A338316 Odd numbers whose distinct prime indices are pairwise coprime, where a singleton is always considered coprime.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 55, 59, 61, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 89, 93, 95, 97, 99, 101, 103, 107, 109, 113, 119, 121, 123, 125, 127, 131, 135, 137, 139, 141, 143, 145, 149, 151

Offset: 1

Author

Gus Wiseman, Oct 24 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.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions. a(n) gives the n-th Heinz number of an integer partition with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime (A338317).

Examples

			The sequence of terms together with their prime indices begins:
      1: {}          33: {2,5}       71: {20}
      3: {2}         35: {3,4}       73: {21}
      5: {3}         37: {12}        75: {2,3,3}
      7: {4}         41: {13}        77: {4,5}
      9: {2,2}       43: {14}        79: {22}
     11: {5}         45: {2,2,3}     81: {2,2,2,2}
     13: {6}         47: {15}        83: {23}
     15: {2,3}       49: {4,4}       85: {3,7}
     17: {7}         51: {2,7}       89: {24}
     19: {8}         53: {16}        93: {2,11}
     23: {9}         55: {3,5}       95: {3,8}
     25: {3,3}       59: {17}        97: {25}
     27: {2,2,2}     61: {18}        99: {2,2,5}
     29: {10}        67: {19}       101: {26}
     31: {11}        69: {2,9}      103: {27}
		

Crossrefs

A338315 does not consider singletons coprime, with Heinz numbers A337987.

A338317 counts the partitions with these Heinz numbers.

A337694 is a pairwise non-coprime instead of pairwise coprime version.

A007359 counts singleton or pairwise coprime partitions with no 1's, with Heinz numbers A302568.

A101268 counts pairwise coprime or singleton compositions, ranked by A335235.

A302797 lists squarefree numbers whose distinct parts are pairwise coprime.

A304709 counts partitions whose distinct parts are pairwise coprime, with Heinz numbers A304711.

A327516 counts pairwise coprime partitions, ranked by A302696.

A337485 counts pairwise coprime partitions with no 1's, with Heinz numbers A337984.

A337561 counts pairwise coprime strict compositions.

A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.

A337697 counts pairwise coprime compositions with no 1's.

Cf. A051424, A056239, A112798, A220377, A289509, A302569, A303282, A318719, A328673, A328867.

Programs

Mathematica

Select[Range[1,100,2],#==1||PrimePowerQ[#]||CoprimeQ@@Union[PrimePi/@First/@FactorInteger[#]]&]