega(n)
def omega(n): return len(sympy.primefactors(n))
print([n for n in range(1, 1000) if bigomega(n) == 4 and omega(n) == 2])

A350767 a(1)=1. Thereafter, a(n+1) is the least unused number k such that either d(j(n)) properly divides d(k) or d(k) properly divides d(j(n)), where j(n) = a(n)+1 and d is the divisor counting function A000005.

Original entry on oeis.org

1, 6, 8, 12, 10, 14, 2, 15, 48, 18, 20, 3, 28, 21, 5, 7, 11, 4, 22, 24, 32, 13, 17, 9, 19, 23, 26, 29, 27, 25, 30, 33, 31, 37, 40, 34, 41, 35, 49, 43, 47, 16, 38, 42, 39, 46, 44, 53, 51, 59, 45, 54, 56, 60, 50, 61, 66, 52, 55, 57, 67, 71, 58, 62, 72, 63, 192, 65

Offset: 1

David James Sycamore, Jan 14 2022

nonn

If d(j(n)) is prime p then d(a(n+1)) must be properly divisible by p. In practice the proper divisor for computation of a(n+1) toggles between d(j(n)) and d(k).
Conjecture: This is a permutation of the positive integers. Numbers with the same number (tau) of divisors appear in their natural orders (e.g., primes, semiprimes, squares).
The plot, after the first few terms, resolves itself into points tightly packed on and around a straight line of slope 1, with exceptional points appearing as significant upward or downward "spikes".
When d(j(n)) is prime p appearing for the first time in the sequence J = {d(j(a(n)), n>=1}, then a(n+1) is the smallest number with 2p divisors, which produces a significantly large upward spike above the straight line (6, 12, 48, 192, 3072, 12288, ...).
When d(j(a(n)) is 2p, seen for the first time in J, then a(n+1) is the smallest number with p divisors, which produces a large downward spike, below the straight line (2, 4, 16, 64, 1024, 4096, ...).
The sequence of fixed points starts: 1, 46, 69, 74, 110, 140, 142, 152, 154, 178, ... apparently becoming denser as n increases.

			a(1)=1, so j(1)=2, d(j(1))=2, a prime, so we need the smallest unused k such that d(k) is properly divisible by 2, hence a(2)=6.
a(2)=6, j(2)=4, d(j(2))=3, a prime so we need the smallest unused k such that d(k) is properly divisible by 3, hence a(3)=8.

Michael De Vlieger, Table of n, a(n) for n = 1..10001
Michael De Vlieger, Scatterplot of a(n), n = 1..256, color coded to show primes in red, evens in blue, 2 in magenta, odd composites in black. A gold highlight indicates terms such that (a(n)+1) | a(n+1). Outlying points are annotated.
Michael De Vlieger, Log-log scatterplot of a(n), n = 1..2^16. Terms in A3680(p) such that p is prime appear annotated in red, and those in A061286 appear in blue.

Cf. A000040, A001248, A030514, A054753, A030516, A030626, A085986, A178739, A030629, A030630, A030631, A030632, A030633, A030634, A005179, A061286, A003680.

Block[{c, d, j, k, u, nn}, nn = 120; j = u = 2; c[] = 0; c[1] = 1; Do[d[i] = DivisorSigma[0, i], {i, 2^(Ceiling@ Log2[nn] + 3)}]; {1}~Join~Reap[Do[k = u; While[Nand[Or[Divisible[d[j], d[k]], Divisible[d[k], d[j]]], d[j] != d[k], c[k] == 0], k++]; Sow[k]; c[k] = i; j = k + 1; If[k == u, While[c[u] != 0, u++]], {i, nn}] ][[-1, -1]] ] (* _Michael De Vlieger, Jan 14 2022 *)

isok(k, last, set) = if (!setsearch(set, k), my(ndk=numdiv(k), ndl=numdiv(last+1)); (ndl != ndk) && ((!(ndk % ndl)) || (!(ndl % ndk))));
lista(nn) = {my(last = 1, list = List(last), set = Set(list)); for (n=2, nn, my(k=1); while (!isok(k, last, set), k++); listput(list, k); set = Set(list); last = k;); Vec(list);} \\ Michel Marcus, Jan 15 2022

More terms from Michael De Vlieger, Jan 14 2022

A355462 Powerful numbers divisible by exactly 2 distinct primes.

Original entry on oeis.org

36, 72, 100, 108, 144, 196, 200, 216, 225, 288, 324, 392, 400, 432, 441, 484, 500, 576, 648, 675, 676, 784, 800, 864, 968, 972, 1000, 1089, 1125, 1152, 1156, 1225, 1296, 1323, 1352, 1372, 1444, 1521, 1568, 1600, 1728, 1936, 1944, 2000, 2025, 2116, 2304, 2312, 2500

Offset: 1

Amiram Eldar, Jul 03 2022

nonn

First differs from A286708 at n = 25.
Number of the form p^i * q^j, where p != q are primes and i,j > 1.
Numbers k such that A001221(k) = 2 and A051904(k) >= 2.
The possible values of the number of the divisors (A000005) of terms in this sequence is any composite number that is not 8 or twice a prime (A264828 \ {1, 8}).
675 = 3^3*5^2 and 676 = 2^2*13^2 are 2 consecutive integers in this sequence. There are no other such pairs below 10^22 (the lesser members of such pairs are terms of A060355).

			36 is a term since 36 = 2^2 * 3^2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index to sequences related to prime signature.

Intersection of A001694 and A007774.
Subsequence of A286708.
Subsequences: A085986, A143610, A162142, A179646, A179666, A179671, A179689, A179694, A179699, A179702, A179705, A189988, A189990, A189991, A190464, A190465, A303661.
Cf. A000005, A001221, A051904, A060355, A136141, A264828.

Select[Range[2500], Length[(e = FactorInteger[#][[;; , 2]])] == 2 && Min[e] > 1 &]

is(n) = {my(f=factor(n)); #f~ == 2 && vecmin(f[,2]) > 1};

Sum_{n>=1} 1/a(n) = ((Sum_{p prime} (1/(p*(p-1))))^2 - Sum_{p prime} (1/(p^2*(p-1)^2)))/2 = 0.1583860791... .

A369209 Numbers whose number of divisors has the largest prime factor 3.

Original entry on oeis.org

4, 9, 12, 18, 20, 25, 28, 32, 36, 44, 45, 49, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 96, 98, 99, 100, 108, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 160, 164, 169, 171, 172, 175, 180, 188, 196, 198, 200, 204, 207, 212, 220, 224, 225, 228

Offset: 1

Amiram Eldar, Jan 16 2024

Subsequence of A059269 and first differs from it at n = 36: A059269(136) = 44 has 15 = 3 * 5 divisors and thus is not a term of this sequence.
Numbers k such that A000005(k) is in A065119.
Numbers k such that A071188(k) = 3.
Equals the complement of A354181, without the terms of A036537 (i.e., complement(A354181) \ A036537).
The asymptotic density of this sequence is Product_{p prime} (1-1/p) * (Sum_{k>=1} 1/p^(A003586(k)-1)) - A327839 = 0.26087647470200496716... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A003586, A006530, A036537, A065119, A336595, A071188, A211337, A211338, A327839, A354181.
Subsequence of A013929 and A059269.
Subsequences: A001248, A030627, A050997, A054753, A062503, A067259, A079395, A085986, A085987, A086975, A095990, A096156, A138032, A162143, A179643, A179645.

gpf[n_] := FactorInteger[n][[-1, 1]]; Select[Range[300], gpf[DivisorSigma[0, #]] == 3 &]

gpf(n) = if(n == 1, 1, vecmax(factor(n)[, 1]));
is(n) = gpf(numdiv(n)) == 3;

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

A355571 Complement of A007956: numbers not of the form P(k)/k where P(n) is the product of the divisors of n.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 24, 25, 28, 30, 32, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 121, 124, 126, 128, 130, 132, 135, 136, 138, 140, 147, 148, 150, 152

Offset: 1

Luca Onnis, Jul 07 2022

nonn

There are no primes in the sequence, since A007956(p^2) = p for all primes p.
There are infinitely many terms, in fact p^2 is a term for all primes p.
If 8k+1 is not a perfect square, then p^k is a term for all primes p.
Depends only on the prime signature: n is in this sequence if and only if A046523(n) is in this sequence. - Charles R Greathouse IV, Jul 08 2022
Contains all the weak numbers (A052485) aside from the primes (A000040) and squarefree semiprimes (A006881). - Charles R Greathouse IV, Jul 08 2022

			4 is a term of this sequence because there are no numbers k such that A007956(k) = 4.
2^10 is not a term of this sequence because A007956(32) = 1024 (Note that 8*10+1=81=9^2 is a perfect square).
p^4 belongs to this sequence for all primes p, in fact 8*4+1=33 is not a perfect square, so there are no numbers h such that A007956(h) = p^4.

Wacław Sierpiński, Elementary Theory of Numbers, Ex. 2 p. 174, Warsaw, 1964.

Subsequences by prime signature: A001248 (p^2), A054753 (p^2*q), A030514 (p^4), A065036 (p^3*q), A007304 (p*q*r), A050997 (p^5), A085986 (p^2*q^2).

Cf. A007956, A052485, A106543.

Complement[Complement[Table[n, {n, 2, 1000}], Select[NumericalSort[Table[Times @@ Most[Divisors[n]], {n, 1000000}]], # != 1 && # < 1000 &]], Select[Table[Prime[n], {n, 1, 1000}], # < 1000 &]]

a(n) = n + O(n log log n/log n). - Charles R Greathouse IV, Jul 08 2022

cp's OEIS Frontend

A307342 Products of four primes, except fourth powers of primes.

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A307682 Products of four primes, two of which are distinct.

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A350767 a(1)=1. Thereafter, a(n+1) is the least unused number k such that either d(j(n)) properly divides d(k) or d(k) properly divides d(j(n)), where j(n) = a(n)+1 and d is the divisor counting function A000005.

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A355462 Powerful numbers divisible by exactly 2 distinct primes.

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A369209 Numbers whose number of divisors has the largest prime factor 3.

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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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A355571 Complement of A007956: numbers not of the form P(k)/k where P(n) is the product of the divisors of n.

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