cp's OEIS Frontend

A337533 1 together with nonsquares whose square part's square root is in the sequence.

%I A337533 #22 Feb 16 2025 08:34:00
%S A337533 1,2,3,5,6,7,8,10,11,12,13,14,15,17,18,19,20,21,22,23,24,26,27,28,29,
%T A337533 30,31,33,34,35,37,38,39,40,41,42,43,44,45,46,47,50,51,52,53,54,55,56,
%U A337533 57,58,59,60,61,62,63,65,66,67,68
%N A337533 1 together with nonsquares whose square part's square root is in the sequence.
%C A337533 The appearance of a number is determined by its prime signature.
%C A337533 Every squarefree number is present, as the square root of the square part of a squarefree number is 1. Other 4th-power-free numbers are present if and only if they are nonsquare.
%C A337533 If the square part of nonsquarefree k is a 4th power, k does not appear.
%C A337533 Every positive integer k is the product of a unique subset S_k of the terms of A050376, which are arranged in array form in A329050 (primes in column 0, squares of primes in column 1, 4th powers of primes in column 2 and so on). k > 1 is in this sequence if and only if the members of S_k occur in consecutive columns of A329050, starting with column 0.
%C A337533 If the qualifying condition in the previous paragraph was based on the rows instead of the columns of A329050, we would get A055932. The self-inverse function defined by A225546 transposes A329050. A225546 also has multiplicative properties such that if we consider A055932 and this sequence as sets, A225546(.) maps the members of either set 1:1 onto the other set.
%H A337533 Robert Israel, <a href="/A337533/b337533.txt">Table of n, a(n) for n = 1..10000</a>
%H A337533 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquarePart.html">Square part</a>
%H A337533 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F A337533 Numbers m such that A209229(A267116(m) + 1) = 1.
%F A337533 If A008835(a(n)) > 1 then A335324(a(n)) > 1.
%F A337533 If A008833(a(n)) > 1 then A007913(a(n)) > 1.
%e A337533 4 is square and not 1, so 4 is not in the sequence.
%e A337533 12 = 3 * 2^2 is nonsquare, and has square part 4, whose square root (2) is in the sequence. So 12 is in the sequence.
%e A337533 32 = 2 * 4^2 is nonsquare, but has square part 16, whose square root (4) is not in the sequence. So 32 is not in the sequence.
%p A337533 S:= {1}:
%p A337533 for n from 2 to 100 do
%p A337533   if not issqr(n) then
%p A337533     F:= ifactors(n)[2];
%p A337533     s:= mul(t[1]^floor(t[2]/2),t=F);
%p A337533     if member(s,S) then S:= S union {n} fi
%p A337533   fi
%p A337533 od:
%p A337533 sort(convert(S,list)); # _Robert Israel_, Jan 07 2025
%t A337533 pow2Q[n_] := n == 2^IntegerExponent[n, 2]; Select[Range[100], # == 1 || pow2Q[1 + BitOr @@ (FactorInteger[#][[;; , 2]])] &] (* _Amiram Eldar_, Sep 18 2020 *)
%Y A337533 Complement of A337534.
%Y A337533 Closed under A000188(.).
%Y A337533 A209229, A267116 are used in a formula defining this sequence.
%Y A337533 Subsequences: A005117, A036537, A179646, A179666, A179669, A189987, A191554, A191555, A298207, A317090.
%Y A337533 Subsequence of A164514.
%Y A337533 A007913, A008833, A008835, A335324 give the squarefree, square and comparably related parts of a number.
%Y A337533 Cf. A050376, A329050.
%Y A337533 Related to A055932 via A225546.
%K A337533 nonn,easy
%O A337533 1,2
%A A337533 _Peter Munn_, Aug 31 2020