A383535 Heinz number of the positive first differences of the 0-prepended prime indices of n.

1, 2, 3, 2, 5, 4, 7, 2, 3, 6, 11, 4, 13, 10, 6, 2, 17, 4, 19, 6, 9, 14, 23, 4, 5, 22, 3, 10, 29, 8, 31, 2, 15, 26, 10, 4, 37, 34, 21, 6, 41, 12, 43, 14, 6, 38, 47, 4, 7, 6, 33, 22, 53, 4, 15, 10, 39, 46, 59, 8, 61, 58, 9, 2, 25, 20, 67, 26, 51, 12, 71, 4, 73

Offset: 1

Author

Gus Wiseman, May 21 2025

Keywords

• nonn

Comments

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Also Heinz number of the first differences of the distinct 0-prepended prime indices of n.

Examples

			The terms together with their prime indices begin:
     1: {}        2: {1}        31: {11}       38: {1,8}
     2: {1}      17: {7}         2: {1}        47: {15}
     3: {2}       4: {1,1}      15: {2,3}       4: {1,1}
     2: {1}      19: {8}        26: {1,6}       7: {4}
     5: {3}       6: {1,2}      10: {1,3}       6: {1,2}
     4: {1,1}     9: {2,2}       4: {1,1}      33: {2,5}
     7: {4}      14: {1,4}      37: {12}       22: {1,5}
     2: {1}      23: {9}        34: {1,7}      53: {16}
     3: {2}       4: {1,1}      21: {2,4}       4: {1,1}
     6: {1,2}     5: {3}         6: {1,2}      15: {2,3}
    11: {5}      22: {1,5}      41: {13}       10: {1,3}
     4: {1,1}     3: {2}        12: {1,1,2}    39: {2,6}
    13: {6}      10: {1,3}      43: {14}       46: {1,9}
    10: {1,3}    29: {10}       14: {1,4}      59: {17}
     6: {1,2}     8: {1,1,1}     6: {1,2}       8: {1,1,1}
		

Crossrefs

For multiplicities instead of differences we have A181819.

Positions of first appearances are A358137.

Positions of squarefree numbers are A383512, counted by A098859.

Positions of nonsquarefree numbers are A383513, counted by A336866.

These are Heinz numbers of rows of A383534.

A000040 lists the primes, differences A001223.

A048767 is the Look-and-Say transform, union A351294, complement A351295.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

A320348 counts strict partitions with distinct 0-appended differences, ranks A325388.

A325324 counts partitions with distinct 0-appended differences, ranks A325367.

Cf. A122111, A124010 (prime signature), A130091, A287352, A325351, A325366, A325368, A355536, A381431, A384008, A384009.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@DeleteCases[Differences[Prepend[prix[n],0]],0],{n,100}]

Formula

A001222(a(n)) = A001221(n).

A056239(a(n)) = A061395(n).

A055396(a(n)) = A055396(n).

A061395(a(n)) = A241919(n).