cp's OEIS Frontend

A358332 Numbers whose prime indices have arithmetic and geometric mean differing by one.

%I A358332 #22 Mar 01 2025 08:37:36
%S A358332 57,228,1064,1150,1159,2405,3249,7991,29785,29999,30153,35378,51984,
%T A358332 82211,133931,185193,187039,232471,242592,374599,404225,431457,685207,
%U A358332 715129,927288,1132096,1165519,1322500,1343281,1555073,1872413,2016546,2873687,3468319,4266421,4327344
%N A358332 Numbers whose prime indices have arithmetic and geometric mean differing by one.
%C A358332 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.
%H A358332 Max Alekseyev, <a href="/A358332/b358332.txt">Table of n, a(n) for n = 1..1458</a>
%H A358332 Wikipedia, <a href="https://en.wikipedia.org/wiki/Geometric_mean">Geometric mean</a>
%e A358332 The terms together with their prime indices begin:
%e A358332       57: {2,8}
%e A358332      228: {1,1,2,8}
%e A358332     1064: {1,1,1,4,8}
%e A358332     1150: {1,3,3,9}
%e A358332     1159: {8,18}
%e A358332     2405: {3,6,12}
%e A358332     3249: {2,2,8,8}
%e A358332     7991: {18,32}
%e A358332    29785: {3,4,9,12}
%e A358332    29999: {32,50}
%e A358332    30153: {2,8,9,9}
%e A358332    35378: {1,4,4,8,8}
%e A358332    51984: {1,1,1,1,2,2,8,8}
%e A358332    82211: {50,72}
%e A358332   133931: {4,8,8,16}
%e A358332   185193: {2,2,2,8,8,8}
%e A358332   187039: {72,98}
%e A358332   232471: {12,18,27}
%t A358332 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A358332 Select[Range[10000],Mean[primeMS[#]]==1+GeometricMean[primeMS[#]]&]
%o A358332 (PARI) isok(k) = if (k>1, my(f=factor(k), vf=List()); for (i=1, #f~, for (j=1, f[i,2], listput(vf, primepi(f[i,1])))); my(v = Vec(vf)); vecsum(v)/#v == 1 + sqrtn(vecprod(v), #v);); \\ _Michel Marcus_, Nov 11 2022
%Y A358332 The partitions with these Heinz numbers are counted by A358331.
%Y A358332 A000040 lists the primes.
%Y A358332 A001222 counts prime indices, distinct A001221.
%Y A358332 A003963 multiplies together prime indices.
%Y A358332 A056239 adds up prime indices.
%Y A358332 A067538 counts partitions with integer average, ranked by A316413.
%Y A358332 A067539 counts partitions with integer geometric mean, ranked by A326623.
%Y A358332 A078175 lists numbers whose prime factors have integer average.
%Y A358332 A320322 counts partitions whose product is a perfect power.
%Y A358332 Cf. A000720, A051293, A111233, A215366, A289508, A289509, A326027, A326624, A326028, A326645, A357710.
%K A358332 nonn
%O A358332 1,1
%A A358332 _Gus Wiseman_, Nov 09 2022
%E A358332 More terms from _Michel Marcus_, Nov 11 2022