cp's OEIS Frontend

A140831 Numbers in whose canonical prime factorization the powers of the primes do not form an increasing sequence.

%I A140831 #42 May 02 2025 09:07:31
%S A140831 12,24,40,45,48,56,60,63,80,84,90,96,112,120,126,132,135,144,156,160,
%T A140831 168,175,176,180,189,192,204,208,224,228,240,252,264,270,275,276,280,
%U A140831 288,297,300,312,315,320,325,336,348,350,351,352,360,372,378,384,405
%N A140831 Numbers in whose canonical prime factorization the powers of the primes do not form an increasing sequence.
%C A140831 Previous name was: Let p^b(n,p) be the largest power of the prime p that divides n. The integer n is included if the list of p^b(n,p)'s, where each p is a distinct prime divisor of n, arranged by size of each p^b(n,p) is not in the same order as the list of p^b(n,p)'s arranged by size of each prime p.
%C A140831 This sequence contains no squarefree integers.
%C A140831 90 is the smallest integer in this sequence but not in sequence A126855.
%C A140831 The number of terms < 10^n: 0, 12, 151, 1575, 16154, 161630, 1617052, ..., . - _Robert G. Wilson v_, Aug 31 2008
%C A140831 If k is in the sequence, then all powers of k are in the sequence. - _Mike Jones_, Jun 16 2022
%C A140831 If k is in the sequence then k*A020639(k)^m is in the sequence for m >= 0. - _David A. Corneth_, Jun 16 2022
%C A140831 Conjecture: There are infinitely many terms k such that k+1 is also a term. - _Mike Jones_, Jun 18 2022
%H A140831 David A. Corneth, <a href="/A140831/b140831.txt">Table of n, a(n) for n = 1..10000</a>
%e A140831 The prime factorization of 90 is, when arranged by size of the distinct primes, 2^1 * 3^2 * 5^1. Since 3^2 is > 5^1, even though 5 > 3, 90 is in the sequence.
%t A140831 fQ[n_] := Block[{f = First@# ^ Last@# & /@ FactorInteger@n}, f != Sort@f]; Select[ Range@ 407, fQ@# &] (* _Robert G. Wilson v_, Aug 31 2008 *)
%o A140831 (PARI) is(n) = { my(f = factor(n)); for(i = 1, #f~-1, if(f[i,1]^f[i,2] > f[i+1,1]^f[i+1,2], return(1) ) ); 0 } \\ _David A. Corneth_, Jun 16 2022
%Y A140831 Complement of A383397.
%Y A140831 Cf. A020639, A141809, A141810, A126855.
%K A140831 nonn,easy
%O A140831 1,1
%A A140831 _Leroy Quet_, Jul 18 2008
%E A140831 More terms from _Robert G. Wilson v_, Aug 31 2008
%E A140831 Simpler name from _Mike Jones_, Jun 15 2022