A343337 Numbers with no prime index divisible by all the other prime indices.

1, 15, 30, 33, 35, 45, 51, 55, 60, 66, 69, 70, 75, 77, 85, 90, 91, 93, 95, 99, 102, 105, 110, 119, 120, 123, 132, 135, 138, 140, 141, 143, 145, 150, 153, 154, 155, 161, 165, 170, 175, 177, 180, 182, 186, 187, 190, 198, 201, 203, 204, 205, 207, 209, 210, 215

Offset

1,2

Comments

Alternative name: 1 and numbers whose greatest prime index is not divisible by all the other prime indices.

First differs from A318992 in lacking 195.

First differs from A343343 in lacking 195.

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 (counted by A343341). 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..56.

Example

The sequence of terms together with their prime indices begins:
      1: {}            90: {1,2,2,3}      141: {2,15}
     15: {2,3}         91: {4,6}          143: {5,6}
     30: {1,2,3}       93: {2,11}         145: {3,10}
     33: {2,5}         95: {3,8}          150: {1,2,3,3}
     35: {3,4}         99: {2,2,5}        153: {2,2,7}
     45: {2,2,3}      102: {1,2,7}        154: {1,4,5}
     51: {2,7}        105: {2,3,4}        155: {3,11}
     55: {3,5}        110: {1,3,5}        161: {4,9}
     60: {1,1,2,3}    119: {4,7}          165: {2,3,5}
     66: {1,2,5}      120: {1,1,1,2,3}    170: {1,3,7}
     69: {2,9}        123: {2,13}         175: {3,3,4}
     70: {1,3,4}      132: {1,1,2,5}      177: {2,17}
     75: {2,3,3}      135: {2,2,2,3}      180: {1,1,2,2,3}
     77: {4,5}        138: {1,2,9}        182: {1,4,6}
     85: {3,7}        140: {1,1,3,4}      186: {1,2,11}
For example, 195 has prime indices {2,3,6}, and 6 is divisible by both 2 and 3, so 195 does not belong to the sequence.

Mathematica

Select[Range[1000], #==1||With[{p=PrimePi/@First/@FactorInteger[#]}, !And@@IntegerQ/@(Max@@p/p)]&]

Crossrefs

The complement is counted by A130689.

The dual version is A342193.

The case with smallest prime index not dividing all the others is A343338.

The case with smallest prime index dividing by all the others is A343340.

These are the Heinz numbers of the partitions counted by A343341.

Including the dual version gives A343343.

A000005 counts divisors.

A000070 counts partitions with a selected part.

A006128 counts partitions with a selected position.

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, A257993, A338470, A341450, A343342, A343345, A343346, A343377, A343379, A343381, A343382.

Sequence in context: A115801 A343343 A318992 * A347455 A324970 A033898

Adjacent sequences: A343334 A343335 A343336 * A343338 A343339 A343340

Keyword

nonn

Author

Gus Wiseman, Apr 13 2021

Status

approved