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.

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, 52, 53, 59, 61, 63, 67, 68, 71, 73, 76, 79, 80, 83, 89, 92, 97, 99, 101, 103, 107, 108, 109, 112, 113, 116, 117, 124, 125, 127, 128, 131, 137, 139, 148, 149, 151, 153

Offset: 1

Author

Gus Wiseman, Aug 10 2021

Keywords

• nonn

Comments

This is the sorted version of A342768(n) = position of first appearance of n in A346701 (but A346703 works also).

Examples

			The terms together with their prime indices begin:
     1: {}          31: {11}            71: {20}
     2: {1}         32: {1,1,1,1,1}     73: {21}
     3: {2}         37: {12}            76: {1,1,8}
     5: {3}         41: {13}            79: {22}
     7: {4}         43: {14}            80: {1,1,1,1,3}
     8: {1,1,1}     44: {1,1,5}         83: {23}
    11: {5}         45: {2,2,3}         89: {24}
    12: {1,1,2}     47: {15}            92: {1,1,9}
    13: {6}         48: {1,1,1,1,2}     97: {25}
    17: {7}         52: {1,1,6}         99: {2,2,5}
    19: {8}         53: {16}           101: {26}
    20: {1,1,3}     59: {17}           103: {27}
    23: {9}         61: {18}           107: {28}
    27: {2,2,2}     63: {2,2,4}        108: {1,1,2,2,2}
    28: {1,1,4}     67: {19}           109: {29}
    29: {10}        68: {1,1,7}        112: {1,1,1,1,4}
		

Crossrefs

Removing 1 gives a subset of A026424.

The unsorted even version is A129597.

The unsorted version is A342768(n) = A342767(n,n).

Except the first term, the even version is 2*a(n).

A000290 lists squares.

A001221 counts distinct prime factors.

A001222 counts all prime factors.

A006530 gives the greatest prime factor.

A061395 gives the greatest prime index.

A027193 counts partitions of odd length.

A056239 adds up prime indices, row sums of A112798.

A209281 = odd bisection sum of standard compositions (even: A346633).

A316524 = alternating sum of prime indices (sign: A344617, rev.: A344616).

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A344606 counts alternating permutations of prime indices.

A346697 = odd bisection sum of prime indices (weights of A346703).

A346699 = odd bisection sum of reversed prime indices (weights of A346701).

Cf. A028260, A033942, A035363, A037143, A341446, A344653, A345957, A345958, A345959, A346698, A346700, A346704.

Programs

Maple

filter:= proc(n) issqr(n/max(numtheory:-factorset(n))) end proc:
filter(1):= true:
select(filter, [$1..200]); # Robert Israel, Nov 26 2022

Mathematica

sqrQ[n_]:=IntegerQ[Sqrt[n]];
Select[Range[100],sqrQ[#*FactorInteger[#][[-1,1]]]&]

PARI

isok(m) = (m==1) || issquare(m/vecmax(factor(m)[,1])); \\ Michel Marcus, Aug 12 2021

Formula

a(n) = A129597(n)/2 for n > 1.