A352831 Numbers whose weakly increasing prime indices y have exactly one fixed point y(i) = i.

2, 4, 8, 9, 10, 12, 14, 16, 22, 24, 26, 27, 28, 32, 34, 36, 38, 40, 44, 46, 48, 52, 58, 60, 62, 63, 64, 68, 70, 72, 74, 75, 76, 80, 81, 82, 86, 88, 92, 94, 96, 98, 99, 104, 106, 108, 110, 112, 116, 117, 118, 120, 122, 124, 125, 128, 130, 132, 134, 135, 136

Offset

1,1

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.

Links

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

Example

The terms together with their prime indices begin:
      2: {1}             36: {1,1,2,2}         74: {1,12}
      4: {1,1}           38: {1,8}             75: {2,3,3}
      8: {1,1,1}         40: {1,1,1,3}         76: {1,1,8}
      9: {2,2}           44: {1,1,5}           80: {1,1,1,1,3}
     10: {1,3}           46: {1,9}             81: {2,2,2,2}
     12: {1,1,2}         48: {1,1,1,1,2}       82: {1,13}
     14: {1,4}           52: {1,1,6}           86: {1,14}
     16: {1,1,1,1}       58: {1,10}            88: {1,1,1,5}
     22: {1,5}           60: {1,1,2,3}         92: {1,1,9}
     24: {1,1,1,2}       62: {1,11}            94: {1,15}
     26: {1,6}           63: {2,2,4}           96: {1,1,1,1,1,2}
     27: {2,2,2}         64: {1,1,1,1,1,1}     98: {1,4,4}
     28: {1,1,4}         68: {1,1,7}           99: {2,2,5}
     32: {1,1,1,1,1}     70: {1,3,4}          104: {1,1,1,6}
     34: {1,7}           72: {1,1,1,2,2}      106: {1,16}
For example, 63 is in the sequence because its prime indices {2,2,4} have a unique fixed point at the second position.

Mathematica

pq[y_]:=Length[Select[Range[Length[y]], #==y[[#]]&]];
Select[Range[100], pq[Flatten[Cases[FactorInteger[#], {p_, k_}:>Table[PrimePi[p], {k}]]]]==1&]

Crossrefs

* = unproved

These are the positions of 1's in A352822.

*The reverse version for no fixed points is A352826, counted by A064428.

*The reverse version is A352827, counted by A001522 (strict A352829).

The version for no fixed points is A352830, counted by A238394.

These partitions are counted by A352832, compositions A240736.

Allowing more than one fixed point gives A352872, counted by A238395.

A000700 counts self-conjugate partitions, ranked by A088902.

A001222 counts prime indices, distinct A001221.

A008290 counts permutations by fixed points, nonfixed A098825.

A056239 adds up prime indices, row sums of A112798 and A296150.

A115720 and A115994 count partitions by their Durfee square.

A238349 counts compositions by fixed points, complement A352523.

A238352 counts reversed partitions by fixed points.

A352833 counts partitions by fixed points.

Cf. A062457, A064410, A065770, A177510, A257990, A342192, A349158, A351983, A352520, A352823, A352824.

Sequence in context: A131625 A196000 A044952 * A047466 A003485 A072602

Adjacent sequences: A352828 A352829 A352830 * A352832 A352833 A352834

Keyword

nonn

Author

Gus Wiseman, Apr 08 2022

Status

approved