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A346635
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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.
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7
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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
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OFFSET
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1,2
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COMMENTS
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This is the sorted version of A342768(n) = position of first appearance of n in A346701 (but A346703 works also).
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LINKS
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FORMULA
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EXAMPLE
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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}
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MAPLE
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filter:= proc(n) issqr(n/max(numtheory:-factorset(n))) end proc:
filter(1):= true:
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MATHEMATICA
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sqrQ[n_]:=IntegerQ[Sqrt[n]];
Select[Range[100], sqrQ[#*FactorInteger[#][[-1, 1]]]&]
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PROG
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(PARI) isok(m) = (m==1) || issquare(m/vecmax(factor(m)[, 1])); \\ Michel Marcus, Aug 12 2021
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CROSSREFS
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Removing 1 gives a subset of A026424.
The unsorted even version is A129597.
Except the first term, the even version is 2*a(n).
A001221 counts distinct prime factors.
A006530 gives the greatest prime factor.
A061395 gives the greatest prime index.
A027193 counts partitions of odd length.
A344606 counts alternating permutations of prime indices.
A346699 = odd bisection sum of reversed prime indices (weights of A346701).
Cf. A028260, A033942, A035363, A037143, A341446, A344653, A345957, A345958, A345959, A346698, A346700, A346704.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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