A080718 - OEIS

%I #32 Apr 21 2025 02:35:15

%S 1,1255,12955,17482,25105,100255,101299,105295,107329,117067,124483,

%T 127417,129595,132565,145273,146137,149782,174082,174298,174793,

%U 174982,250105,256315,263155,295105,297463,307183,325615,371893,536539

%N 1, together with numbers n that are the product of two primes p and q such that the multiset of the digits of n coincides with the multiset of the digits of p and q.

%C Except for 1, this sequence is a subsequence of A280928. More specifically, members of A280928 are also members of this sequence if and only if they are semiprime. - _Ely Golden_, Jan 11 2017

%C This sequence has no equivalent in odd bases. This is because any equivalent of A280928 in an odd base must have all terms having at least 3 prime factors. - _Ely Golden_, Jan 11 2017

%C All entries other than 1 are congruent to 4 mod 9, because p*q == p + q mod 9 (with p and q not both divisible by 3) implies p*q == 4 mod 9. - _Robert Israel_, May 05 2014

%H Michael S. Branicky, <a href="/A080718/b080718.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..145 from Robert Israel)

%e 1255 = 5*251, 12955 = 5*2591, 17482 = 2*8741, 100255 = 5*20051, 146137=317*461, etc.

%p filter:= proc(n) local F,p,q,Ln,Lpq;

%p   F:= ifactors(n)[2];

%p   if nops(F) > 2 or convert(F,`+`)[2]<>2 then return false fi;

%p   p:= F[1][1];

%p   if nops(F) = 2 then q:= F[2][1] else q:= F[1][1] fi;

%p   Ln:= sort(convert(n,base,10));

%p   Lpq:= sort([op(convert(p,base,10)),op(convert(q,base,10))]);

%p   evalb(Ln = Lpq);

%p end proc:

%p filter(1):= true:

%p A080718:= select(filter,[1, seq(4+9*i,i=1..10^6)]); # _Robert Israel_, May 04 2014

%t ptpQ[n_]:=Module[{sidn=Sort[IntegerDigits[n]],fi=Transpose[ FactorInteger[ n]]}, fi[[2]]=={1,1}&&Sort[Flatten[ IntegerDigits/@ fi[[1]]]]==sidn]; Join[{1}, Select[Range[4,550000,9],ptpQ]] (* _Harvey P. Dale_, Jun 22 2014 *)

%o (Python)

%o from sympy import factorint

%o from itertools import count, islice

%o def agen(): # generator

%o     yield 1

%o     for k in count(4, 9):

%o         t = sorted(str(k))

%o         f = factorint(k)

%o         if sum(f.values()) == 2:

%o             p, q = min(f), max(f)

%o             if t == sorted(str(p)+str(q)):

%o                 yield k

%o print(list(islice(agen(), 30))) # _Michael S. Branicky_, Apr 20 2025

%Y The following sequences are all closely related: A020342, A014575, A080718, A280928, A048936, A144563.

%K nonn,base

%O 1,2

%A _Jeff Heleen_, Mar 06 2003

%E Edited by _N. J. A. Sloane_, Jan 03 2009

%E Incorrect entry 163797 removed by _Robert Israel_, May 04 2014