A282231 First term of A175304 with a given prime signature.

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

Author

Vladimir Shevelev, Feb 09 2017

Keywords

• nonn

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

Examples

			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)
		

Links

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

Crossrefs

Cf. A025487, A037916, A175304, A282175.

Programs

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 *)

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

Extensions

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