A352872 - OEIS

%I #8 May 15 2022 11:49:49

%S 2,4,6,8,9,10,12,14,16,18,20,22,24,26,27,28,30,32,34,36,38,40,42,44,

%T 45,46,48,50,52,54,56,58,60,62,63,64,66,68,70,72,74,75,76,78,80,81,82,

%U 84,86,88,90,92,94,96,98,99,100,102,104,106,108,110,112,114

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

%C First differs from A118672 in having 75.

%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}           28: {1,1,4}         56: {1,1,1,4}

%e       4: {1,1}         30: {1,2,3}         58: {1,10}

%e       6: {1,2}         32: {1,1,1,1,1}     60: {1,1,2,3}

%e       8: {1,1,1}       34: {1,7}           62: {1,11}

%e       9: {2,2}         36: {1,1,2,2}       63: {2,2,4}

%e      10: {1,3}         38: {1,8}           64: {1,1,1,1,1,1}

%e      12: {1,1,2}       40: {1,1,1,3}       66: {1,2,5}

%e      14: {1,4}         42: {1,2,4}         68: {1,1,7}

%e      16: {1,1,1,1}     44: {1,1,5}         70: {1,3,4}

%e      18: {1,2,2}       45: {2,2,3}         72: {1,1,1,2,2}

%e      20: {1,1,3}       46: {1,9}           74: {1,12}

%e      22: {1,5}         48: {1,1,1,1,2}     75: {2,3,3}

%e      24: {1,1,1,2}     50: {1,3,3}         76: {1,1,8}

%e      26: {1,6}         52: {1,1,6}         78: {1,2,6}

%e      27: {2,2,2}       54: {1,2,2,2}       80: {1,1,1,1,3}

%e For example, the multiset {2,3,3} with Heinz number 75 has a fixed point at position 3, so 75 is in the sequence.

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

%t Select[Range[100],pq[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]>0&]

%Y * = unproved

%Y These partitions are counted by A238395, strict A096765.

%Y These are the nonzero positions in A352822.

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

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

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

%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 A114088 counts partitions by excedances.

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

%Y A122111 represents partition conjugation using Heinz numbers.

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

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Apr 06 2022