A085155 -id:A085155 - cp's OEIS frontend

A128200 Numbers of divisors of semiprime powers: A000005(A085155(n)).

1, 3, 4, 3, 4, 4, 4, 5, 4, 4, 3, 4, 4, 4, 4, 9, 4, 4, 4, 3, 4, 4, 4, 4, 4, 7, 4, 4, 4, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 9, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 9, 4, 4, 4, 4, 4, 4, 4, 4, 4, 16, 4, 4, 4, 4, 9, 4, 4, 4, 4, 4, 4, 4, 9, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4

Offset: 1

Reinhard Zumkeller, Mar 04 2007

nonn

The range of this sequence is the union of squares and the odd numbers: A128201.

R. Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A005408, A000290.

sppQ[n_] := With[{f = FactorInteger[n][[All, 2]]}, n == 1 || Length[f] == 1 && EvenQ[f[[1]]] || Length[f] == 2 && f[[1]] == f[[2]]];
DivisorSigma[0, #]& /@ Select[Range[1000], sppQ] (* Jean-François Alcover, Nov 21 2020 *)

a(n) = (1+A001222(A085155(n))/A001221(A085155(n)))^A001221(A085155(n)).

A085156 Powers of primes or powers of semiprimes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 67, 69, 71, 73, 74, 77, 79, 81, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 100, 101

Offset: 1

Reinhard Zumkeller, Jun 21 2003

nonn

m is a term iff A067029(m) = A071178(m) and A001221(m) <= 2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Union of A000961 and A085155.

Cf. A001221, A067029, A071178.

q[n_] := Module[{e = FactorInteger[n][[;;, 2]]}, Length[e] == 1 || (Length[e] == 2 && Equal @@ e)]; Select[Range[100], q] (* Amiram Eldar, Apr 19 2025 *)

is(n)=my(f=factor(n)[, 2]); #f<2 || (#f==2 && f[1]==f[2]) \\ Charles R Greathouse IV, Oct 19 2015

A303661 Powers of squarefree semiprimes that are not squarefree.

Original entry on oeis.org

36, 100, 196, 216, 225, 441, 484, 676, 1000, 1089, 1156, 1225, 1296, 1444, 1521, 2116, 2601, 2744, 3025, 3249, 3364, 3375, 3844, 4225, 4761, 5476, 5929, 6724, 7225, 7396, 7569, 7776, 8281, 8649, 8836, 9025, 9261, 10000, 10648, 11236, 12321, 13225, 13924, 14161, 14884

Offset: 1

Ilya Gutkovskiy, Apr 28 2018

nonn

			1089 is in the sequence because 1089 = 3^2*11^2.
1296 is in the sequence because 1296 = 2^4*3^4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Squarefree.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A006881, A013929, A072774, A072777, A085155, A123711, A200511, A246547, A303606.

Select[Range[15000], Length[Union[FactorInteger[#][[All, 2]]]] == 1 && PrimeNu[#] == 2 && ! SquareFreeQ[#] &]
seq[max_] := Module[{sp = Select[Range[Floor@Sqrt[max]], SquareFreeQ[#] && PrimeNu[#] == 2 &], s = {}}, Do[s = Join[s, sp[[k]]^Range[2, Floor@Log[sp[[k]], max]]], {k, 1, Length[sp]}]; Union@s]; seq[10000] (* Amiram Eldar, Feb 12 2021 *)

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A303661(n):
    def g(x): return int(-(t:=primepi(s:=isqrt(x)))-(t*(t-1)>>1)+sum(primepi(x//k) for k in primerange(1, s+1)))
    def f(x): return n-1+x-sum(g(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 19 2024

Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A006881(n)-1)*A006881(n)) = Sum_{k>=2} (P(k)^2 - P(2*k))/2 = 0.07160601536406295068..., where P(k) is the prime zeta function. - Amiram Eldar, Feb 12 2021

A122443 Least prime factor of powers of semiprimes.

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 3, 2, 3, 2, 5, 2, 3, 2, 5, 2, 2, 3, 2, 7, 3, 5, 3, 2, 2, 2, 5, 3, 2, 7, 3, 2, 5, 2, 3, 7, 3, 2, 5, 2, 2, 3, 5, 2, 7, 11, 2, 3, 3, 7, 2, 3, 2, 11, 5, 2, 5, 2, 3, 7, 2, 13, 3, 2, 3, 5, 11, 2, 2, 3, 2, 7, 5, 2, 11, 3, 2, 5, 2, 7, 2, 3, 13, 3, 2, 5, 3, 13

Offset: 1

Jonathan Vos Post, Sep 06 2006

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A020639, A076396, A085155.
Cf. A122444 (greatest prime factor of powers of semiprimes).
Cf. A076396 (smallest prime factor of n-th perfect power).

FactorInteger[#][[1, 1]] & /@ Select[Range@ 250, Function[n, Or[n == 1, And[Length@ # == 1, EvenQ@ First@ #], And[Length@ # == 2, SameQ @@ #]] &[FactorInteger[n][[All, -1]]]]] (* Michael De Vlieger, Mar 04 2017 *)

is(n)=my(f=factor(n)[, 2]); #f==0 || (#f==2 && f[1]==f[2]) || (#f==1 && f[1]%2==0);
spf(n) = if (n==1, 1, factor(n)[1,1]);
lista(nn) = {for (n=1, nn, if (is(n), print1(spf(n), ", ")););} \\ Michel Marcus, Mar 04 2017

a(n) = A020639(A085155(n)) = least prime factor of A085155 powers of semiprimes.

More terms from Michel Marcus, Mar 04 2017

A122444 Greatest prime factor of powers of semiprimes.

Original entry on oeis.org

1, 2, 3, 3, 5, 7, 5, 2, 7, 11, 5, 13, 11, 17, 7, 3, 19, 13, 23, 7, 17, 11, 19, 29, 31, 2, 13, 23, 37, 11, 3, 41, 17, 43, 29, 13, 31, 47, 19, 5, 53, 37, 23, 59, 17, 11, 61, 41, 43, 19, 67, 47, 71, 13, 29, 73, 31, 79, 53, 23, 83

Offset: 1

Jonathan Vos Post, Sep 06 2006

A122443 is least prime factor of powers of semiprimes. Cf. A076397 Largest prime factor of n-th perfect power.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358, A006530, A020639, A076396, A076397, A085155, A122443.

FactorInteger[#][[-1, 1]] & /@ Select[Range@ 168, Function[n, Or[n == 1, And[Length@ # == 1, EvenQ@ First@ #], And[Length@ # == 2, SameQ @@ #]] &[FactorInteger[n][[All, -1]]]]] (* Michael De Vlieger, Mar 04 2017 *)

a(n) = A006530(A085155(n)) = greatest prime factor of A085155 powers of semiprimes.

A180868 Numbers n such that n and n+1 are semiprime powers.

Original entry on oeis.org

9, 14, 15, 21, 25, 33, 34, 35, 38, 57, 64, 81, 85, 86, 93, 94, 118, 121, 122, 133, 141, 142, 145, 158, 177, 201, 202, 205, 213, 214, 215, 216, 217, 218, 225, 253, 298, 301, 302, 326, 334, 361, 381, 393, 394, 445, 446, 453, 481, 484, 501

Offset: 1

Jonathan Vos Post, Jan 22 2011

This is to semiprimes A001358 and powers of semiprimes A085155 as A006549 is to primes A000040 and powers of primes A000961.

			15 is in the sequence because 15 = (3*5)^1 and 15+1 = 16 = (2*2)^2 are both semiprime powers.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000961, A001358, A006549, A085155.

spp:= proc(n) option remember; local l;
        if n<2 or isprime(n) then false
        else l:= ifactors(n)[2];
             if nops(l)>2 then false
           elif nops(l)=2 then evalb(l[1][2]=l[2][2])
           else evalb(irem(l[1][2], 2)=0)
             fi
        fi
      end:
a:= proc(n) option remember; local k;
      for k from 1+ `if`(n=1, 8, a(n-1))
        while not spp(k) or not spp(k+1)
      do od; k
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 22 2011

sppQ[n_] := With[{f = FactorInteger[n][[All, 2]]}, n==1 || Length[f]==1 && EvenQ[f[[1]]] || Length[f]==2 && f[[1]]==f[[2]]];
Select[Range[1000], sppQ[#] && sppQ[#+1]&] (* Jean-François Alcover, Nov 21 2020 *)

{ n : {n,n+1} is subset of {A085155} } = { n : n = A001358(i)^j and n+1 = A001358(k)^m }.

More terms and edited by Alois P. Heinz, Jan 22 2011

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A128200 Numbers of divisors of semiprime powers: A000005(A085155(n)).

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A085156 Powers of primes or powers of semiprimes.

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A303661 Powers of squarefree semiprimes that are not squarefree.

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A122443 Least prime factor of powers of semiprimes.

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A122444 Greatest prime factor of powers of semiprimes.

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A180868 Numbers n such that n and n+1 are semiprime powers.

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