A379307 Positive integers whose prime indices include no squarefree numbers.

1, 7, 19, 23, 37, 49, 53, 61, 71, 89, 97, 103, 107, 131, 133, 151, 161, 173, 193, 197, 223, 227, 229, 239, 251, 259, 263, 281, 307, 311, 337, 343, 359, 361, 371, 379, 383, 409, 419, 427, 433, 437, 457, 463, 479, 497, 503, 521, 523, 529, 541, 569, 593, 613, 623

Offset

1,2

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..55.

Example

The terms together with their prime indices begin:
    1: {}
    7: {4}
   19: {8}
   23: {9}
   37: {12}
   49: {4,4}
   53: {16}
   61: {18}
   71: {20}
   89: {24}
   97: {25}
  103: {27}
  107: {28}
  131: {32}
  133: {4,8}
  151: {36}
  161: {4,9}
  173: {40}

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], Length[Select[prix[#], SquareFreeQ]]==0&]

Crossrefs

Partitions of this type are counted by A114374, strict A256012.

Positions of zero in A379306.

For a unique squarefree part we have A379316, counted by A379308 (strict A379309).

A000040 lists the primes, differences A001223.

A005117 lists the squarefree numbers, differences A076259.

A008966 is the characteristic function for the squarefree numbers.

A013929 lists the nonsquarefree numbers, differences A078147.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

A061398 counts squarefree numbers between primes, zeros A068360.

A377038 gives k-th differences of squarefree numbers.

Other counts of prime indices:

- A257994 prime, see A002095, A096258, A320628, A331386, A331915, A379304, A379305.

- A257991 odd, see A000041, A000070, A066207, A349158.

- A257992 even, see A000009, A038348, A066208, A379317.

- A330944 nonprime, see A000586, A000607, A076610, A330945.

- A379300 composite, see A023895, A034891, A036497, A302540, A379301.

- A379310 nonsquarefree, see A302478.

- A379311 old prime, see A204389, A320629, A379312-A379315.

Cf. A000720, A013928, A038550, A057627, A068361, A070321, A071403, A072284, A087436, A112929, A377430, A378086.

Sequence in context: A353046 A032680 A141831 * A176182 A292349 A065749

Adjacent sequences: A379304 A379305 A379306 * A379308 A379309 A379310

Keyword

nonn

Author

Gus Wiseman, Dec 27 2024

Status

approved