cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A179702 Numbers of the form p^4*q^5 where p and q are two distinct primes.

Original entry on oeis.org

2592, 3888, 20000, 50000, 76832, 151875, 253125, 268912, 468512, 583443, 913952, 1361367, 2576816, 2672672, 3557763, 4170272, 5940688, 6940323, 7503125, 8954912, 10504375, 13045131, 20295603, 22632992, 22717712, 29552672, 30074733
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

Keywords

Comments

Subsequence of A046312 and of A137493. - R. J. Mathar, Jul 27 2010

Links

Crossrefs

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688, A179689, A179690, A179691, A179692, A179693, A179694, A179695, A179696, A179698, A179699, A179700.
Cf. A085964, A085965, A085969.

Programs

  • Mathematica
    fQ[n_] := Sort[Last /@ FactorInteger @n] == {4, 5}; Select[ Range@ 31668000, fQ] (* fixed by Robert G. Wilson v, Aug 26 2010 *)
lst = {}; Do[ If[p != q, AppendTo[lst, Prime@p^4*Prime@q^5]], {p, 12}, {q, 10}]; Take[ Sort@ Flatten@ lst, 27] (* Robert G. Wilson v, Aug 26 2010 *)
Take[Union[First[#]^4 Last[#]^5&/@Flatten[Permutations/@Subsets[ Prime[ Range[30]],{2}],1]],30] (* Harvey P. Dale, Jan 01 2012 *)
  • PARI
    list(lim)=my(v=List(),t);forprime(p=2, (lim\16)^(1/5), t=p^5;forprime(q=2, (lim\t)^(1/4), if(p==q, next);listput(v,t*q^4))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011
  • Python
    from sympy import primepi, integer_nthroot, primerange
def A179702(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(integer_nthroot(x//p**5,4)[0]) for p in primerange(integer_nthroot(x,5)[0]+1))+primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Formula

Sum_{n>=1} 1/a(n) = P(4)*P(5) - P(9) = A085964 * A085965 - A085969 = 0.000748..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

Extensions

Edited and extended by Ray Chandler and R. J. Mathar, Jul 26 2010

A190464 Numbers with prime factorization p^4*q^6.

Original entry on oeis.org

5184, 11664, 40000, 153664, 250000, 455625, 937024, 1265625, 1750329, 1827904, 1882384, 5345344, 8340544, 9529569, 10673289, 17909824, 20820969, 28344976, 37515625, 45265984, 59105344, 60886809, 73530625, 77228944, 95004009, 119946304, 143496441, 180848704, 204004089, 218803264
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 10 2011

Keywords

Comments

A subsequence of A175745 (Numbers with 35 divisors).
First different term in A175745 is 17179869184(=2^34).

Links

Crossrefs

Cf. A030629, A179692, A179699, A179705, A175745.
Cf. A085964, A085966.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={4,6}; Select[Range[50000000],f] (*and*) lst={};Do[Do[If[n!=m,AppendTo[lst,Prime[n]^6*Prime[m]^4]], {n,50}],{m,50}]; Take[Union@lst,50]
  • PARI
    list(lim)=my(v=List(),t);forprime(p=2, (lim\16)^(1/6), t=p^6;forprime(q=2, (lim\t)^(1/4), if(p==q, next);listput(v,t*q^4))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

Formula

Sum_{n>=1} 1/a(n) = P(4)*P(6) - P(10) = A085964 * A085966 - P(10) = 0.000320..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

A190465 Numbers with prime factorization p^5q^5.

Original entry on oeis.org

7776, 100000, 537824, 759375, 4084101, 5153632, 11881376, 39135393, 45435424, 52521875, 79235168, 90224199, 205962976, 345025251, 503284375, 601692057, 656356768, 916132832, 1160290625, 1564031349, 2219006624, 2706784157, 3707398432
Offset: 1

Views

Author

Vladimir Joseph Stephan Orlovsky, May 10 2011

Keywords

Links

Crossrefs

Cf. A179699, A179705, A190464.
Cf. A085965.

Programs

  • Mathematica
    f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1}; Select[Range[10000],f]^5
  • PARI
    list(lim)=my(v=List(),t);forprime(p=2, lim^(1/10), t=p^5;forprime(q=p+1, (lim\t)^(1/5), listput(v,t*q^5))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

Formula

Sum_{n>=1} 1/a(n) = (P(5)^2 - P(10))/2 = (A085965^2 - P(10))/2 = 0.000142..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

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

Views

Author

Amiram Eldar, Jul 03 2022

Keywords

Comments

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).

Examples

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

Links

Crossrefs

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.

Programs

  • Mathematica
    Select[Range[2500], Length[(e = FactorInteger[#][[;; , 2]])] == 2 && Min[e] > 1 &]
  • PARI
    is(n) = {my(f=factor(n)); #f~ == 2 && vecmin(f[,2]) > 1};

Formula

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... .

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

Views

Author

Amiram Eldar, Feb 19 2025

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

Formula

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... .
Showing 1-5 of 5 results.

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