A339562 Squarefree numbers with no prime index dividing all the other prime indices.

1, 15, 33, 35, 51, 55, 69, 77, 85, 91, 93, 95, 105, 119, 123, 141, 143, 145, 155, 161, 165, 177, 187, 195, 201, 203, 205, 209, 215, 217, 219, 221, 231, 247, 249, 253, 255, 265, 285, 287, 291, 295, 299, 301, 309, 323, 327, 329, 335, 341, 345, 355, 357, 377, 381

Offset: 1

Author

Gus Wiseman, Apr 10 2021

Keywords

• nonn

Comments

First differs from A342193 in lacking 45.

Alternative name: 1 and squarefree numbers with smallest prime index not dividing all the other prime indices.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Also 1 and Heinz numbers of strict integer partitions with smallest part not dividing all the others (counted by A341450). The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

Examples

			The sequence of terms together with their prime indices begins:
      1: {}         141: {2,15}     219: {2,21}
     15: {2,3}      143: {5,6}      221: {6,7}
     33: {2,5}      145: {3,10}     231: {2,4,5}
     35: {3,4}      155: {3,11}     247: {6,8}
     51: {2,7}      161: {4,9}      249: {2,23}
     55: {3,5}      165: {2,3,5}    253: {5,9}
     69: {2,9}      177: {2,17}     255: {2,3,7}
     77: {4,5}      187: {5,7}      265: {3,16}
     85: {3,7}      195: {2,3,6}    285: {2,3,8}
     91: {4,6}      201: {2,19}     287: {4,13}
     93: {2,11}     203: {4,10}     291: {2,25}
     95: {3,8}      205: {3,13}     295: {3,17}
    105: {2,3,4}    209: {5,8}      299: {6,9}
    119: {4,7}      215: {3,14}     301: {4,14}
    123: {2,13}     217: {4,11}     309: {2,27}
		

Crossrefs

The squarefree complement is A339563.

These partitions are counted by A341450.

The not necessarily squarefree version is A342193.

A000005 counts divisors.

A000070 counts partitions with a selected part.

A001221 counts distinct prime factors.

A005117 lists squarefree numbers.

A006128 counts partitions with a selected position (strict: A015723).

A056239 adds up prime indices (row sums of A112798).

A083710 counts partitions with a dividing part (strict: A097986).

Cf. A253249, A264401, A257993, A338470, A343337, A343338, A343339, A343379, A343380, A343382.

Programs

Mathematica

Select[Range[100],#==1||SquareFreeQ[#]&&With[{p=PrimePi/@First/@FactorInteger[#]},!And@@IntegerQ/@(p/Min@@p)]&]