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 1-4 of 4 results.

A368682 Products of primorials that are perfect powers but not prime powers.

Original entry on oeis.org

36, 144, 216, 576, 900, 1296, 1728, 2304, 3600, 5184, 7776, 9216, 13824, 14400, 20736, 27000, 32400, 36864, 44100, 46656, 57600, 82944, 110592, 129600, 147456, 176400, 186624, 216000, 230400, 248832, 279936, 331776, 373248, 518400, 589824, 705600, 746496, 810000
Offset: 1

Views

Author

Michael De Vlieger, Jan 02 2024

Keywords

Comments

Intersection of A025487 and A131605.
Proper subset of A286708.
Contains A365308 (perfect powers of composite primorials) and A368508 (perfect powers of composite superprimorials).
These numbers are perfect powers of some smaller product of primorials.

Examples

			b(n) = A025487(n).
a(1) = b(11) = 36 = 6^2 = b(4)^2,
a(2) = b(19) = 144 = 12^2 = b(6)^2,
a(3) = b(23) = 216 = 6^3 = b(4)^3,
a(4) = b(33) = 576 = 24^2 = b(8)^2,
a(5) = b(38) = 900 = 30^2 = b(9)^2, etc.

Links

Crossrefs

Cf. A000079, A000961, A001597, A002110, A025487, A131605, A286708, A365308, A368508, A368681.

Programs

  • Mathematica
    Select[Range[36, 2^18, 2], And[Union@ Differences@ PrimePi@ #1 == {1}, AllTrue[Union@ Differences@ #2, # <= 0 &], GCD @@ #2 > 1] & @@ Transpose@ FactorInteger[#] &]

Formula

This sequence is { A368681 \ A000079 }.

A368507 Powers of superprimorials.

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 32, 64, 128, 144, 256, 360, 512, 1024, 1728, 2048, 4096, 8192, 16384, 20736, 32768, 65536, 75600, 129600, 131072, 248832, 262144, 524288, 1048576, 2097152, 2985984, 4194304, 8388608, 16777216, 33554432, 35831808, 46656000, 67108864, 134217728
Offset: 1

Views

Author

Michael De Vlieger, Dec 28 2023

Keywords

Comments

Numbers k = Product_{i=1..j} p_i^e_(m*(j-i+1)) for m >= 0 and j >= 1.
Let b(n) = A006939(n) and let P(n) = A002110(n).
This sequence contains {1}, A000079, A006939, certain k in A364710 (intersection of A126706 and A025487), and certain m in A364930 (intersection of A286708 and A025487).
The only prime in this sequence is 2.
Prime powers in this sequence are powers of 2.
Outside of {1, 2}, superprimorials are in A364710.
Squareful numbers in this sequence contain {2^k, k > 1}, which are in A000079, a proper subset of A246547, and {b(k)^m, k > 1, m > 1}, which are in A364930, a proper subset of A286708.

Examples

			Powers of 2 are in the sequence since 2 = P(1).
Powers of 12 are terms, since 12 = P(1)*P(2).
Powers of 360 are terms, since 360 = P(1)*P(2)*P(3), etc.

Links

Crossrefs

Cf. A000079, A002110, A006939, A025487, A126706, A246547, A286708, A364710, A364930, A368508.

Programs

  • Mathematica
    nn = 2^60; k = 1; P = 2; Q = 2; {1}~Join~Union@ Reap[While[j = 1; While[Q^j < nn, Sow[Q^j]; j++]; j > 1, k++; P *= Prime[k]; Q *= P] ][[-1, 1]]

A368681 Products of primorials that are perfect powers.

Original entry on oeis.org

1, 4, 8, 16, 32, 36, 64, 128, 144, 216, 256, 512, 576, 900, 1024, 1296, 1728, 2048, 2304, 3600, 4096, 5184, 7776, 8192, 9216, 13824, 14400, 16384, 20736, 27000, 32400, 32768, 36864, 44100, 46656, 57600, 65536, 82944, 110592, 129600, 131072, 147456, 176400, 186624
Offset: 1

Views

Author

Michael De Vlieger, Jan 02 2024

Keywords

Comments

Intersection of A025487 and A001597.
Contains A365308 (perfect powers of composite primorials), A368508 (perfect powers of composite superprimorials), and A368682.
These numbers are perfect powers of some smaller product of primorials.

Examples

			Let b(n) = A025487(n).
a(1) = b(1) = 1 = 1^k = b(1)^k, k >= 2,
a(2) = b(3) = 4 = 2^2 = b(2)^2,
a(3) = b(5) = 8 = 2^3 = b(2)^3,
a(6) = b(11) = 36 = 6^2 = b(4)^2,
a(9) = b(19) = 144 = 12^2 = b(6)^2, etc.
2 is not in the sequence since 2 is squarefree and not in A001597.

Links

Crossrefs

Cf. A000079, A001597, A002110, A025487, A131605, A365308, A368508, A368682.

Programs

  • Mathematica
    {1}~Join~Select[Range[4, 200000, 2], Or[PrimePowerQ[#], And[Union@ Differences@ PrimePi@ #1 == {1}, AllTrue[Union@ Differences@ #2, # <= 0 &], GCD @@ #2 > 1] & @@ Transpose@ FactorInteger[#]] &]

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
Showing 1-4 of 4 results.

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