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 A175304 A positive integer n is included if d(n+d(n)) = d(n), where d(n) is the number of divisors of n. 17
 3, 5, 6, 10, 11, 12, 17, 22, 29, 34, 35, 41, 44, 51, 58, 59, 60, 65, 70, 71, 72, 82, 84, 87, 91, 92, 96, 101, 102, 107, 111, 115, 118, 119, 125, 128, 129, 130, 137, 141, 142, 147, 149, 155, 174, 179, 182, 183, 191, 197, 201, 202, 205, 209, 213, 214, 215, 217, 222 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS The sequence contains the smaller member of every pair of twin primes (A001359) and all squarefree semiprimes m such that m+4 is also a squarefree semiprime (A255746). Can one prove that this is an infinite sequence? - Vladimir Shevelev, Jul 11 2015 The sequence does not contain perfect squares. Indeed, let a(m)=k^2. Then d(k^2)+d(k^2))=d(k^2). Note that d(k^2) is odd. On the other hand, it is known (A046522) that d(k^2)<2*k. Hence, (k+1)^2 - k^2 > d(k^2). Thus k^2 p (if p^t+t+1= say q^l*r^m, then (l+1)*(m+1)=t+1 which is impossible by the condition). But q>=p+2 and p^t+t+1>=p^t+2*t*p^(t-1) or t+1>=2*t*p^(t-1) which trivially has only solution t=1; however, by the condition t>=2. - Vladimir Shevelev, Feb 18 2017 If an odd integer k is in this sequence, so is 2k. - Charlie Neder, Jan 14 2019 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..10000 EXAMPLE 10 is in the sequence because d(10)=4 and d(10+d(10))=d(14)=4. - Emeric Deutsch, Apr 08 2010 MAPLE with(numtheory): a := proc (n) if tau(n+tau(n)) = tau(n) then n else end if end proc: seq(a(n), n = 1 .. 230); # Emeric Deutsch, Apr 08 2010 MATHEMATICA Select[Range@ 224, Function[n, DivisorSigma[0, n + #] == # &@ DivisorSigma[0, n]]](* Michael De Vlieger, Sep 27 2015 *) Position[#, 0][[All, 1]] &@ Table[DivisorSigma[0, n + DivisorSigma[0, n]] - DivisorSigma[0, n], {n, 222}] (* Michael De Vlieger, May 21 2017 *) PROG (PARI) is(n)=numdiv(n+n=numdiv(n))==n \\ M. F. Hasler, Sep 27 2015 CROSSREFS Cf. A000005, A062249, A001359, A255746, A259934, A282175, A282231, A286529. Positions of zeros in A286530. Sequence in context: A331915 A065512 A066147 * A140951 A190243 A187127 Adjacent sequences:  A175301 A175302 A175303 * A175305 A175306 A175307 KEYWORD nonn AUTHOR Leroy Quet, Mar 24 2010 EXTENSIONS More terms from Emeric Deutsch, Apr 08 2010 STATUS approved

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Last modified July 10 05:09 EDT 2020. Contains 335572 sequences. (Running on oeis4.)