A179644 -id:A179644 - cp's OEIS frontend

A189983 Numbers with prime factorization pqrst^2.

4620, 5460, 6930, 7140, 7980, 8190, 8580, 9660, 10710, 11220, 11550, 11970, 12012, 12180, 12540, 12870, 13020, 13260, 13650, 14490, 14820, 15180, 15540, 15708, 16170, 16830, 17220, 17556, 17850, 17940, 18018, 18060, 18270, 18564, 18810, 19110, 19140, 19380

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 03 2011

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures

Cf. A178740, A179644, A179670.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,1,1,2}; Select[Range[30000],f]

list(lim)=my(v=List(),t1,t2,t3,t4); forprime(p=2,sqrtint(lim\210), t1=p^2; forprime(q=2,lim\(30*t1), if(q==p, next); t2=q*t1; forprime(r=2,lim\(6*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\(2*t3), if(s==p || s==q || s==r, next); t4=s*t3; forprime(t=2,lim\t4, if(t==p || t==q || t==r || t==s, next); listput(v, t4*t)))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

A381311 Numbers whose powerful part (A057521) is a power of a prime with an even exponent >= 2.

Original entry on oeis.org

4, 9, 12, 16, 18, 20, 25, 28, 44, 45, 48, 49, 50, 52, 60, 63, 64, 68, 75, 76, 80, 81, 84, 90, 92, 98, 99, 112, 116, 117, 121, 124, 126, 132, 140, 147, 148, 150, 153, 156, 162, 164, 169, 171, 172, 175, 176, 188, 192, 198, 204, 207, 208, 212, 220, 228, 234, 236

Offset: 1

Amiram Eldar, Feb 19 2025

Numbers k whose largest unitary divisor that is a square, A350388(k), is a prime power (A246655), or equivalently, A350388(k) is in A056798 \ {1}.
Numbers having exactly one non-unitary prime factor and its multiplicity is even.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., 2*m} with m >= 1, i.e., any number (including zero) of 1's and then a single even number.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} p/((p-1)*(p+1)^2) = 0.24200684327095676029... .

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

Complement of A381312 within A190641.
Cf. A057521, A118914, A246655, A350388.
Subsequence of: A377816.
Subsequences: A001248, A030514, A030516, A030629, A030631, A054753, A056798 \ {1}, A085987, A178739, A179644, A179645, A179668, A179672, A179693, A179747, A189982, A189983, A189985, A189987, A190110, A190292, A190388.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;;,2]]]}, EvenQ[e[[1]]] && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000],q]

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); !(e[1] % 2) && (#e == 1 || e[2] == 1));

A258618 a(n) = (4n+9)n^2.

Original entry on oeis.org

0, 13, 68, 189, 400, 725, 1188, 1813, 2624, 3645, 4900, 6413, 8208, 10309, 12740, 15525, 18688, 22253, 26244, 30685, 35600, 41013, 46948, 53429, 60480, 68125, 76388, 85293, 94864, 105125, 116100, 127813, 140288, 153549, 167620, 182525, 198288, 214933

Offset: 0

Garrett Frandson, Jun 05 2015

Consider a natural number r such that r has 19 proper divisors and 6 prime factors. (Note that these prime factors do not have to be distinct.) The difference between these two values, say d(r), is in this case 13. Where n is a positive integer, d(r^n)=(4*n+9)*n^2.

The integers that satisfy the proper-divisor-prime-factor requirement are those of A179644.

			The smallest integer that satisfies this is 240: It has 19 proper divisors (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120) and 6 prime factors (2, 2, 2, 2, 3, 5), so d(240)=13. The square of 240, 57600, we would expect to have a difference of 68 between the number of its proper divisors and prime factors, and with respectively 80 and 12, d(57600)=68 indeed. Checking this with further integer powers of 240 will continue to generate terms in this sequence.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A179644.

[(4*n+9)*n^2: n in [0..40]]; // Vincenzo Librandi, Jun 06 2015

Table[(4 n + 9) n^2, {n, 0, 40}] (* Vincenzo Librandi, Jun 06 2015 *)
LinearRecurrence[{4,-6,4,-1},{0,13,68,189},40] (* Harvey P. Dale, Sep 12 2020 *)

vector(50,n,n--;(4*n+9)*n^2) \\ Derek Orr, Jun 21 2015

G.f.: x*(13+16*x-5*x^2)/(1-x)^4. - Vincenzo Librandi, Jun 06 2015

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Vincenzo Librandi, Jun 06 2015

More terms from Vincenzo Librandi, Jun 06 2015

A381316 Numbers whose powerful part (A057521) is a power of a prime with an exponent >= 3 (A246549).

Original entry on oeis.org

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 80, 81, 88, 96, 104, 112, 120, 125, 128, 135, 136, 152, 160, 162, 168, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 264, 270, 272, 280, 296, 297, 304, 312, 320, 328, 336, 343, 344, 351, 352, 368, 375, 376, 378

Offset: 1

Amiram Eldar, Feb 19 2025

First differs from A344653 and A345193 at n = 17: a(17) = 120 is not a term of these sequences.
Numbers whose prime signature (A118914) is of the form {1, 1, ..., m} with m >= 3, i.e., any number (including zero) of 1's and then a single number >= 3.
The asymptotic density of this sequence is (1/zeta(2)) * Sum_{p prime} 1/(p*(p^2-1)) = A369632 / A013661 = 0.13463358553764438661... .

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

Complement of A060687 within A190641.
Cf. A013661, A057521, A246549, A369632.
Cf. A344653, A345193.
Subsequences: A030078, A030514, A030516, A030629, A030631, A050997, A056824, A065036, A079395, A092759, A138031, A178739, A178740, A179644, A179645, A179664, A179665, A179667, A179668, A179670, A179672, A179692, A179693, A179696, A179704, A179747, A189975, A189984, A189987, A190110, A190292, A190378, A190383, A190388, A190473, A381312.

q[n_] := Module[{e = ReverseSort[FactorInteger[n][[;; , 2]]]}, e[[1]] > 2 && (Length[e] == 1 || e[[2]] == 1)]; Select[Range[1000], q]

isok(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2], , 4)); e[1] > 2 && (#e == 1 || e[2] == 1));

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

cp's OEIS Frontend

A189983 Numbers with prime factorization pqrst^2.

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A381311 Numbers whose powerful part (A057521) is a power of a prime with an even exponent >= 2.

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A258618 a(n) = (4*n+9)*n^2.

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A381316 Numbers whose powerful part (A057521) is a power of a prime with an exponent >= 3 (A246549).

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

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A258618 a(n) = (4n+9)n^2.