A342193 Numbers with no prime index dividing all the other prime indices.

1, 15, 33, 35, 45, 51, 55, 69, 75, 77, 85, 91, 93, 95, 99, 105, 119, 123, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 195, 201, 203, 205, 207, 209, 215, 217, 219, 221, 225, 231, 245, 247, 249, 253, 255, 265, 275, 279, 285, 287, 291, 295, 297, 299

Offset: 1

Author

Gus Wiseman, Apr 11 2021

Keywords

• nonn

Comments

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

First differs from A339562 in having 45.

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 integer partitions with smallest part not dividing all the others (counted by A338470). 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: {}         105: {2,3,4}      201: {2,19}
     15: {2,3}      119: {4,7}        203: {4,10}
     33: {2,5}      123: {2,13}       205: {3,13}
     35: {3,4}      135: {2,2,2,3}    207: {2,2,9}
     45: {2,2,3}    141: {2,15}       209: {5,8}
     51: {2,7}      143: {5,6}        215: {3,14}
     55: {3,5}      145: {3,10}       217: {4,11}
     69: {2,9}      153: {2,2,7}      219: {2,21}
     75: {2,3,3}    155: {3,11}       221: {6,7}
     77: {4,5}      161: {4,9}        225: {2,2,3,3}
     85: {3,7}      165: {2,3,5}      231: {2,4,5}
     91: {4,6}      175: {3,3,4}      245: {3,4,4}
     93: {2,11}     177: {2,17}       247: {6,8}
     95: {3,8}      187: {5,7}        249: {2,23}
     99: {2,2,5}    195: {2,3,6}      253: {5,9}
		

Crossrefs

The complement is counted by A083710 (strict: A097986).

The complement with no 1's is A083711 (strict: A098965).

These partitions are counted by A338470 (strict: A341450).

The squarefree case is A339562, with squarefree complement A339563.

The case with maximum prime index not divisible by all others is A343338.

The case with maximum prime index divisible by all others is A343339.

A000005 counts divisors.

A000070 counts partitions with a selected part.

A001221 counts distinct prime factors.

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

A056239 adds up prime indices, row sums of A112798.

A299702 lists Heinz numbers of knapsack partitions.

A339564 counts factorizations with a selected factor.

Cf. A066637, A072774, A098743, A253249, A264401, A257993, A342050, A342051, A343344.

Programs

Mathematica

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