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

57, 228, 1064, 1150, 1159, 2405, 3249, 7991, 29785, 29999, 30153, 35378, 51984, 82211, 133931, 185193, 187039, 232471, 242592, 374599, 404225, 431457, 685207, 715129, 927288, 1132096, 1165519, 1322500, 1343281, 1555073, 1872413, 2016546, 2873687, 3468319, 4266421, 4327344

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..36.

Wikipedia, Geometric mean

Example

The terms together with their prime indices begin:
      57: {2,8}
     228: {1,1,2,8}
    1064: {1,1,1,4,8}
    1150: {1,3,3,9}
    1159: {8,18}
    2405: {3,6,12}
    3249: {2,2,8,8}
    7991: {18,32}
   29785: {3,4,9,12}
   29999: {32,50}
   30153: {2,8,9,9}
   35378: {1,4,4,8,8}
   51984: {1,1,1,1,2,2,8,8}
   82211: {50,72}
  133931: {4,8,8,16}
  185193: {2,2,2,8,8,8}
  187039: {72,98}
  232471: {12,18,27}

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[10000], Mean[primeMS[#]]==1+GeometricMean[primeMS[#]]&]

Prog

(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

Crossrefs

The partitions with these Heinz numbers are counted by A358331.

A000040 lists the primes.

A001222 counts prime indices, distinct A001221.

A003963 multiplies together prime indices.

A056239 adds up prime indices.

A067538 counts partitions with integer average, ranked by A316413.

A067539 counts partitions with integer geometric mean, ranked by A326623.

A078175 lists numbers whose prime factors have integer average.

A320322 counts partitions whose product is a perfect power.

Cf. A000720, A051293, A111233, A215366, A289508, A289509, A326027, A326624, A326028, A326645, A357710.

Sequence in context: A336191 A277805 A158660 * A158668 A145296 A176635

Adjacent sequences: A358329 A358330 A358331 * A358333 A358334 A358335

Keyword

nonn

Author

Gus Wiseman, Nov 09 2022

Extensions

More terms from Michel Marcus, Nov 11 2022

Status

approved