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.

A076467 Perfect powers m^k where m is a positive integer and k >= 3.

Original entry on oeis.org

1, 8, 16, 27, 32, 64, 81, 125, 128, 216, 243, 256, 343, 512, 625, 729, 1000, 1024, 1296, 1331, 1728, 2048, 2187, 2197, 2401, 2744, 3125, 3375, 4096, 4913, 5832, 6561, 6859, 7776, 8000, 8192, 9261, 10000, 10648, 12167, 13824, 14641, 15625, 16384, 16807
Offset: 1

Views

Author

Robert G. Wilson v, Oct 14 2002

Keywords

Comments

If p|n with p prime then p^3|n.

Links

Crossrefs

Subsequence of A036966.
Cf. A001597, A076468, A076469, A076470.

Programs

  • Haskell
    a076467 n = a076467_list !! (n-1)
a076467_list = 1 : filter ((> 2) . foldl1 gcd . a124010_row) [2..]
-- Reinhard Zumkeller, Apr 13 2012
  • Haskell
    import qualified Data.Set as Set (null)
import Data.Set (empty, insert, deleteFindMin)
a076467 n = a076467_list !! (n-1)
a076467_list = 1 : f [2..] empty where
   f xs'@(x:xs) s | Set.null s || m > x ^ 3 = f xs $ insert (x ^ 3, x) s
                  | m == x ^ 3  = f xs s
                  | otherwise = m : f xs' (insert (m * b, b) s')
                  where ((m, b), s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 18 2013
  • Maple
    N:= 10^5: # to get all terms <= N
S:= {1, seq(seq(m^k, m = 2 .. floor(N^(1/k))),k=3..ilog2(N))}:
sort(convert(S,list)); # Robert Israel, Sep 30 2015
  • Mathematica
    a = {1}; Do[ If[ Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]] > 2, a = Append[a, n]; Print[n]], {n, 2, 17575}]; a
(* Second program: *)
n = 10^5; Join[{1}, Table[m^k, {k, 3, Floor[Log[2, n]]}, {m, 2, Floor[n^(1/k)]}] // Flatten // Union] (* Jean-François Alcover, Feb 13 2018, after Robert Israel *)
  • PARI
    is(n)=ispower(n)>2||n==1 \\ Charles R Greathouse IV, Sep 03 2015, edited for n=1 by M. F. Hasler, May 26 2018
  • PARI
    A076467(lim)={my(L=List(1),lim2=logint(lim,2),m,k);for(k=3,lim2, for(m=2,sqrtnint(lim,k),listput(L, m^k);));listsort(L,1);L}
b076467(lim)={my(L=A076467(lim)); for(i=1,#L,print(i ," ",L[i]));} \\ Anatoly E. Voevudko, Sep 29 2015, edited by M. F. Hasler, May 25 2018
  • PARI
    A076467_vec(LIM,S=List(1))={for(x=2,sqrtnint(LIM,3),for(k=3, logint(LIM, x), listput(S, x^k))); Set(S)} \\ M. F. Hasler, May 25 2018
  • Python
    from sympy import mobius, integer_nthroot
def A076467(n):
    def f(x): return int(n-1+x-integer_nthroot(x,4)[0]+sum(mobius(k)*(integer_nthroot(x,k)[0]+integer_nthroot(x,k<<1)[0]-2) for k in range(3,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 14 2024

Formula

For n > 1: GCD(exponents in prime factorization of a(n)) > 2, cf. A124010. - Reinhard Zumkeller, Apr 13 2012
Sum_{n>=1} 1/a(n) = 2 - zeta(2) + Sum_{k>=2} mu(k)*(2 - zeta(k) - zeta(2*k)) = 1.3300056287... - Amiram Eldar, Jul 02 2022

Extensions

Edited by Robert Israel, Sep 30 2015

A122181 Numbers k that can be written as k = x*y*z with 1 < x < y < z (A122180(k) > 0).

Original entry on oeis.org

24, 30, 36, 40, 42, 48, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 128, 130, 132, 135, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 165, 168, 170, 174, 176, 180, 182, 184, 186, 189, 190, 192, 195
Offset: 1

Views

Author

Rick L. Shepherd, Aug 24 2006

Keywords

Comments

Equivalently, numbers k with at least 7 divisors (A000005(k) > 6). Equivalently, numbers k with at least 5 proper divisors (A070824(k) > 4). Equivalently, numbers k such that i) k has at least three distinct prime factors (A000977), ii) k has two distinct prime factors and four or more total prime factors (k = p^j*q^m, p,q primes, j+m >= 4), or iii) k = p^m, a perfect power (A001597) but restricted to prime p and m >= 6 [= 1+2+3] (some terms of A076470).

Examples

			a(1) = 24 = 2*3*4, a product of three distinct proper divisors (omega(24) = 2, bigomega(24) = 4).
a(2) = 30 = 2*3*5, a product of three distinct prime factors (omega(30) = 3).
a(10) = 64 = 2*4*8 [= 2^1*2^2*2^3] (omega(64) = 1, bigomega(64) = 6).

Links

Crossrefs

Cf. A122179, A122180, A000005, A070824, A000977, A001221, A001222, A001597, A076470.

Programs

  • Mathematica
    Select[Range[200], DivisorSigma[0, #] > 6 &] (* Amiram Eldar, Oct 05 2024 *)
  • PARI
    isok(n) = numdiv(n)>6
  • PARI
    isok(n) = (omega(n)==1 && bigomega(n)>5) || (omega(n)==2 && bigomega(n)>3) || (omega(n)>2)

A069493 6-full numbers: if p divides n then so does p^6.

Original entry on oeis.org

1, 64, 128, 256, 512, 729, 1024, 2048, 2187, 4096, 6561, 8192, 15625, 16384, 19683, 32768, 46656, 59049, 65536, 78125, 93312, 117649, 131072, 139968, 177147, 186624, 262144, 279936, 373248, 390625, 419904, 524288, 531441, 559872, 746496, 823543, 839808
Offset: 1

Views

Author

Benoit Cloitre, Apr 15 2002

Keywords

Comments

a(m) mod prime(n) > 0 for m < A258603(n); a(A258600(n)) = A030516(n) = prime(n)^6. - Reinhard Zumkeller, Jun 06 2015

Examples

			2^7*3^6 = 93312 is a member (although not of A076470).

Links

Crossrefs

Cf. A036967, A036966, A001694. Different from A076470.
Cf. A030516, A258603.

Programs

  • Haskell
    import Data.Set (singleton, deleteFindMin, fromList, union)
a069493 n = a069493_list !! (n-1)
a069493_list = 1 : f (singleton z) [1, z] zs where
   f s q6s p6s'@(p6:p6s)
     | m < p6 = m : f (union (fromList $ map (* m) ps) s') q6s p6s'
     | otherwise = f (union (fromList $ map (* p6) q6s) s) (p6:q6s) p6s
     where ps = a027748_row m
           (m, s') = deleteFindMin s
   (z:zs) = a030516_list
-- Reinhard Zumkeller, Jun 03 2015
  • Mathematica
    Join[{1},Select[Range[900000],Min[FactorInteger[#][[All,2]]]>5&]] (* Harvey P. Dale, Mar 03 2018 *)
  • PARI
    for(n=1,560000,if(n*sumdiv(n,d,isprime(d)/d^6)==floor(n*sumdiv(n,d,isprime(d)/d^6)),print1(n,",")))
  • Python
    from math import gcd
from sympy import factorint, integer_nthroot
def A069493(n):
    def f(x):
        c = n+x
        for y1 in range(1,integer_nthroot(x,11)[0]+1):
            if all(d<=1 for d in factorint(y1).values()):
                for y2 in range(1,integer_nthroot(z2:=x//y1**11,10)[0]+1):
                    if gcd(y2,y1)==1 and all(d<=1 for d in factorint(y2).values()):
                        for y3 in range(1,integer_nthroot(z3:=z2//y2**10,9)[0]+1):
                            if gcd(y3,y1)==1 and gcd(y3,y2)==1 and all(d<=1 for d in factorint(y3).values()):
                                for y4 in range(1,integer_nthroot(z4:=z3//y3**9,8)[0]+1):
                                    if gcd(y4,y1)==1 and gcd(y4,y2)==1 and gcd(y4,y3)==1 and all(d<=1 for d in factorint(y4).values()):
                                        for y5 in range(1,integer_nthroot(z5:=z4//y4**8,7)[0]+1):
                                            if gcd(y5,y1)==1 and gcd(y5,y2)==1 and gcd(y5,y3)==1 and gcd(y5,y4)==1 and all(d<=1 for d in factorint(y5).values()):
                                                c -= integer_nthroot(z5//y5**7,6)[0]
        return c
    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
    return bisection(f,n,n) # Chai Wah Wu, Sep 10 2024

Formula

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^5*(p-1))) = 1.0334657852594050612296726462481884631303137561267151463866539131591664... - Amiram Eldar, Jul 09 2020

A076468 Perfect powers m^k where k >= 4.

Original entry on oeis.org

1, 16, 32, 64, 81, 128, 243, 256, 512, 625, 729, 1024, 1296, 2048, 2187, 2401, 3125, 4096, 6561, 7776, 8192, 10000, 14641, 15625, 16384, 16807, 19683, 20736, 28561, 32768, 38416, 46656, 50625, 59049, 65536, 78125, 83521, 100000, 104976, 117649
Offset: 1

Views

Author

Robert G. Wilson v, Oct 14 2002

Keywords

Comments

If p|n then at least p^4|n.
Subsequence of A036967. - R. J. Mathar, May 27 2011

Links

Crossrefs

Cf. A001597, A036967, A076467, A076469, A076470.
Cf. A002117, A013661.

Programs

  • Haskell
    import qualified Data.Set as Set (null)
import Data.Set (empty, insert, deleteFindMin)
a076468 n = a076468_list !! (n-1)
a076468_list = 1 : f [2..] empty where
   f xs'@(x:xs) s | Set.null s || m > x ^ 4 = f xs $ insert (x ^ 4, x) s
                  | m == x ^ 4  = f xs s
                  | otherwise = m : f xs' (insert (m * b, b) s')
                  where ((m, b), s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 19 2013
  • Mathematica
    a = {1}; Do[ If[ Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]] > 3, a = Append[a, n]; Print[n]], {n, 2, 131071}]; a
  • Python
    from sympy import mobius, integer_nthroot
def A076468(n):
    def f(x): return int(n+2+x-integer_nthroot(x,4)[0]-(integer_nthroot(x,6)[0]<<1)-integer_nthroot(x,9)[0]+sum(mobius(k)*(integer_nthroot(x,k)[0]+integer_nthroot(x,k<<1)[0]+integer_nthroot(x,3*k)[0]-3) for k in range(5,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 14 2024

Formula

Sum_{n>=1} 1/a(n) = 3 - zeta(2) - zeta(3) + Sum_{k>=2} mu(k)*(3 - zeta(k) - zeta(2*k) - zeta(3*k)) = 1.1473274274... . - Amiram Eldar, Dec 03 2022

A076469 Perfect powers m^k where k >= 5.

Original entry on oeis.org

1, 32, 64, 128, 243, 256, 512, 729, 1024, 2048, 2187, 3125, 4096, 6561, 7776, 8192, 15625, 16384, 16807, 19683, 32768, 46656, 59049, 65536, 78125, 100000, 117649, 131072, 161051, 177147, 248832, 262144, 279936, 371293, 390625, 524288, 531441
Offset: 1

Views

Author

Robert G. Wilson v, Oct 14 2002

Keywords

Comments

If p|n when at least p^5|n.

Links

Crossrefs

Cf. A001597, A076467, A076468, A076470.
Cf. A002117, A013661, A013662.

Programs

  • Mathematica
    a = {1}; Do[ If[ Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]] > 4, a = Append[a, n]; Print[n]], {n, 2, 537823}]; a
  • Python
    from sympy import mobius, integer_nthroot
def A076469(n):
    def f(x): return int(n+3+x-(integer_nthroot(x,6)[0]<<1)-integer_nthroot(x,8)[0]-integer_nthroot(x,9)[0]-integer_nthroot(x,12)[0]+sum(mobius(k)*(integer_nthroot(x,k)[0]+integer_nthroot(x,k<<1)[0]+integer_nthroot(x,3*k)[0]+integer_nthroot(x,k<<2)[0]-4) for k in range(5,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 14 2024

Formula

Sum_{n>=1} 1/a(n) = 4 - zeta(2) - zeta(3) - zeta(4) + Sum_{k>=2} mu(k)*(4 - zeta(k) - zeta(2*k) - zeta(3*k) - zeta(4*k)) = 1.06932853458... . - Amiram Eldar, Dec 03 2022
Showing 1-5 of 5 results.

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