A345193 Heinz numbers of non-twin (x,x) inseparable partitions.

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 80, 81, 88, 96, 104, 112, 125, 128, 135, 136, 144, 152, 160, 162, 176, 184, 189, 192, 208, 224, 232, 240, 243, 248, 250, 256, 272, 288, 296, 297, 304, 320, 324, 328, 336, 343, 344, 351, 352, 368, 375, 376, 384, 400, 405

Offset

1,1

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.

A multiset is separable if it has an anti-run permutation (no adjacent parts equal). This is equivalent to having maximal multiplicity greater than one plus the sum of the remaining multiplicities. For example, the partition (3,2,2,2,1) has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2), so is separable.

Links

Table of n, a(n) for n=1..55.

Formula

Complement of A001248 in A335448.

Example

The sequence of terms together with their prime indices begins:
      8: {1,1,1}          112: {1,1,1,1,4}        232: {1,1,1,10}
     16: {1,1,1,1}        125: {3,3,3}            240: {1,1,1,1,2,3}
     24: {1,1,1,2}        128: {1,1,1,1,1,1,1}    243: {2,2,2,2,2}
     27: {2,2,2}          135: {2,2,2,3}          248: {1,1,1,11}
     32: {1,1,1,1,1}      136: {1,1,1,7}          250: {1,3,3,3}
     40: {1,1,1,3}        144: {1,1,1,1,2,2}      256: {1,1,1,1,1,1,1,1}
     48: {1,1,1,1,2}      152: {1,1,1,8}          272: {1,1,1,1,7}
     54: {1,2,2,2}        160: {1,1,1,1,1,3}      288: {1,1,1,1,1,2,2}
     56: {1,1,1,4}        162: {1,2,2,2,2}        296: {1,1,1,12}
     64: {1,1,1,1,1,1}    176: {1,1,1,1,5}        297: {2,2,2,5}
     80: {1,1,1,1,3}      184: {1,1,1,9}          304: {1,1,1,1,8}
     81: {2,2,2,2}        189: {2,2,2,4}          320: {1,1,1,1,1,1,3}
     88: {1,1,1,5}        192: {1,1,1,1,1,1,2}    324: {1,1,2,2,2,2}
     96: {1,1,1,1,1,2}    208: {1,1,1,1,6}        328: {1,1,1,13}
    104: {1,1,1,6}        224: {1,1,1,1,1,4}      336: {1,1,1,1,2,4}

Mathematica

Select[Range[100], !MatchQ[FactorInteger[#], {{_, 2}}]&&Select[Permutations[Flatten[ConstantArray@@@FactorInteger[#]]], !MatchQ[#, {___, x_, x_, ___}]&]=={}&]

Crossrefs

A000041 counts integer partitions.

A001248 lists Heinz numbers of twins (x,x).

A001250 counts wiggly permutations.

A003242 counts anti-run compositions.

A025047 counts wiggly compositions (ascend: A025048, descend: A025049).

A056239 adds up prime indices, row sums of A112798.

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A344740 counts twins and partitions w/ wiggly permutation, rank: A344742.

A345164 counts wiggly permutations of prime indices (with twins: A344606).

A345165 counts partitions without a wiggly permutation, ranked by A345171.

A345170 counts partitions with a wiggly permutation, ranked by A345172.

A345192 counts non-wiggly compositions.

Cf. A001222, A071321, A316523, A316524, A335126, A344614, A344616, A344652, A344654, A345169, A345173.

Sequence in context: A375145 A381316 A344653 * A365866 A033859 A177899

Adjacent sequences: A345190 A345191 A345192 * A345194 A345195 A345196

Keyword

nonn

Author

Gus Wiseman, Jun 17 2021

Status

approved