A343338 Numbers with no prime index dividing or divisible by 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, 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, 301

Offset: 1

Author

Gus Wiseman, Apr 13 2021

Keywords

• nonn

Comments

Alternative name: 1 and numbers whose smallest prime index does not divide all the other prime indices, nor whose greatest prime index is divisible by all the other prime indices.

First differs from A302697 in having 91.

First differs from A337987 in having 91.

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 Heinz numbers of partitions with greatest part not divisible by all the others and smallest part not dividing all the others (counted by A343342). 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}      203: {4,10}
     15: {2,3}      119: {4,7}        205: {3,13}
     33: {2,5}      123: {2,13}       207: {2,2,9}
     35: {3,4}      135: {2,2,2,3}    209: {5,8}
     45: {2,2,3}    141: {2,15}       215: {3,14}
     51: {2,7}      143: {5,6}        217: {4,11}
     55: {3,5}      145: {3,10}       219: {2,21}
     69: {2,9}      153: {2,2,7}      221: {6,7}
     75: {2,3,3}    155: {3,11}       225: {2,2,3,3}
     77: {4,5}      161: {4,9}        231: {2,4,5}
     85: {3,7}      165: {2,3,5}      245: {3,4,4}
     91: {4,6}      175: {3,3,4}      247: {6,8}
     93: {2,11}     177: {2,17}       249: {2,23}
     95: {3,8}      187: {5,7}        253: {5,9}
     99: {2,2,5}    201: {2,19}       255: {2,3,7}
For example, the prime indices of 975 are {2,3,3,6}, all of which divide 6, but not all of which are multiples of 2, so 975 is not in the sequence.
		

Crossrefs

The first condition alone gives A342193.

The second condition alone gives A343337.

The half-opposite versions are A343339 and A343340.

The partitions with these Heinz numbers are counted by A343342.

The opposite version is the complement of A343343.

A000005 counts divisors.

A000070 counts partitions with a selected part.

A001055 counts factorizations.

A056239 adds up prime indices, row sums of A112798.

A067824 counts strict chains of divisors starting with n.

A253249 counts strict chains of divisors.

A339564 counts factorizations with a selected factor.

Cf. A083710, A130689, A338470, A339562, A341450, A343341, A343346, A343347, A343348, A343377, A343379, A343382.

Programs

Mathematica

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

Formula

Intersection of A342193 and A343337.