A339563 Squarefree numbers > 1 whose smallest prime index divides all the other prime indices.

2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 37, 38, 39, 41, 42, 43, 46, 47, 53, 57, 58, 59, 61, 62, 65, 66, 67, 70, 71, 73, 74, 78, 79, 82, 83, 86, 87, 89, 94, 97, 101, 102, 103, 106, 107, 109, 110, 111, 113, 114, 115, 118, 122, 127

Offset

1,1

Comments

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 strict integer partitions whose smallest part divides all the others (counted by A097986). 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..63.

Example

The sequence of terms together with their prime indices begins:
      2: {1}       29: {10}        59: {17}
      3: {2}       30: {1,2,3}     61: {18}
      5: {3}       31: {11}        62: {1,11}
      6: {1,2}     34: {1,7}       65: {3,6}
      7: {4}       37: {12}        66: {1,2,5}
     10: {1,3}     38: {1,8}       67: {19}
     11: {5}       39: {2,6}       70: {1,3,4}
     13: {6}       41: {13}        71: {20}
     14: {1,4}     42: {1,2,4}     73: {21}
     17: {7}       43: {14}        74: {1,12}
     19: {8}       46: {1,9}       78: {1,2,6}
     21: {2,4}     47: {15}        79: {22}
     22: {1,5}     53: {16}        82: {1,13}
     23: {9}       57: {2,8}       83: {23}
     26: {1,6}     58: {1,10}      86: {1,14}

Mathematica

Select[Range[2, 100], SquareFreeQ[#]&&With[{p=PrimePi/@First/@FactorInteger[#]}, And@@IntegerQ/@(p/Min@@p)]&]

Crossrefs

These partitions are counted by A097986 (non-strict: A083710).

The case with no 1's is counted by A098965 (non-strict: A083711).

The squarefree complement is A339562, ranked by A341450.

The complement of the not necessarily squarefree version is A342193.

A000005 counts divisors.

A000070 counts partitions with a selected part.

A001055 counts factorizations.

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.

A338470 counts partitions with no dividing part.

Cf. A130714, A253249, A257993, A299702, A339564, A343340, A343378.

Sequence in context: A327906 A333866 A325362 * A325396 A326533 A144147

Adjacent sequences: A339560 A339561 A339562 * A339564 A339565 A339566

Keyword

nonn

Author

Gus Wiseman, Apr 10 2021

Status

approved