A352830 Numbers whose weakly increasing prime indices y have no fixed points y(i) = i.

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141

Offset

1,2

Comments

First differs from A325128 in lacking 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.

All terms are odd.

Links

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

Example

The terms together with their prime indices begin:
      1: {}        35: {3,4}     69: {2,9}     105: {2,3,4}
      3: {2}       37: {12}      71: {20}      107: {28}
      5: {3}       39: {2,6}     73: {21}      109: {29}
      7: {4}       41: {13}      77: {4,5}     111: {2,12}
     11: {5}       43: {14}      79: {22}      113: {30}
     13: {6}       47: {15}      83: {23}      115: {3,9}
     15: {2,3}     49: {4,4}     85: {3,7}     119: {4,7}
     17: {7}       51: {2,7}     87: {2,10}    121: {5,5}
     19: {8}       53: {16}      89: {24}      123: {2,13}
     21: {2,4}     55: {3,5}     91: {4,6}     127: {31}
     23: {9}       57: {2,8}     93: {2,11}    129: {2,14}
     25: {3,3}     59: {17}      95: {3,8}     131: {32}
     29: {10}      61: {18}      97: {25}      133: {4,8}
     31: {11}      65: {3,6}    101: {26}      137: {33}
     33: {2,5}     67: {19}     103: {27}      139: {34}

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 A238394, strict A025147.

These are the zeros of A352822.

*The reverse version is A352826, counted by A064428 (strict A352828).

*The complement reverse version is A352827, counted by A001522.

The complement is 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.

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, A342192, A352486, A352823, A352824, A352825, A352831.

Sequence in context: A343011 A100933 A325128 * A320056 A175679 A088828

Adjacent sequences: A352827 A352828 A352829 * A352831 A352832 A352833

Keyword

nonn

Author

Gus Wiseman, Apr 06 2022

Status

approved