A284342 - OEIS

%I #34 Apr 30 2021 12:39:25

%S 12,18,24,36,40,45,48,50,54,56,60,63,72,75,80,84,90,96,98,100,108,112,

%T 120,126,132,135,144,147,150,156,160,162,168,175,176,180,189,192,196,

%U 198,200,204,208,216,224,225,228,234,240,242,245,250,252,264,270,275,276,280,288,294,297,300

%N Numbers n such that A065642(n) < n*lpf(n), where lpf = least prime factor (A020639).

%C Numbers n for which A065642(n) < A285109(n). Positions of terms > 1 in A285337. - _Antti Karttunen_, Apr 19 2017

%C For any n in this sequence, k*n is also in this sequence. No term is squarefree. For any distinct primes p and q with p > q, p^2*q and p*q^(ceiling(log_q(p))) are in this sequence. - _Charlie Neder_, Oct 29 2018

%H Antti Karttunen, <a href="/A284342/b284342.txt">Table of n, a(n) for n = 1..10000</a>

%t Select[Range[2, 300], Function[{n, c, lpf}, SelectFirst[Range[n + 1, n^2], Times @@ FactorInteger[#][[All, 1]] == c &] < n lpf] @@ {#1, Times @@ #2, #2[[1]]} & @@ {#, FactorInteger[#][[All, 1]]} &] (* _Michael De Vlieger_, Oct 31 2018 *)

%o (PARI) for(n=1,300,for(k=1,n^2-n,a=factorback(factorint(n)[,1]); b=factorback(factorint(n+k)[,1]); c=vecmin(factor(n)[,1]); if(a==b&&n+k<n*c&!print1(n", ")&break)))

%o (PARI)

%o A020639(n) = if(1==n,n,vecmin(factor(n)[, 1]));

%o A007947(n) = factorback(factorint(n)[, 1]); \\ From _Andrew Lelechenko_, May 09 2014

%o A065642(n) = { my(r=A007947(n)); if(1==n,n,n = n+r; while(A007947(n) <> r, n = n+r); n); };

%o isA284342(n) = (A065642(n) < n*A020639(n));

%o n=0; k=1; while(k <= 10000, n=n+1; if(isA284342(n),write("b284342.txt", k, " ", n);k=k+1));

%o \\ _Antti Karttunen_, Apr 19 2017

%o (Python)

%o from operator import mul

%o from sympy import primefactors

%o from functools import reduce

%o def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))

%o def a065642(n):

%o     if n==1: return 1

%o     r=a007947(n)

%o     n = n + r

%o     while a007947(n)!=r:

%o         n+=r

%o     return n

%o print([n for n in range(10, 301) if a065642(n)<n*min(primefactors(n))]) # _Indranil Ghosh_, Apr 20 2017

%Y Cf. A007947, A020639, A065642, A285100 (complement), A285109, A285337.

%K nonn

%O 1,1

%A _Gionata Neri_, Mar 25 2017