A339562 - OEIS

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

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OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..55.

EXAMPLE

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}

MATHEMATICA

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

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.

Sequence in context: A243592 A089966 A327784 * A338468 A337984 A050384

Adjacent sequences: A339559 A339560 A339561 * A339563 A339564 A339565

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 10 2021

STATUS

approved