A282175 -id:A282175 - cp's OEIS frontend

A175304 A positive integer n is included if d(n+d(n)) = d(n), where d(n) is the number of divisors of n.

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

Offset: 1

Leroy Quet, Mar 24 2010

nonn

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^2Vladimir 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^(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

			10 is in the sequence because d(10)=4 and d(10+d(10))=d(14)=4. - _Emeric Deutsch_, Apr 08 2010

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000005, A062249, A001359, A255746, A259934, A282175, A282231, A286529.

Positions of zeros in A286530.

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

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 *)

is(n)=numdiv(n+n=numdiv(n))==n \\ M. F. Hasler, Sep 27 2015

More terms from Emeric Deutsch, Apr 08 2010

A282231 First term of A175304 with a given prime signature.

Original entry on oeis.org

3, 6, 12, 60, 70, 72, 96, 125, 128, 250, 264, 450, 480, 756, 1152, 1380, 1458, 1980, 2030, 2048, 3640, 4860, 6552, 7776, 10648, 11448, 11907, 12348, 14960, 17664, 18432, 27540, 31620, 34200, 40500, 42978, 58140, 65000, 75776, 102240, 131328, 146529, 153120

Offset: 1

Vladimir Shevelev, Feb 09 2017

nonn

Conjecturally the sequence is infinite.
The sequence of the corresponding prime signatures begins  p, p*q, p^2*q, p^2*q*r, p*q*r, p^3*q^2, p^5*q, p^3, p^7, ...
There are no prime signatures of perfect squares. Indeed, A175304 contains no squares (see our comment there). - Vladimir Shevelev, Feb 10 2017
A037916(a(n)) gives a numerical version of the second comment: {1,11,21,211,111,32,51,3,7,31,311,221,511,321,72,2111,61,2211,1111,...}, however due to the limitations of the notation in A037916, we cannot represent a(20)=2048 since A037916(2^10)=digit 10, which is not a valid decimal digit. A037916 is useful if we refrain from rendering the multiplicities as decimal digits, instead maintaining them as a list. - Michael De Vlieger, Feb 10 2017

			From _Michael De Vlieger_, Feb 10 2017: (Start)
a(1) = 3 since 3 is prime and has a prime signature of "1"; it is the very first prime in the sequence, followed by {5,11,17,29,41,...}. The prime signature "1" is the first distinct signature encountered in the sequence
a(2) = 6 since it is a squarefree semiprime with prime signature "11"; it is the very first such number in the sequence, followed by {10,22,34, 35,51,...}. This prime signature is the second distinct signature encountered in the sequence.
a(3) = 12 since it has a prime signature of "21" (i.e., the exponents of  p^2*q^1, A037916(12)=21) and this signature is the third distinct signature encountered. It is the very first number with this signature, followed by {44,92,147,236,332,...}. (End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..234

Cf. A025487, A037916, A175304, A282175.

Map[#[[1, 1]] &, GatherBy[#, Last]] &@ Map[{#, Reverse@ Sort@ FactorInteger[#][[All, -1]]} &, Select[Range[10^6], Function[n, DivisorSigma[0, n + #] == # &@ DivisorSigma[0, n]]]] (* Michael De Vlieger, Feb 10 2017 *)

sig(n)=vecsort(factor(n)[,2]~,,4)
has(n)=my(d=numdiv(n)); d==numdiv(n+d)
try(n)=my(t); has(n) && !mapisdefined(m,t=sig(n)) && (mapput(m,t,0) || 1)
v=List();for(n=3,1e9,if(try(n), listput(v,n); print(#v" "n))) \\ Charles R Greathouse IV, Feb 20 2017

More terms from Peter J. C. Moses, Feb 09 2017

cp's OEIS Frontend

A175304 A positive integer n is included if d(n+d(n)) = d(n), where d(n) is the number of divisors of n.

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