reviewed

approved

proposed

reviewed

editing

proposed

(PARI) isok(k) = my(f=factor(k)); issquarefree(f) && (bigomega(f) != 2); \\ Michel Marcus, Oct 07 2023

approved

editing

proposed

approved

editing

proposed

Cf. A000009, A004526, A008967, `A035363, `A078408, A365659, ~A365825, ~A365826.

Complement of A006881 A001358 in A005117.

Cf. A000009 ptns_into_odd, A004526 ptns_len2, A008967 prs_upto_n_sum_k, A035363 ptns_use_ev, A078408 odd_ptns, A365659 strptns_w_binsum_w_reps, A365825 ptns_not_len2_no_half, A365826 strptns_not_len2_no_half.

Cf. A000009, A004526, A008967, `A035363, `A078408, A365659, ~A365825, ~A365826.

The terms together with their prime indices begin:

[meh]The complement appears to first differ from A174891 in lacking 210.

allocated for Gus Wiseman

Squarefree non-semiprimes.

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47, 53, 59, 61, 66, 67, 70, 71, 73, 78, 79, 83, 89, 97, 101, 102, 103, 105, 107, 109, 110, 113, 114, 127, 130, 131, 137, 138, 139, 149, 151, 154, 157, 163, 165, 167, 170, 173, 174, 179, 181, 182, 186

1,2

First differs from A030059 in having 210.

Intersection of A005117 and A100959.

Complement of A006881 in A005117.

The terms together with their prime indices begin:

1: {} 43: {14} 102: {1,2,7}

2: {1} 47: {15} 103: {27}

3: {2} 53: {16} 105: {2,3,4}

5: {3} 59: {17} 107: {28}

7: {4} 61: {18} 109: {29}

11: {5} 66: {1,2,5} 110: {1,3,5}

13: {6} 67: {19} 113: {30}

17: {7} 70: {1,3,4} 114: {1,2,8}

19: {8} 71: {20} 127: {31}

23: {9} 73: {21} 130: {1,3,6}

29: {10} 78: {1,2,6} 131: {32}

30: {1,2,3} 79: {22} 137: {33}

31: {11} 83: {23} 138: {1,2,9}

37: {12} 89: {24} 139: {34}

41: {13} 97: {25} 149: {35}

42: {1,2,4} 101: {26} 151: {36}

Select[Range[100], SquareFreeQ[#]&&PrimeOmega[#]!=2&]

First condition alone is A005117 (squarefree).

Second condition alone is A100959 (non-semiprime).

The nonprime case is 1 followed by A350352.

Partitions of this type are counted by A365827, non-strict A058984.

[meh]The complement appears to first differ from A174891 in lacking 210.

A001358 lists semiprimes, squarefree A006881.

Cf. A000009 ptns_into_odd, A004526 ptns_len2, A008967 prs_upto_n_sum_k, A035363 ptns_use_ev, A078408 odd_ptns, A365659 strptns_w_binsum_w_reps, A365825 ptns_not_len2_no_half, A365826 strptns_not_len2_no_half.

allocated

nonn

Gus Wiseman, Oct 05 2023

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editing