cp's OEIS Frontend

A324525 Numbers divisible by prime(k)^k for each prime index k.

%I A324525 #20 Nov 23 2020 02:05:48
%S A324525 1,2,4,8,9,16,18,27,32,36,54,64,72,81,108,125,128,144,162,216,243,250,
%T A324525 256,288,324,432,486,500,512,576,625,648,729,864,972,1000,1024,1125,
%U A324525 1152,1250,1296,1458,1728,1944,2000,2048,2187,2250,2304,2401,2500,2592
%N A324525 Numbers divisible by prime(k)^k for each prime index k.
%C A324525 These are a kind of self-describing numbers (cf. A001462, A304679).
%C A324525 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. The prime signature of a number is the multiset of multiplicities (or exponents) in its prime factorization.
%C A324525 Also Heinz numbers of integer partitions where the multiplicity of k is at least k (A117144). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%H A324525 Amiram Eldar, <a href="/A324525/b324525.txt">Table of n, a(n) for n = 1..10000</a>
%F A324525 Closed under multiplication.
%F A324525 Sum_{n>=1} 1/a(n) = Product_{k>=1} 1 + 1/(prime(k)^(k-1) * (prime(k)-1)) = 2.35782843100111139159... - _Amiram Eldar_, Nov 23 2020
%e A324525 The sequence of terms together with their prime indices begins as follows. For example, 36 = prime(1) * prime(1) * prime(2) * prime(2) is a term because the prime multiplicities are {2,2}, which are greater than or equal to the prime indices {1,2}.
%e A324525     1: {}
%e A324525     2: {1}
%e A324525     4: {1,1}
%e A324525     8: {1,1,1}
%e A324525     9: {2,2}
%e A324525    16: {1,1,1,1}
%e A324525    18: {1,2,2}
%e A324525    27: {2,2,2}
%e A324525    32: {1,1,1,1,1}
%e A324525    36: {1,1,2,2}
%e A324525    54: {1,2,2,2}
%e A324525    64: {1,1,1,1,1,1}
%e A324525    72: {1,1,1,2,2}
%e A324525    81: {2,2,2,2}
%e A324525   108: {1,1,2,2,2}
%e A324525   125: {3,3,3}
%e A324525   128: {1,1,1,1,1,1,1}
%p A324525 q:= n-> andmap(i-> i[2]>=numtheory[pi](i[1]), ifactors(n)[2]):
%p A324525 select(q, [$1..3000])[];  # _Alois P. Heinz_, Mar 08 2019
%t A324525 Select[Range[1000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k>=PrimePi[p]]&]
%t A324525 seq[max_] := Module[{ps = {2}, p, s = {1}, s1, s2, emax}, While[ps[[-1]]^Length[ps] < max, AppendTo[ps, NextPrime[ps[[-1]]]]]; Do[p = ps[[k]]; emax = Floor[Log[p, max]]; s1 = Join[{1}, p^Range[k, emax]]; s2 = Select[Union[Flatten[Outer[Times, s, s1]]], # <= max &]; s = Union[s, s2], {k, 1, Length[ps]}]; s]; seq[3000] (* _Amiram Eldar_, Nov 23 2020 *)
%Y A324525 Cf. A001156, A033461, A056239, A062457, A112798, A117144, A118914 (prime signature), A124010, A181819, A276078, A304679.
%Y A324525 Cf. A109298, A324524, A324571, A324572, A324587, A324588.
%Y A324525 Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.
%K A324525 nonn
%O A324525 1,2
%A A324525 _Gus Wiseman_, Mar 08 2019