A356845 Odd numbers with gapless prime indices.

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 35, 37, 41, 43, 45, 47, 49, 53, 59, 61, 67, 71, 73, 75, 77, 79, 81, 83, 89, 97, 101, 103, 105, 107, 109, 113, 121, 125, 127, 131, 135, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 175, 179, 181, 191

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.

A sequence is gapless if it covers an interval of positive integers.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Example

The terms together with their prime indices begin:
    1: {}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   35: {3,4}
   37: {12}
   41: {13}
   43: {14}

Maple

q:= n-> (s-> s={} or nops(s)=max(s)-min(s)+1)(
    map(numtheory[pi], {ifactors(n)[2][.., 1][]})):
select(q, [2*i-1$i=1..96])[];  # Alois P. Heinz, Feb 02 2026

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
nogapQ[m_]:=Or[m=={}, Union[m]==Range[Min[m], Max[m]]];
Select[Range[1, 100, 2], nogapQ[primeMS[#]]&]

Crossrefs

Consists of the odd terms of A073491.

These partitions are counted by A264396.

The strict case is A294674, counted by A136107.

The version for compositions is A356843, counted by A251729.

A001221 counts distinct prime factors, sum A001414.

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

A356069 counts gapless divisors, initial A356224 (complement A356225).

A356230 ranks gapless factorization lengths, firsts A356603.

A356233 counts factorizations into gapless numbers.

Cf. A003963, A034296, A055932, A073493, A107428, A287170, A289508, A325160, A356231, A356234, A356841.

Sequence in context: A193414 A138217 A074775 * A225105 A374909 A143451

Adjacent sequences: A356842 A356843 A356844 * A356846 A356847 A356848

Keyword

nonn

Author

Gus Wiseman, Sep 03 2022

Status

approved