A352872 Numbers whose weakly increasing prime indices y have a fixed point y(i) = i.

2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 40, 42, 44, 45, 46, 48, 50, 52, 54, 56, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 94, 96, 98, 99, 100, 102, 104, 106, 108, 110, 112, 114

Offset

1,1

Comments

First differs from A118672 in having 75.

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..64.

Example

The terms together with their prime indices begin:
      2: {1}           28: {1,1,4}         56: {1,1,1,4}
      4: {1,1}         30: {1,2,3}         58: {1,10}
      6: {1,2}         32: {1,1,1,1,1}     60: {1,1,2,3}
      8: {1,1,1}       34: {1,7}           62: {1,11}
      9: {2,2}         36: {1,1,2,2}       63: {2,2,4}
     10: {1,3}         38: {1,8}           64: {1,1,1,1,1,1}
     12: {1,1,2}       40: {1,1,1,3}       66: {1,2,5}
     14: {1,4}         42: {1,2,4}         68: {1,1,7}
     16: {1,1,1,1}     44: {1,1,5}         70: {1,3,4}
     18: {1,2,2}       45: {2,2,3}         72: {1,1,1,2,2}
     20: {1,1,3}       46: {1,9}           74: {1,12}
     22: {1,5}         48: {1,1,1,1,2}     75: {2,3,3}
     24: {1,1,1,2}     50: {1,3,3}         76: {1,1,8}
     26: {1,6}         52: {1,1,6}         78: {1,2,6}
     27: {2,2,2}       54: {1,2,2,2}       80: {1,1,1,1,3}
For example, the multiset {2,3,3} with Heinz number 75 has a fixed point at position 3, so 75 is in the sequence.

Mathematica

pq[y_]:=Length[Select[Range[Length[y]], #==y[[#]]&]];
Select[Range[100], pq[Flatten[Cases[FactorInteger[#], {p_, k_}:>Table[PrimePi[p], {k}]]]]>0&]

Crossrefs

* = unproved

These partitions are counted by A238395, strict A096765.

These are the nonzero positions in A352822.

*The complement reverse version is A352826, counted by A064428.

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

The complement is A352830, counted by A238394 (strict A025147).

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.

A114088 counts partitions by excedances.

A115720 and A115994 count partitions by their Durfee square.

A122111 represents partition conjugation using Heinz numbers.

A124010 gives prime signature, sorted A118914, conjugate rank A238745.

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, A093641, A257990, A325187, A342192, A352486, A352823, A352824, A352825, A352831, A352832.

Sequence in context: A182339 A110262 A176693 * A118672 A100417 A333337

Adjacent sequences: A352869 A352870 A352871 * A352873 A352874 A352875

Keyword

nonn

Author

Gus Wiseman, Apr 06 2022

Status

approved