

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
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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<k^2+d(k^2)<(k+1)^2 and k^2+d(k^2) cannot be perfect squares. So, k^2+d(k^2) is even and we have a contradiction.  Vladimir Shevelev, Feb 10 2017
If p is prime and t+1 is odd prime, then p^t is not in the sequence. Indeed, if d(p^t+t+1)=t+1, then p^t+t+1=q^t, where q is prime > 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^(t1) or t+1>=2*t*p^(t1) 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: A182637 A065512 A066147 * A140951 A190243 A187127
Adjacent sequences: A175301 A175302 A175303 * A175305 A175306 A175307


KEYWORD

nonn,changed


AUTHOR

Leroy Quet, Mar 24 2010


EXTENSIONS

More terms from Emeric Deutsch, Apr 08 2010


STATUS

approved



