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.

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A386762 Perfect powers of nonsquarefree numbers k that are not squareful.

Original entry on oeis.org

144, 324, 400, 576, 784, 1600, 1728, 1936, 2025, 2304, 2500, 2704, 2916, 3136, 3600, 3969, 4624, 5625, 5776, 5832, 6400, 7056, 7744, 8000, 8100, 8464, 9216, 9604, 9801, 10816, 12544, 13456, 13689, 13824, 14400, 15376, 15876, 17424, 18225, 18496, 19600, 20736, 21609
Offset: 1

Views

Author

Michael De Vlieger, Aug 02 2025

Keywords

Comments

A131605 is the union of this sequence, A303606, and A383394, where the three sequences do not intersect one another.
A001597 is the union of A131605 and A246547.
Superset of A368508 (i.e., perfect powers of superprimorials that are not powers of 2).

Examples

			Table of n, a(n) for n = 1..12:
 n    a(n)
-----------------------------
 1    144 = 12^2 = 2^4 *  3^2
 2    324 = 18^2 = 2^2 *  3^4
 3    400 = 20^2 = 2^4 *  5^2
 4    576 = 24^2 = 2^6 *  3^2
 5    784 = 28^2 = 2^4 *  7^2
 6   1600 = 40^2 = 2^6 *  5^2
 7   1728 = 12^3 = 2^6 *  3^3
 8   1936 = 44^2 = 2^4 * 11^2
 9   2025 = 45^2 = 3^4 *  5^2
10   2304 = 48^2 = 2^8 *  3^2
11   2500 = 50^2 = 2^2 *  5^4
12   2704 = 52^2 = 2^4 * 13^2

Links

Crossrefs

Cf. A001597, A126706, A131605, A246547, A286708, A303606, A368508, A383394.

Programs

  • Mathematica
    nn = 2^15; i = 1; k = 2; MapIndexed[Set[S[First[#2]], #1] &, Select[Range@ Sqrt[nn], 1 == Min[#] < Max[#] &@ FactorInteger[#][[All, -1]] &] ]; Union@ Reap[While[j = 2; While[S[i]^j < nn, Sow[S[i]^j]; j++]; j > 2, k++; i++] ][[-1, 1]]
  • Python
    from math import isqrt
from sympy import mobius, integer_nthroot
def A386762(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while f(kmin) < kmin: kmin >>= 1		
        kmin = max(kmin,kmax >> 1)
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x):
        c, l, j = 1+x-squarefreepi(integer_nthroot(x,3)[0])-squarefreepi(x), 0, isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c += j*(l-w)
            l, j = w, isqrt(x//k2**3)
        return c+l
    def f(x): return n+x-sum(g(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))
    return bisection(f,n,n) # Chai Wah Wu, Aug 11 2025

A365745 Powers s^m, m > 1, where s is a composite squarefree number that is not a primorial.

Original entry on oeis.org

100, 196, 225, 441, 484, 676, 1000, 1089, 1156, 1225, 1444, 1521, 1764, 2116, 2601, 2744, 3025, 3249, 3364, 3375, 3844, 4225, 4356, 4761, 4900, 5476, 5929, 6084, 6724, 7225, 7396, 7569, 8281, 8649, 8836, 9025, 9261, 10000, 10404, 10648, 11025, 11236, 12100, 12321
Offset: 1

Views

Author

Michael De Vlieger, Dec 10 2023

Keywords

Examples

			100 is the first term in this sequence since it is 10^2; 10 is a squarefree composite number. All powers 10^m, m > 1 are in the sequence.
36 is not in the sequence since it is 6^2, where 6 is a product of the smallest 2 primes; none of the powers 6^m, m > 1 are in the sequence.

Links

Crossrefs

Cf. A002110, A120944, A303606, A365308.

Programs

  • Mathematica
    nn = 2^30; ss = Floor@ Sqrt[nn]; p = 3;
s = Complement[
  Select[Range[ss], And[SquareFreeQ[#], CompositeQ[#]] &],
  NestWhileList[(Set[p, NextPrime[p]]; # p) &, 6, # <= ss &] ];
Union@ Reap[Do[k = 2; While[s[[i]]^k <= nn, Sow[s[[i]]^k]; k++],
  {i, Length[s]}] ][[-1, 1]]

Formula

This sequence is {A303606 \ A365308}.
This sequence contains powers s^m, m > 1, for s in {A120944 \ A002110}.

A367511 Highly composite numbers h(k) = A002182(k) such that h >= rad(h)^2, where rad() = A007947().

Original entry on oeis.org

1, 4, 36, 48, 45360, 50400
Offset: 1

Views

Author

Michael De Vlieger, Feb 08 2024

Keywords

Comments

Alternatively, this sequence lists h(k) such that A301413(k) >= A002110(A108602(k)), where A301413 is the "variable part" v described on page 5 of 12 of the Siano paper.
This sequence is likely finite and full. See Chapter III regarding the structure of "Highly Composite Numbers".
Terms larger than 36 are in A366250; A366250 is in A364702, which is in turn a proper subset of A332785, itself contained in A126706.
36 is in A365308, a proper subset of A303606, contained in A131605, in turn contained in A286708.

Examples

			Let P(n) = A002110(n).
a(1) = h(1) = 1 since 1 >= 1^2.
a(2) = h(3) = 4 since 4 >= P(1)^2, 4 >= 2^2.
a(3) = h(7) = 36 since 36 >= P(2)^2, 36 >= 6^2.
a(4) = h(8) = 48 since 48 >= P(2)^2, 48 >= 6^2.
a(5) = h(26) = 43560 since 43560 >= P(4)^2, where P(4) = 210, and 210^2 = 44100.
a(6) = h(27) = 50400 since 50400 >= P(4)^2.
Let V(i) = A301414(i) and let P(j) = A002110(j).
Plot of highly composite h = V(i)*P(j) at (x,y) = (j,i), i = 1..16, j = 1..7, showing h in this sequence in parentheses, and h in A168263 marked with an asterisk (*):
V(i)\P(j) 1   2    6   30   210    2310    30030 ...
        +---------------------------------------
      1 |(1*) 2*   6*
      2 |    (4*) 12*  60*
      4 |         24* 120*  840*
      6 |        (36) 180* 1260*
      8 |        (48) 240  1680*
     12 |             360  2520   27720*
     24 |             720  5040   55440   720720
     36 |                  7560   83160  1081080
     48 |                 10080  110880  1441440
     72 |                 15120  166320  2162160
     96 |                 20160  221760  2882880
    120 |                 25200  277200  3603600
    144 |                        332640  4324320
    216 |                (45360) 498960  6486480
    240 |                (50400) 554400  7207200
    ...

Links

Crossrefs

Cf. A001221, A002110, A002182, A007947, A025487, A108602, A126706, A131605, A168263, A286708, A301413, A301414, A303606, A332785, A365308, A362702, A366250.

Programs

  • Mathematica
    (* First load function f at A025487, then run the following: *)
s = Union@ Flatten@ f[12];
t = Map[DivisorSigma[0, #] &, s];
h = Map[s[[FirstPosition[t, #][[1]]]] &, Union@ FoldList[Max, t]];
Reap[Do[If[# >= Product[Prime[j], {j, PrimeNu[#]}]^2, Sow[#]] &[ h[[i]] ],
  {i, Length[h]}] ][[-1, 1]]
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