A081060 -id:A081060 - cp's OEIS frontend

A081062 Neither 3-smooth numbers nor prime powers.

10, 14, 15, 20, 21, 22, 26, 28, 30, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 50, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 110, 111, 112, 114, 115, 116

Offset: 1

Reinhard Zumkeller, Mar 04 2003

A081060(m) > 1 iff m = a(k) for some k. - corrected by Gionata Neri, Jul 30 2016
Complement of A081061.
Composites with smallest prime factor^largest prime factor > largest prime factor^smallest prime factor. - Juri-Stepan Gerasimov, Jan 04 2009

			12 = 2^2*3 is not in the sequence because it is 3-smooth (all prime factors are 3 or less). 17 = 17^1 and 49 = 7^2 are not in the sequence because they are prime powers. - _Michael B. Porter_, Jul 31 2016

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003586, A000961.

filter:= proc(n) local f; f:= numtheory:-factorset(n); nops(f) > 1 and max(f) > 3 end proc:
select(filter, [$1..1000]); # Robert Israel, Jul 31 2016

Select[Range@ 120, Nor[PrimePowerQ@ #, 3 EulerPhi[6 #] == 6 #] &] (* Michael De Vlieger, Aug 02 2016, after Robert G. Wilson v at A003586 *)

from sympy import integer_log, primepi, integer_nthroot
def A081062(n):
    def f(x): return int(n+1-(a:=x.bit_length())-(b:=integer_log(x,3)[0])+sum((x//3**i).bit_length() for i in range(b+1))+sum(primepi(integer_nthroot(x, k)[0]) for k in range(1, a)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Sep 16 2024

A081061 Union of 3-smooth numbers and prime powers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 23, 24, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 48, 49, 53, 54, 59, 61, 64, 67, 71, 72, 73, 79, 81, 83, 89, 96, 97, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 162, 163, 167

Offset: 1

Reinhard Zumkeller, Mar 04 2003

nonn

A081060(m)=1 iff m=a(k) for some k.

Complement of A081062.

Jean-François Alcover, Table of n, a(n) for n = 1..10000

Cf. A003586, A000961, A081060, A081062, A081063.

smooth3Q[n_] := n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3] == 1;
Select[Range[1000], PrimePowerQ[#] || smooth3Q[#]&] (* Jean-François Alcover, Oct 14 2021 *)

from sympy import integer_log, primepi, integer_nthroot
def A081061(n):
    def f(x): return int(n+x-1+(a:=x.bit_length())+(b:=integer_log(x,3)[0])-sum((x//3**i).bit_length() for i in range(b+1))-sum(primepi(integer_nthroot(x, k)[0]) for k in range(1, a)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Sep 16 2024

A178921 Product of distances between successive distinct prime divisors of n; zero if n has only 1 distinct prime factor.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 5, 2, 0, 0, 1, 0, 3, 4, 9, 0, 1, 0, 11, 0, 5, 0, 2, 0, 0, 8, 15, 2, 1, 0, 17, 10, 3, 0, 4, 0, 9, 2, 21, 0, 1, 0, 3, 14, 11, 0, 1, 6, 5, 16, 27, 0, 2, 0, 29, 4, 0, 8, 8, 0, 15, 20, 6, 0, 1, 0, 35, 2, 17, 4, 10, 0, 3, 0, 39, 0, 4, 12, 41, 26, 9, 0, 2, 6, 21, 28, 45, 14, 1, 0, 5, 8, 3, 0, 14, 0, 11

Offset: 1

Alex Ratushnyak, Aug 18 2012

For n <= 41, a(n) = A049087(n).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A081060, A049087.

Cf. also A137795.

f[n_] := Module[{ps}, If[n <= 1, 0, ps = Transpose[FactorInteger[n]][[1]]; Times @@ Differences[ps]]]; Table[f[n], {n, 100}] (* T. D. Noe, Aug 20 2012 *)
Array[Apply[Times, Differences@ FactorInteger[#][[All, 1]] /. {} -> 0] &, 105] (* Michael De Vlieger, Sep 10 2018 *)

A178921(n) = if(1>=omega(n), 0, my(ps = factor(n)[,1], m = 1); for(i=2, #ps, m *= (ps[i]-ps[i-1])); (m)); \\ Antti Karttunen, Sep 07 2018

from sympy import primerange
primes = list(primerange(2,500))
for n in range(1,100):
    d = n
    prev = 0
    product = 1
    for p in primes:
        if d%p==0:
            if prev:
                product *= p-prev
            while d%p==0:
                d//=p
            if d==1:
                break
            prev = p
    if prev==0:
        product = 0
    print(product, end=',')

More terms from Antti Karttunen, Sep 07 2018

cp's OEIS Frontend

A081062 Neither 3-smooth numbers nor prime powers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

A081061 Union of 3-smooth numbers and prime powers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

A178921 Product of distances between successive distinct prime divisors of n; zero if n has only 1 distinct prime factor.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions