A388715 Numbers with superdiagonal prime signature.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 36, 37, 41, 43, 47, 49, 50, 53, 54, 59, 61, 64, 67, 71, 72, 73, 75, 79, 81, 83, 89, 97, 98, 100, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 144, 147, 149, 151, 157

Offset

1,2

Comments

First differs from the complement of A080363 in lacking 150, 294, 300, ...

A sequence (y_1, ..., y_k) is superdiagonal iff y_i >= i for all i = 1..k.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Links

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

Example

The prime signature of 300 is (2,1,2), which are not superdiagonal, so 300 is not in the sequence.
The prime signature of 750 is (1,1,3), which are not superdiagonal, so 750 is not in the sequence.
The prime signature of 4500 is (2,2,3), which are superdiagonal, so 4500 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   18: {1,2,2}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   37: {12}

Maple

q:= n-> (l-> is(min(l-[$1..nops(l)])>=0))(sort(ifactors(n)[2])[.., 2]):
select(q, [$1..200])[];  # Alois P. Heinz, Sep 23 2025

Mathematica

suppQ[mset_]:=And@@Table[mset[[i]]>=i, {i, Length[mset]}];
Select[Range[100], suppQ[Last/@FactorInteger[#]]&]

Prog

(PARI) isok(k) = my(f=factor(k)[, 2]); for (i=1, #f~, if (f[i] < i, return(0))); return(1); \\ Michel Marcus, Sep 24 2025
(Python)
from sympy import factorint
def ok(n): return all(e >= i for i, (p, e) in enumerate(factorint(n).items(), 1))
print([k for k in range(1, 160) if ok(k)]) # Michael S. Branicky, Sep 30 2025

Crossrefs

The squarefree case is A008578.

For prime factors instead of exponents we appear to have A387112.

Partitions of this type are counted by A388714, reverse A388720.

The strictly superdiagonal case is A388719, counted by A388716.

A001222 counts prime indices, distinct A001221.

A003106 counts strictly superdiagonal partitions, strict A237979, ranks A352830.

A003114 counts superdiagonal partitions, reverse A238873, complement A387118.

A124010 gives prime signature, sorted A118914, conjugate rank A238745.

Cf. A025157, A052335, A052337, A088902, A352828, A355529, A387576, A387578, A388710, A388711, A388724.

Sequence in context: A046684 A374666 A082377 * A392696 A166401 A329131

Adjacent sequences: A388712 A388713 A388714 * A388716 A388717 A388718

Keyword

nonn

Author

Gus Wiseman, Sep 22 2025

Status

approved