A081061 -id:A081061 - 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

A081060 Product of differences of distinct prime factors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 5, 2, 1, 1, 1, 1, 3, 4, 9, 1, 1, 1, 11, 1, 5, 1, 6, 1, 1, 8, 15, 2, 1, 1, 17, 10, 3, 1, 20, 1, 9, 2, 21, 1, 1, 1, 3, 14, 11, 1, 1, 6, 5, 16, 27, 1, 6, 1, 29, 4, 1, 8, 72, 1, 15, 20, 30, 1, 1, 1, 35, 2, 17, 4, 110, 1, 3, 1, 39, 1, 20, 12, 41, 26, 9, 1, 6, 6, 21, 28

Offset: 1

Reinhard Zumkeller, Mar 04 2003

nonn

a(n)=1 iff n is 3-smooth (A003586) or n is a prime power (A000961), see A081061;
a(A006881(k)) > 1 for k > 1; if a(n) > 1 then A079275(n) > 0.
From Robert G. Wilson v, Aug 06 2018: (Start)
First occurrence of k, k=1,2,3,... or 0 if impossible: 1, 15, 10, 21, 14, 30, 0, 33, 22, 39, 26, 85, 0, 51, 34, 57, 38, 115, ..., ;
Impossible values: 7, 13, 19, 23, 25, 31, 33, 37, 43, 47, 49, 53, 55, 61, 63, 67, 73, 75, 79, 83, 85, 89, 91, 93, 97, ..., ;
Records: 1, 3, 5, 9, 11, 15, 17, 20, 21, 27, 29, 72, 110, 210, 272, 420, 540, 702, 812, 1190, 1482, 1640, 1980, 2262, 2550, 2592, 3192, 3422, 5280, 5760, 5852, ..., .
(End).

			a(42) = a(2*3*7) = |2-3|*|2-7|*|3-7| = 1*5*4 = 20.

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

a[n_] := Times @@ Flatten[Differences@# & /@ Subsets[First@# & /@ FactorInteger@n, {2}]]; Array[a, 90] (* Robert G. Wilson v, Aug 06 2018 *)

A081060(n) = if(omega(n)<=1,1,my(ps = factor(n)[, 1]~, m=1); for(i=1,(#ps)-1,for(j=i+1,#ps, m *= (ps[j]-ps[i]))); (m)); \\ Antti Karttunen, Aug 06 2018

a(n) = Product(abs(p-q): p, q distinct prime factors of n).

A081063 Number of numbers <= n that are 3-smooth or prime powers.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 12, 12, 13, 14, 15, 16, 16, 16, 16, 17, 18, 19, 19, 20, 20, 21, 21, 22, 23, 23, 23, 23, 24, 25, 25, 25, 25, 26, 26, 27, 27, 27, 27, 28, 29, 30, 30, 30, 30, 31, 32, 32, 32, 32, 32, 33, 33, 34, 34, 34, 35, 35, 35, 36, 36, 36, 36, 37, 38, 39

Offset: 1

Reinhard Zumkeller, Mar 04 2003

nonn

a(n) = #{A081061(k): 1<=k<=n}.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A003586, A000961.

Accumulate[Table[If[PrimePowerQ[n]||Max[FactorInteger[n][[All,1]]]<5,1,0],{n,80}]] (* Harvey P. Dale, Nov 29 2020 *)

from sympy import integer_log, primepi, integer_nthroot
def A081063(n): return int(1-(a:=n.bit_length())-(b:=integer_log(n,3)[0])+sum((n//3**i).bit_length() for i in range(b+1))+sum(primepi(integer_nthroot(n, k)[0]) for k in range(1, a))) # Chai Wah Wu, Sep 16 2024

a(n)=A071521(n)+A065515(n)-A000523(n)-A062153(n)+1.

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A081062 Neither 3-smooth numbers nor prime powers.

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A081060 Product of differences of distinct prime factors of n.

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A081063 Number of numbers <= n that are 3-smooth or prime powers.

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