A338331 Numbers whose set of distinct prime indices (A304038) is pairwise coprime, where a singleton is always considered coprime.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Author

Gus Wiseman, Oct 31 2020

Keywords

• nonn

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 partitions whose set of distinct parts is a singleton or pairwise coprime. 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: {}          16: {1,1,1,1}     32: {1,1,1,1,1}
      2: {1}         17: {7}           33: {2,5}
      3: {2}         18: {1,2,2}       34: {1,7}
      4: {1,1}       19: {8}           35: {3,4}
      5: {3}         20: {1,1,3}       36: {1,1,2,2}
      6: {1,2}       22: {1,5}         37: {12}
      7: {4}         23: {9}           38: {1,8}
      8: {1,1,1}     24: {1,1,1,2}     40: {1,1,1,3}
      9: {2,2}       25: {3,3}         41: {13}
     10: {1,3}       26: {1,6}         43: {14}
     11: {5}         27: {2,2,2}       44: {1,1,5}
     12: {1,1,2}     28: {1,1,4}       45: {2,2,3}
     13: {6}         29: {10}          46: {1,9}
     14: {1,4}       30: {1,2,3}       47: {15}
     15: {2,3}       31: {11}          48: {1,1,1,1,2}
		

Crossrefs

A302798 is the squarefree case.

A304709 counts partitions with pairwise coprime distinct parts, with ordered version A337665 and Heinz numbers A304711.

A304711 does not consider singletons relatively prime, except for (1).

A304712 counts the partitions with these Heinz numbers.

A316476 is the version for indivisibility instead of relative primality.

A328867 is the pairwise non-coprime instead of pairwise coprime version.

A337600 counts triples of this type, with ordered version A337602.

A338330 is the complement.

A000961 lists powers of primes.

A051424 counts pairwise coprime or singleton partitions.

A304038 gives the distinct prime indices of each positive integer.

A327516 counts pairwise coprime partitions.

A333228 ranks compositions whose distinct parts are pairwise coprime.

Cf. A000837, A047968, A056239, A112798, A289509, A302797, A305148, A318716, A318719, A337664, A337695.

Programs

Mathematica

Select[Range[100],#==1||PrimePowerQ[#]||CoprimeQ@@PrimePi/@First/@FactorInteger[#]&]

Formula

Equals A304711 \/ A000961.