A388719 Numbers with strictly superdiagonal prime signature.

1, 4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 108, 121, 125, 128, 169, 216, 243, 256, 289, 324, 343, 361, 432, 500, 512, 529, 625, 648, 729, 841, 864, 961, 972, 1000, 1024, 1125, 1296, 1331, 1369, 1372, 1681, 1728, 1849, 1944, 2000, 2048, 2187, 2197, 2209, 2401

Offset

1,2

Comments

A sequence (y_1, ..., y_k) is strictly 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

Table of n, a(n) for n=1..51.

Example

The prime signature of 36 is (2,2), which is not strictly superdiagonal, so 36 is not in the sequence.
The prime signature of 72 is (3,2), which is not strictly superdiagonal, so 72 is not in the sequence.
The prime signature of 108 is (2,3), which is strictly superdiagonal, so 108 is in the sequence.
The prime signature of 216 is (3,3), which is strictly superdiagonal, so 216 is in the sequence.
The terms together with their prime indices begin:
     1: {}
     4: {1,1}
     8: {1,1,1}
     9: {2,2}
    16: {1,1,1,1}
    25: {3,3}
    27: {2,2,2}
    32: {1,1,1,1,1}
    49: {4,4}
    64: {1,1,1,1,1,1}
    81: {2,2,2,2}
   108: {1,1,2,2,2}
   121: {5,5}
   125: {3,3,3}
   128: {1,1,1,1,1,1,1}
   169: {6,6}
   216: {1,1,1,2,2,2}
   243: {2,2,2,2,2}
   256: {1,1,1,1,1,1,1,1}

Maple

q:= n-> (l-> is(min(l-[$1..nops(l)])>0))(sort(ifactors(n)[2])[.., 2]):
select(q, [$1..2500])[];  # Alois P. Heinz, Sep 23 2025

Mathematica

suppstrQ[mset_]:=And@@Table[mset[[i]]>i, {i, Length[mset]}];
Select[Range[1000], suppstrQ[Last/@If[#==1, {}, 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, 2500) if ok(k)]) # Michael S. Branicky, Sep 30 2025

Crossrefs

For prime factors instead of exponents we have A352830, non-strict A387112.

The non-strict version is A388715, counted by A388714, reverse A388720.

Partitions of this type are counted by A388716, reverse A388721.

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: A306531 A363722 A088949 * A134600 A134601 A227476

Adjacent sequences: A388716 A388717 A388718 * A388720 A388721 A388722

Keyword

nonn

Author

Gus Wiseman, Sep 23 2025

Status

approved