A331914 - OEIS

%I #8 Jun 10 2020 14:06:26

%S 1,2,3,4,5,6,7,8,10,11,12,13,14,16,17,19,20,21,22,23,24,26,28,29,31,

%T 32,34,35,37,38,39,40,41,42,43,44,46,47,48,49,52,53,56,57,58,59,61,62,

%U 64,65,67,68,69,70,71,73,74,76,77,78,79,80,82,83,84,86,87

%N Numbers with at most one prime prime index, counted with multiplicity.

%C First differs from A324935 in having 39.

%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 sequence of terms together with their prime indices begins:

%e    1: {}           24: {1,1,1,2}      52: {1,1,6}

%e    2: {1}          26: {1,6}          53: {16}

%e    3: {2}          28: {1,1,4}        56: {1,1,1,4}

%e    4: {1,1}        29: {10}           57: {2,8}

%e    5: {3}          31: {11}           58: {1,10}

%e    6: {1,2}        32: {1,1,1,1,1}    59: {17}

%e    7: {4}          34: {1,7}          61: {18}

%e    8: {1,1,1}      35: {3,4}          62: {1,11}

%e   10: {1,3}        37: {12}           64: {1,1,1,1,1,1}

%e   11: {5}          38: {1,8}          65: {3,6}

%e   12: {1,1,2}      39: {2,6}          67: {19}

%e   13: {6}          40: {1,1,1,3}      68: {1,1,7}

%e   14: {1,4}        41: {13}           69: {2,9}

%e   16: {1,1,1,1}    42: {1,2,4}        70: {1,3,4}

%e   17: {7}          43: {14}           71: {20}

%e   19: {8}          44: {1,1,5}        73: {21}

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

%e   21: {2,4}        47: {15}           76: {1,1,8}

%e   22: {1,5}        48: {1,1,1,1,2}    77: {4,5}

%e   23: {9}          49: {4,4}          78: {1,2,6}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],Count[primeMS[#],_?PrimeQ]<=1&]

%Y These are numbers n such that A257994(n) <= 1.

%Y Prime-indexed primes are A006450, with products A076610.

%Y The number of distinct prime prime indices is A279952.

%Y Numbers with at least one prime prime index are A331386.

%Y The set S of numbers with at most one prime index in S are A331784.

%Y The set S of numbers with at most one distinct prime index in S are A331912.

%Y Numbers with exactly one prime prime index are A331915.

%Y Numbers with exactly one distinct prime prime index are A331916.

%Y Numbers with at most one distinct prime prime index are A331995.

%Y Cf. A000040, A000720, A007097, A018252, A112798, A320628, A330945, A331785.

%K nonn

%O 1,2

%A _Gus Wiseman_, Feb 08 2020