A358137 Heinz number of the partial sums of the prime indices of n.

1, 2, 3, 6, 5, 10, 7, 30, 21, 14, 11, 42, 13, 22, 33, 210, 17, 110, 19, 66, 39, 26, 23, 330, 65, 34, 273, 78, 29, 130, 31, 2310, 51, 38, 85, 546, 37, 46, 57, 390, 41, 170, 43, 102, 357, 58, 47, 2730, 133, 238, 69, 114, 53, 1870, 95, 510, 87, 62, 59, 714, 61

Offset

1,2

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.

Links

Table of n, a(n) for n=1..61.

Formula

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

Example

The terms together with their prime indices begin:
      1: {}
      2: {1}
      3: {2}
      6: {1,2}
      5: {3}
     10: {1,3}
      7: {4}
     30: {1,2,3}
     21: {2,4}
     14: {1,4}
     11: {5}
     42: {1,2,4}
     13: {6}
     22: {1,5}
     33: {2,5}
    210: {1,2,3,4}
     17: {7}
    110: {1,3,5}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Times@@Prime/@Accumulate[primeMS[n]], {n, 100}]

Crossrefs

The sorted version is A325362.

The prime indices are rows of A358136, partial sums of rows of A112798.

A000040 lists the primes.

A000041 counts partitions, strict A000009.

A003963 multiplies prime indices.

A056239 adds up prime indices.

Cf. A000720, A001221, A001222, A215366, A296150, A318283, A355536, A358134.

Sequence in context: A064924 A196330 A332462 * A256662 A055944 A331633

Adjacent sequences: A358134 A358135 A358136 * A358138 A358139 A358140

Keyword

nonn

Author

Gus Wiseman, Oct 31 2022

Status

approved