A190106 -id:A190106 - cp's OEIS frontend

A190108 Numbers with prime factorization pqr^3*s^3 (where p, q, r, s are distinct primes).

7560, 11880, 14040, 16632, 18360, 19656, 20520, 21000, 24840, 25704, 28728, 30888, 31320, 33000, 33480, 34776, 39000, 39960, 40392, 41160, 43848, 44280, 45144, 46440, 46872, 47250, 47736, 50760, 51000, 53352, 54648, 55944, 57000, 57240, 61992, 63720, 64584

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 04 2011

nonn

A050326(a(n)) = 11. - Reinhard Zumkeller, May 03 2013

			From _Petros Hadjicostas_, Oct 26 2019: (Start)
a(1) = (2^3)*(3^3)*5*7 = 7560;
a(2) = (2^3)*(3^3)*5*11 = 11880;
a(3) = (2^3)*(3^3)*5*13 = 14040;
a(4) = (2^3)*(3^3)*7*11 = 16632.
(End)

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures
Index to sequences related to prime signature

Cf. A179704, A190106, A190107.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,3,3};Select[Range[150000],f]

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtnint(lim\120, 3), t1=p^3; forprime(q=2,sqrtnint(lim\(6*t1), 3), if(q==p, next); t2=q^3*t1; forprime(r=2,lim\(2*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\t3, if(s==p || s==q || s==r, next); listput(v, t3*s))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

Name edited by Petros Hadjicostas, Oct 26 2019

A190109 Numbers with prime factorization pq^2r^2*s^3 (where p, q, r, s are distinct primes).

Original entry on oeis.org

12600, 17640, 18900, 19800, 23400, 26460, 29400, 29700, 30600, 31500, 34200, 35100, 38808, 41400, 43560, 45864, 45900, 49500, 51300, 52200, 55800, 58212, 58500, 59976, 60840, 60984, 61740, 62100, 65340, 66150, 66600, 67032, 68796, 72600, 73500, 73800, 76500

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 04 2011

nonn

That is, numbers with prime signature {1,2,2,3}.

			From _Petros Hadjicostas_, Oct 26 2019: (Start)
a(1) = (2^3)*(3^2)*(5^2)*7 = 12600;
a(2) = (2^3)*(3^2)*5*(7^2) = 17640;
a(3) = (2^2)*(3^3)*(5^2)*7 = 18900;
a(4) = (2^3)*(3^2)*(5^2)*11 = 19800.
(End)

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime signature.
Wikipedia, Prime signature.
Will Nicholes, Prime Signatures.
Index to sequences related to prime signature

Cf. A190106, A190107, A190108.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,2,2,3};Select[Range[150000],f]

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtnint(lim\180, 3), t1=p^3; forprime(q=2,sqrtint(lim\(12*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,sqrtint(lim\(2*t2)), if(r==p || r==q, next); t3=r^2*t2; forprime(s=2,lim\t3, if(s==p || s==q || s==r, next); listput(v, t3*s))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

A190107 Numbers with prime factorization pqr^2s^4.

Original entry on oeis.org

5040, 7920, 8400, 9360, 11088, 11340, 11760, 12240, 13104, 13200, 13680, 15600, 16560, 17136, 17820, 19152, 20400, 20592, 20880, 21060, 22320, 22800, 23184, 24948, 25872, 26640, 26928, 27540, 27600, 28350, 29040, 29232, 29484, 29520

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 04 2011

nonn

A050326(a(n)) = 3. - Reinhard Zumkeller, May 03 2013

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures
Index to sequences related to prime signature

Cf. A111467, A179704, A190106.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,2,4};Select[Range[60000],f]

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtnint(lim\60, 4), t1=p^4; forprime(q=2,sqrtint(lim\(6*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,lim\(2*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\t3, if(s==p || s==q || s==r, next); listput(v, t3*s))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

A381314 Powerful numbers that have a single exponent in their prime factorization that equals 2.

Original entry on oeis.org

4, 9, 25, 49, 72, 108, 121, 144, 169, 200, 288, 289, 324, 361, 392, 400, 500, 529, 576, 675, 784, 800, 841, 961, 968, 972, 1125, 1152, 1323, 1352, 1369, 1372, 1568, 1600, 1681, 1849, 1936, 2025, 2209, 2304, 2312, 2500, 2704, 2809, 2888, 2916, 3087, 3136, 3200

Offset: 1

Amiram Eldar, Feb 19 2025

Number of the form A036966(m)/p, m >= 2, where p is a prime divisor of A036966(m).

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

Subsequence of A001694, A377816 and A377818.
Subsequences: A001248, A143610, A179646, A179689, A179699, A189988, A189990, A190106, A190115, A190470, A190471.
Cf. A036966, A065483.

With[{max = 3200}, Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], Count[FactorInteger[#][[;; , 2]], 2] == 1 &]]

isok(k) = if(k == 1, 0, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> (x==2), e) == 1);

Sum_{n>=1} 1/a(n) = Sum_{p prime}((p-1)/(p^3-p^2+1)) * Product_{p prime} (1 + 1/(p^2*(p-1))) = 0.53045141423939736076... .

cp's OEIS Frontend

A190108 Numbers with prime factorization p*q*r^3*s^3 (where p, q, r, s are distinct primes).

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A190109 Numbers with prime factorization p*q^2*r^2*s^3 (where p, q, r, s are distinct primes).

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A190107 Numbers with prime factorization pqr^2s^4.

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A381314 Powerful numbers that have a single exponent in their prime factorization that equals 2.

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A190108 Numbers with prime factorization pqr^3*s^3 (where p, q, r, s are distinct primes).

A190109 Numbers with prime factorization pq^2r^2*s^3 (where p, q, r, s are distinct primes).