OFFSET
1,1
COMMENTS
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
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..234
EXAMPLE
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)
MATHEMATICA
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 *)
PROG
(PARI) 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
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Shevelev, Feb 09 2017
EXTENSIONS
More terms from Peter J. C. Moses, Feb 09 2017
STATUS
approved