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.

A077270 Duplicate of A076048.

Original entry on oeis.org

1, 4, 11, 26, 52, 109, 237, 556, 1405, 3690, 10085, 28157, 80071, 230568, 670122
Offset: 0

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Author

Keywords

A025475 1 and the prime powers p^m where m >= 2, thus excluding the primes.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1331, 1369, 1681, 1849, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7921, 8192
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Also nonprime n such that sigma(n)*phi(n) > (n-1)^2. - Benoit Cloitre, Apr 12 2002
If p is a term of the sequence, then the index n for which a(n) = p is given by n := b(p) := 1 + Sum_{k>=2} PrimePi(p^(1/k)). Here, the sum has floor(log_2(p)) positive terms. For any m > 0, the greatest number n such that a(n) <= m is also given by b(m), thus, b(m) is the number of such prime powers <= m. - Hieronymus Fischer, May 31 2013
That 8 and 9 are the only two consecutive integers in this sequence is known as Catalan's Conjecture and was proved in 2002 by Preda Mihăilescu. - Geoffrey Critzer, Nov 15 2015

Links

Crossrefs

Subsequence of A000961. - Reinhard Zumkeller, Jun 22 2011
Cf. A001597, A000720, A068982, A136141, A193166.
Differences give A053707.
Cf. A076048 (number of terms < 10^n).
There are four different sequences which may legitimately be called "prime powers": A000961 (p^k, k >= 0), A246655 (p^k, k >= 1), A246547 (p^k, k >= 2), A025475 (p^k, k=0 and k >= 2). When you refer to "prime powers", be sure to specify which of these you mean. Also A001597 is the sequence of nontrivial powers n^k, n >= 1, k >= 2. - N. J. A. Sloane, Mar 24 2018

Programs

  • Haskell
    a025475 n = a025475_list !! (n-1)
a025475_list = filter ((== 0) . a010051) a000961_list
-- Reinhard Zumkeller, Jun 22 2011
  • Maple
    isA025475 := proc(n)
    if n < 1 then
        false;
    elif n = 1 then
        true;
    elif isprime(n) then
        false;
    elif nops(numtheory[factorset](n)) = 1 then
        true;
    else
        false;
    end if;
end proc:
A025475 := proc(n)
    option remember;
    local a;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isA025475(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
# R. J. Mathar, Jun 06 2013
# alternative:
N:= 10^5: # to get all terms <= N
Primes:= select(isprime, [2,(2*i+1 $ i = 1 .. floor((sqrt(N)-1)/2))]):
sort([1,seq(seq(p^i, i=2..floor(log[p](N))),p=Primes)]); # Robert Israel, Jul 27 2015
  • Mathematica
    A025475 = Select[ Range[ 2, 10000 ], ! PrimeQ[ # ] && Mod[ #, # - EulerPhi[ # ] ] == 0 & ]
A025475 = Sort[ Flatten[ Table[ Prime[n]^i, {n, 1, PrimePi[ Sqrt[10^4]]}, {i, 2, Log[ Prime[n], 10^4]}]]]
{1}~Join~Select[Range[10^4], And[! PrimeQ@ #, PrimePowerQ@ #] &] (* Michael De Vlieger, Jul 04 2016 *)
Join[{1},Select[Range[100000],PrimePowerQ[#]&&!PrimeQ[#]&]] (* Harvey P. Dale, Oct 29 2023 *)
  • PARI
    for(n=1,10000,if(sigma(n)*eulerphi(n)*(1-isprime(n))>(n-1)^2,print1(n,",")))
  • PARI
    is_A025475(n)={ ispower(n,,&p) && isprime(p) || n==1 }  \\ M. F. Hasler, Sep 25 2011
  • PARI
    list(lim)=my(v=List([1]),L=log(lim+.5));forprime(p=2,(lim+.5)^(1/3),for(e=3,L\log(p),listput(v,p^e))); vecsort(concat(Vec(v), apply(n->n^2,primes(primepi(sqrtint(lim\1)))))) \\ Charles R Greathouse IV, Nov 12 2012
  • PARI
    list(lim)=my(v=List([1])); for(m=2,logint(lim\=1,2), forprime(p=2,sqrtnint(lim,m), listput(v, p^m))); Set(v) \\ Charles R Greathouse IV, Aug 26 2015
  • Python
    from sympy import primerange
A025475_list, m = [1], 10*2
m2 = m**2
for p in primerange(1,m):
    a = p**2
    while a < m2:
        A025475_list.append(a)
        a *= p
A025475_list = sorted(A025475_list) # Chai Wah Wu, Sep 08 2014
  • Python
    from sympy import primepi, integer_nthroot
def A025475(n):
    if n==1: return 1
    def f(x): return int(n-2+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,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 13 2024

Formula

The number of terms <= N is O(sqrt(N)*log N). [See Weisstein link] - N. J. A. Sloane, May 27 2022
A005171(a(n))*A010055(a(n)) = 1. - Reinhard Zumkeller, Nov 01 2009
A192280(a(n)) = 0 for n > 1. - Reinhard Zumkeller, Aug 26 2011
A014963(a(n)) - A089026(a(n)) = A014963(a(n)) - 1. - Eric Desbiaux, May 18 2013
From Hieronymus Fischer, May 31 2013: (Start)
The greatest number n such that a(n) <= m is given by 1 + Sum_{k>=2} A000720(floor(m^(1/k))).
Example 1: m = 10^10 ==> n = 10085;
Example 2: m = 10^11 ==> n = 28157;
Example 3: m = 10^12 ==> n = 80071;
Example 4: m = 10^15 ==> n = 1962690. (End)
Sum_{n>=2} 1/a(n) = Sum_{p prime} 1/(p*(p-1)) = A136141. - Amiram Eldar, Oct 11 2020
From Amiram Eldar, Jan 28 2021: (Start)
Product_{n>=2} (1 + 1/a(n)) = Product_{k>=2} zeta(k)/zeta(2*k) = 2.0729553047...
Product_{n>=2} (1 - 1/a(n)) = A068982. (End)

Extensions

Edited by Daniel Forgues, Aug 18 2009

A238815 Number of prime powers p^k (k >= 0) (A000961) <= 10^n.

Original entry on oeis.org

1, 8, 36, 194, 1281, 9701, 78735, 665135, 5762860, 50851224, 455062596, 4118082970, 37607992089, 346065767407, 3204942420924, 29844572385359, 279238346816393, 2623557174778439, 24739954338671300, 234057667428388199, 2220819603016308080, 21127269487386615272
Offset: 0

Views

Author

Robert G. Wilson v, Mar 05 2014

Keywords

Crossrefs

Cf. A006880, A076048, A267712.

Programs

  • Mathematica
    f[n_] := Block[{k = t = 1}, While[s = PrimePi[ 10^(n/k)]; s != 0, t = t + s; k++]; t]; Array[f, 15, 0]
  • PARI
    a(n)=sum(k=2,10^n,isprimepower(k)>0)+1 \\ Charles R Greathouse IV, Mar 05 2014
  • PARI
    a(n)=sum(e=1,n*log(10)\log(2),primepi(sqrtnint(10^n,e)))+1 \\ Charles R Greathouse IV, Mar 05 2014

Formula

a(n) = A076048(n) + A006880(n).
a(n) ~ 10^n/(n log 10). - Charles R Greathouse IV, Mar 05 2014
For n > 0, a(n) = A267712(n) + 1. - Jon E. Schoenfield, Apr 19 2018

Extensions

a(15)-a(21) from Charles R Greathouse IV, Mar 05 2014
Showing 1-3 of 3 results.

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