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First term of A175304 with a given prime signature.
4

%I #24 May 09 2021 02:16:16

%S 3,6,12,60,70,72,96,125,128,250,264,450,480,756,1152,1380,1458,1980,

%T 2030,2048,3640,4860,6552,7776,10648,11448,11907,12348,14960,17664,

%U 18432,27540,31620,34200,40500,42978,58140,65000,75776,102240,131328,146529,153120

%N First term of A175304 with a given prime signature.

%C Conjecturally the sequence is infinite.

%C 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, ...

%C There are no prime signatures of perfect squares. Indeed, A175304 contains no squares (see our comment there). - _Vladimir Shevelev_, Feb 10 2017

%C 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

%H Charles R Greathouse IV, <a href="/A282231/b282231.txt">Table of n, a(n) for n = 1..234</a>

%e From _Michael De Vlieger_, Feb 10 2017: (Start)

%e 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

%e 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.

%e 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)

%t 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 *)

%o (PARI) sig(n)=vecsort(factor(n)[,2]~,,4)

%o has(n)=my(d=numdiv(n)); d==numdiv(n+d)

%o try(n)=my(t); has(n) && !mapisdefined(m,t=sig(n)) && (mapput(m,t,0) || 1)

%o v=List();for(n=3,1e9,if(try(n), listput(v,n); print(#v" "n))) \\ _Charles R Greathouse IV_, Feb 20 2017

%Y Cf. A025487, A037916, A175304, A282175.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, Feb 09 2017

%E More terms from _Peter J. C. Moses_, Feb 09 2017