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A346635 Numbers whose division (or multiplication) by their greatest prime factor yields a perfect square. Numbers k such that k*A006530(k) is a perfect square.

%I A346635 #17 Nov 27 2022 10:34:49
%S A346635 1,2,3,5,7,8,11,12,13,17,19,20,23,27,28,29,31,32,37,41,43,44,45,47,48,
%T A346635 52,53,59,61,63,67,68,71,73,76,79,80,83,89,92,97,99,101,103,107,108,
%U A346635 109,112,113,116,117,124,125,127,128,131,137,139,148,149,151,153
%N A346635 Numbers whose division (or multiplication) by their greatest prime factor yields a perfect square. Numbers k such that k*A006530(k) is a perfect square.
%C A346635 This is the sorted version of A342768(n) = position of first appearance of n in A346701 (but A346703 works also).
%F A346635 a(n) = A129597(n)/2 for n > 1.
%e A346635 The terms together with their prime indices begin:
%e A346635      1: {}          31: {11}            71: {20}
%e A346635      2: {1}         32: {1,1,1,1,1}     73: {21}
%e A346635      3: {2}         37: {12}            76: {1,1,8}
%e A346635      5: {3}         41: {13}            79: {22}
%e A346635      7: {4}         43: {14}            80: {1,1,1,1,3}
%e A346635      8: {1,1,1}     44: {1,1,5}         83: {23}
%e A346635     11: {5}         45: {2,2,3}         89: {24}
%e A346635     12: {1,1,2}     47: {15}            92: {1,1,9}
%e A346635     13: {6}         48: {1,1,1,1,2}     97: {25}
%e A346635     17: {7}         52: {1,1,6}         99: {2,2,5}
%e A346635     19: {8}         53: {16}           101: {26}
%e A346635     20: {1,1,3}     59: {17}           103: {27}
%e A346635     23: {9}         61: {18}           107: {28}
%e A346635     27: {2,2,2}     63: {2,2,4}        108: {1,1,2,2,2}
%e A346635     28: {1,1,4}     67: {19}           109: {29}
%e A346635     29: {10}        68: {1,1,7}        112: {1,1,1,1,4}
%p A346635 filter:= proc(n) issqr(n/max(numtheory:-factorset(n))) end proc:
%p A346635 filter(1):= true:
%p A346635 select(filter, [$1..200]); # _Robert Israel_, Nov 26 2022
%t A346635 sqrQ[n_]:=IntegerQ[Sqrt[n]];
%t A346635 Select[Range[100],sqrQ[#*FactorInteger[#][[-1,1]]]&]
%o A346635 (PARI) isok(m) = (m==1) || issquare(m/vecmax(factor(m)[,1])); \\ _Michel Marcus_, Aug 12 2021
%Y A346635 Removing 1 gives a subset of A026424.
%Y A346635 The unsorted even version is A129597.
%Y A346635 The unsorted version is A342768(n) = A342767(n,n).
%Y A346635 Except the first term, the even version is 2*a(n).
%Y A346635 A000290 lists squares.
%Y A346635 A001221 counts distinct prime factors.
%Y A346635 A001222 counts all prime factors.
%Y A346635 A006530 gives the greatest prime factor.
%Y A346635 A061395 gives the greatest prime index.
%Y A346635 A027193 counts partitions of odd length.
%Y A346635 A056239 adds up prime indices, row sums of A112798.
%Y A346635 A209281 = odd bisection sum of standard compositions (even: A346633).
%Y A346635 A316524 = alternating sum of prime indices (sign: A344617, rev.: A344616).
%Y A346635 A325534 counts separable partitions, ranked by A335433.
%Y A346635 A325535 counts inseparable partitions, ranked by A335448.
%Y A346635 A344606 counts alternating permutations of prime indices.
%Y A346635 A346697 = odd bisection sum of prime indices (weights of A346703).
%Y A346635 A346699 = odd bisection sum of reversed prime indices (weights of A346701).
%Y A346635 Cf. A028260, A033942, A035363, A037143, A341446, A344653, A345957, A345958, A345959, A346698, A346700, A346704.
%K A346635 nonn
%O A346635 1,2
%A A346635 _Gus Wiseman_, Aug 10 2021