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%I #58 Aug 18 2019 16:08:01
%S 34,55,919,1241,4149,4188,7170,12566,15086,24882,25020,26610,51836,
%T 53964,59988,77058,143370,150420,167561,170562,205728,215070,220818,
%U 418308,564858,731321,907255,910316,986154,1239870,1569336,1622914,1841861,1887240,1979307,2229012,2262108
%N Numbers k whose nearest neighbors have the same number of divisors, the same number of distinct prime factors, and the same sum of divisors.
%C mu(k-1) = mu(k+1), where mu(k) = A008683(k), since k-1 and k+1 have the same number of distinct prime factors.
%C tau(k-1) = tau(k+1) = abs(phi(k-1) - phi(k+1)) iff abs(phi(k-1) - phi(k+1)) = 4, where phi(j) is A000010. When tau(j) = 4 omega(j) = 2 and phi(j), the product of two even numbers is divisible by 4.
%C For known elements:
%C - sigma(k +- 1) and tau(k +- 1) the greatest common divisor is 4.
%C - sigma(k +- 1) is divisible by tau(k +- 1).
%C - the digital root of sigma(k +- 1) is either 3 or 9.
%C - the prime signature of k +- 1 is the same (see question below).
%C The first prime terms are 919, 110495719, 2587274227, 3908452759, 4020447619, and 9314901619. - _Giovanni Resta_, Feb 12 2018
%C Are the prime signatures of k +- 1 always the same? - _Andrey Zabolotskiy_, Feb 14 2018
%H Charles R Greathouse IV, <a href="/A294173/b294173.txt">Table of n, a(n) for n = 1..1478</a>
%e 34 is in the sequence because tau(33)=tau(35)=4, omega(33)=omega(35)=2, and sigma(33)=sigma(35)=48.
%e 919 is in the sequence because tau(918)=tau(920)=16, omega(918)=omega(920)=3, and sigma(918)=sigma(920)=2160.
%p with(numtheory):
%p select(k->sigma(k-1)=sigma(k+1) and mobius(k-1)=mobius(k+1) and tau(k-1)=tau(k+1), [$2..2000000]); # _Muniru A Asiru_, Feb 17 2018
%t 1 + Position[Partition[Array[{DivisorSigma[0, #], DivisorSigma[1, #], PrimeOmega[#]} &, 10^6], 3, 1], _?(#[[1]] == +#[[-1]] &), {1}, Heads -> False][[All, 1]] (* _Michael De Vlieger_, Feb 17 2018 *)
%o (GAP) Filtered([2..2000000],k->Sigma(k-1)=Sigma(k+1) and Number(FactorsInt(k-1))=Number(FactorsInt(k+1)) and Tau(k-1)=Tau(k+1)); # _Muniru A Asiru_, Feb 17 2018
%o (PARI) list(lim)=my(v=List(),k2=7,s2=sigma(k2),k1=8,s1=sigma(k1),s); forfactored(k=9,1+lim\1, s=sigma(k); if(s==s2 && numdiv(k)==numdiv(k2) && omega(k)==omega(k2), listput(v,k1[1])); k2=k1; k1=k; s2=s1; s1=s); Vec(v) \\ _Charles R Greathouse IV_, Feb 20 2018
%Y Intersection of A067888, A088070, and A055574.
%Y Cf. A229254, A008683, A000010.
%K nonn
%O 1,1
%A _Torlach Rush_, Feb 10 2018
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