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 1-3 of 3 results.

A240591 The smaller of a pair of successive powerful numbers (A001694) without any prime number between them.

Original entry on oeis.org

8, 25, 32, 121, 288, 675, 1331, 1369, 1936, 2187, 2700, 3125, 5324, 6724, 9800, 10800, 12167, 15125, 32761, 39200, 48668, 70225, 79507, 88200, 97336, 107648, 143641, 156800, 212521, 228484, 235224, 280900, 312481, 332928, 456968, 465124, 574564, 674028, 744769, 829921, 830297, 857476, 877952, 940896
Offset: 1

Views

Author

Antonio Roldán, Apr 08 2014

Keywords

Examples

			25 is in the sequence because A001694(6)=25, A001694(7)=27, without primes between them.

Links

Crossrefs

Supersequence of A060355.
Cf. A001694, A240590.

Programs

  • Mathematica
    Select[Partition[Join[{1},Select[Range[10^6],Min@FactorInteger[#][[All, 2]]> 1&]],2,1],PrimePi[#[[1]]]==PrimePi[#[[2]]]&][[All,1]] (* Harvey P. Dale, Mar 28 2018 *)
  • PARI
    ispowerful(n)={local(h);if(n==1,h=1,h=(vecmin(factor(n)[, 2])>1));return(h)}
nextpowerful(n)={local(k);k=n+1;while(!ispowerful(k),k+=1);return(k)}
{for(i=1,10^6,if(ispowerful(i),if(nextprime(i)>=nextpowerful(i),print1(i, ", "))))}
  • Python
    from itertools import count, islice
from math import isqrt
from sympy import mobius, integer_nthroot, nextprime
def A240591_gen(): # generator of terms
    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 kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l, j = x-squarefreepi(integer_nthroot(x,3)[0]), 0, isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    m = 1
    for n in count(2):
        k = bisection(lambda x:f(x)+n,m,m)
        if nextprime(m) > k:
            yield m
        m = k
A240591_list = list(islice(A240591_gen(),30)) # Chai Wah Wu, Sep 14 2024

A378593 Number of squarefree k between consecutive powerful numbers (inclusive).

Original entry on oeis.org

3, 3, 0, 5, 5, 1, 3, 3, 8, 8, 6, 5, 11, 6, 8, 2, 1, 11, 14, 17, 2, 11, 6, 11, 7, 20, 0, 22, 10, 10, 18, 6, 20, 6, 28, 9, 8, 9, 28, 30, 14, 17, 0, 32, 33, 12, 24, 12, 22, 37, 4, 3, 17, 16, 36, 24, 16, 3, 42, 44, 18, 4, 13, 11, 2, 45, 46, 29, 20, 48, 26, 22, 23
Offset: 1

Views

Author

Michael De Vlieger, Dec 09 2024

Keywords

Examples

			a(1) = 3 since {[1], 2, 3, [4]} has 3 squarefree numbers.
a(2) = 3 since {[4], 5, 6, 7, [8]} has 3 squarefree numbers.
a(3) = 0 since {[8], [9]} has no squarefree numbers.
a(4) = 5 since between 9 and 16, {10, 11, 13, 14, 15} are squarefree.
a(5) = 5 since between 16 and 25, {17, 19, 21, 22, 23} are squarefree, etc.

Links

Crossrefs

Cf. A001694, A005117, A076446, A240590.

Programs

  • Mathematica
    With[{nn = 2^12},
  s = Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}];  Table[Count[Range[s[[i]], s[[i + 1]]], _?SquareFreeQ], {i, Length[s] - 1}] ]

Formula

a(n) > A076446(n) for n > 1.
a(n) >= A240590(n).

A378899 Number of primes between successive powerful numbers k that are not prime powers (i.e., k in A286708).

Original entry on oeis.org

11, 9, 5, 3, 6, 10, 2, 1, 1, 13, 5, 11, 1, 5, 2, 7, 3, 10, 13, 4, 0, 15, 2, 11, 4, 9, 1, 4, 13, 7, 2, 1, 9, 10, 6, 1, 2, 9, 12, 7, 4, 18, 5, 4, 17, 0, 8, 3, 13, 23, 2, 23, 10, 1, 15, 0, 7, 18, 3, 13, 7, 4, 7, 5, 4, 13, 2, 6, 10, 11, 29, 4, 2, 11, 1, 28, 2, 14
Offset: 0

Views

Author

Michael De Vlieger, Dec 10 2024

Keywords

Examples

			Let s = A286708.
a(0) = 11 since {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31} are primes less than s(1) = 36.
a(1) = 9 since {37, 41, 43, 47, 53, 59, 61, 67, 71} are primes that exceed s(1) but not s(2) = 72.
a(2) = 5 since {73, 79, 83, 89, 97} are primes p such that s(2) < p < s(3), where s(3) = 100.
a(3) = 3 since {101, 103, 107} are primes p such that s(3) < p < s(4), where s(4) = 108, etc.

Links

Crossrefs

Cf. A000040, A000720, A240590, A286708, A378699.

Programs

  • Mathematica
    s = With[{nn = 5000},
  Select[Rest@ Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}],
  Not@*PrimePowerQ]];
{PrimePi[s[[1]]]}~Join~Differences@ Map[PrimePi, s]

Formula

a(0) = pi(36) = A000720(36) = 11.
For n > 0, a(n) = pi(A286708(n+1)) - pi(A286708(n)).
Showing 1-3 of 3 results.

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