A352831 - OEIS

%I #7 May 15 2022 11:48:35

%S 2,4,8,9,10,12,14,16,22,24,26,27,28,32,34,36,38,40,44,46,48,52,58,60,

%T 62,63,64,68,70,72,74,75,76,80,81,82,86,88,92,94,96,98,99,104,106,108,

%U 110,112,116,117,118,120,122,124,125,128,130,132,134,135,136

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

%C 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.

%e The terms together with their prime indices begin:

%e       2: {1}             36: {1,1,2,2}         74: {1,12}

%e       4: {1,1}           38: {1,8}             75: {2,3,3}

%e       8: {1,1,1}         40: {1,1,1,3}         76: {1,1,8}

%e       9: {2,2}           44: {1,1,5}           80: {1,1,1,1,3}

%e      10: {1,3}           46: {1,9}             81: {2,2,2,2}

%e      12: {1,1,2}         48: {1,1,1,1,2}       82: {1,13}

%e      14: {1,4}           52: {1,1,6}           86: {1,14}

%e      16: {1,1,1,1}       58: {1,10}            88: {1,1,1,5}

%e      22: {1,5}           60: {1,1,2,3}         92: {1,1,9}

%e      24: {1,1,1,2}       62: {1,11}            94: {1,15}

%e      26: {1,6}           63: {2,2,4}           96: {1,1,1,1,1,2}

%e      27: {2,2,2}         64: {1,1,1,1,1,1}     98: {1,4,4}

%e      28: {1,1,4}         68: {1,1,7}           99: {2,2,5}

%e      32: {1,1,1,1,1}     70: {1,3,4}          104: {1,1,1,6}

%e      34: {1,7}           72: {1,1,1,2,2}      106: {1,16}

%e For example, 63 is in the sequence because its prime indices {2,2,4} have a unique fixed point at the second position.

%t pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];

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

%Y * = unproved

%Y These are the positions of 1's in A352822.

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

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

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

%Y These partitions are counted by A352832, compositions A240736.

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

%Y A000700 counts self-conjugate partitions, ranked by A088902.

%Y A001222 counts prime indices, distinct A001221.

%Y A008290 counts permutations by fixed points, nonfixed A098825.

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

%Y A115720 and A115994 count partitions by their Durfee square.

%Y A238349 counts compositions by fixed points, complement A352523.

%Y A238352 counts reversed partitions by fixed points.

%Y A352833 counts partitions by fixed points.

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Apr 08 2022