A123711 -id:A123711 - cp's OEIS frontend

A123709 a(n) is the number of nonzero elements in row n of triangle A123706.

1, 2, 3, 4, 3, 4, 3, 4, 4, 6, 3, 8, 3, 6, 7, 4, 3, 8, 3, 8, 7, 6, 3, 8, 4, 6, 4, 8, 3, 11, 3, 4, 7, 6, 7, 8, 3, 6, 7, 8, 3, 11, 3, 8, 8, 6, 3, 8, 4, 8, 7, 8, 3, 8, 7, 8, 7, 6, 3, 16, 3, 6, 8, 4, 7, 12, 3, 8, 7, 14, 3, 8, 3, 6, 8, 8

Offset: 1

Paul D. Hanna, Oct 09 2006

nonn

Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k]. a(n) = 4 when n is in A123710. a(n) = 8 when n is in A123711. a(n) = 16 when n is in A123712.

			a(n) = 3 when n is an odd prime.
a(n) = 7 when n is the product of two different odd primes.  [Corrected by _M. F. Hasler_, Feb 13 2012]
a(n) = 15 when n is the product of three different odd primes.  [Corrected by _M. F. Hasler_, Feb 13 2012]

M. F. Hasler, Table of n, a(n) for n = 1..500
Peter Luschny, Re: Is the A123706 triangle an extension of the Moebius function?, seqcomp list, Feb 12 2012

Cf. A123706, A123707, A123708, A123710, A123711, A123712; A010766.

Moebius[i_,j_]:=If[Divisible[i,j], MoebiusMu[i/j],0];
A123709[n_]:=Length[Select[Table[Moebius[n,j]-Moebius[n,j+1],{j,1,n}],#!=0&]];
Array[A123709, 500] (* Enrique Pérez Herrero, Feb 13 2012 *)

{a(n)=local(M=matrix(n,n,r,c,if(r>=c,floor(r/c)))^-1); sum(k=1,n,if(M[n,k]==0,0,1))}

A123709(n)=#select((matrix(n, n, r, c, r\c)^-1)[n,],x->x)  \\ M. F. Hasler, Feb 12 2012

A123709(n)={ my(t=moebius(n)); sum(k=2,n, t+0 != t=if(n%k,0,moebius(n\k)))+1}  /* the "t+0 != ..." is required because of a bug in PARI versions <= 2.4.2, maybe beyond, which seems to be fixed in v. 2.5.1 */ \\ M. F. Hasler, Feb 13 2012

a(n) = 2^(m+1) - 1 when n is the product of m distinct odd primes. [Corrected by M. F. Hasler, Feb 13 2012]
For any k>1, a(n)=2^k if, and only if, n is a nonsquarefree number with A001221(n) = k-1 (= omega(n), number of distinct prime factors), with the only exception of a(n=6)=2^2. - M. F. Hasler, Feb 12 2012
A123709(n) = 1 + #{ k in 1..n-1 | Moebius(n,k+1) <> Moebius(n,k) }, where Moebius(n,k)={moebius(n/k) if n=0 (mod k), 0 else}, cf. link to message by P. Luschny. - M. F. Hasler, Feb 13 2012

A123712 Indices n such that 16 = A123709(n) = number of nonzero terms in row n of triangle A123706.

Original entry on oeis.org

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 396, 408, 414, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492

Offset: 1

Paul D. Hanna, Oct 09 2006

nonn

Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k].
a(n) = A178212(n) for n <= 52, possibly more. [Reinhard Zumkeller, May 24 2010]
a(n) = A178212(n) for n <= 2000. - Bill McEachen, Jul 14 2024

G. C. Greubel, Table of n, a(n) for n = 1..1200

Cf. A123706, A123709, A123710, A123711, A010766.

Moebius[i_, j_] := If[Divisible[i, j], MoebiusMu[i/j], 0]; A123709[n_] := Length[Select[Table[Moebius[n, j] - Moebius[n, j + 1], {j, 1, n}], # != 0 &]]; Select[Range[6500], A123709[#] == 16 &] (* G. C. Greubel, Apr 22 2017 *)

is(n)=my(M=matrix(n, n, r, c,r\c)^-1); sum(k=1, n, M[n, k]!=0)==16 \\ Charles R Greathouse IV, Feb 09 2012

A200511 Numbers n with omega(n)=2 and bigomega(n)>2, where omega=A001221=number of distinct prime factors, bigomega=A001222=prime factors counted with multiplicity.

Original entry on oeis.org

12, 18, 20, 24, 28, 36, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 196, 200, 207, 208, 212, 216, 224, 225, 232, 236

Offset: 1

M. F. Hasler, Feb 09 2012

nonn

Equivalently, numbers of the form n=p^k*q^m where k,m>0, k+m>2 and p,q prime.

It appears that this is equal to A123711.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A200521, A178212.

Select[Range[240], PrimeNu[#] == 2 && PrimeOmega[#] > 2 &] (* Jean-François Alcover, Jun 29 2013 *)

for(n=1,999,bigomega(n)>2 & omega(n)==2 & print1(n","))

A123710 Indices n such that 4 = A123709(n) = number of nonzero terms in row n of triangle A123706.

Original entry on oeis.org

4, 6, 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

Offset: 1

Paul D. Hanna, Oct 09 2006

Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k].

Except for a(2)=6, these are proper prime powers, i.e., numbers p^k where k>1, p prime (A025475). - M. F. Hasler, Feb 12 2012

Cf. A123706, A123709, A123711, A123712; A010766.

for(n=1,1e4, A123709(n)==4 & print1(n","))  \\ M. F. Hasler, Feb 12 2012

a(n) = A025475(n) for n>2 (conjectured). - M. F. Hasler, Feb 12 2012

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

cp's OEIS Frontend

A123709 a(n) is the number of nonzero elements in row n of triangle A123706.

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A123712 Indices n such that 16 = A123709(n) = number of nonzero terms in row n of triangle A123706.

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A200511 Numbers n with omega(n)=2 and bigomega(n)>2, where omega=A001221=number of distinct prime factors, bigomega=A001222=prime factors counted with multiplicity.

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A123710 Indices n such that 4 = A123709(n) = number of nonzero terms in row n of triangle A123706.

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A303661 Powers of squarefree semiprimes that are not squarefree.

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