A363131 Number of non-co-modes in the prime factorization of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1

Offset

1,180

Comments

We define a non-co-mode in a multiset to be an element that appears more times than at least one of the others. For example, the non-co-modes in {a,a,b,b,b,c,d,d,d} are {a,b,d}.

Number of prime factors of n whose exponent is larger than the minimal exponent in the prime factorization of n (A051904). - Antti Karttunen, Oct 23 2025

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from exponents in factorization of n.

Example

The prime factorization of 360 is 2*2*2*3*3*5, with non-co-modes {2,3}, so a(360) = 2.

Mathematica

prifacs[n_]:=If[n==1, {}, Flatten[ConstantArray@@@FactorInteger[n]]];
ncomsi[ms_]:=Select[Union[ms], Count[ms, #]>Min@@Length/@Split[ms]&];
Table[Length[ncomsi[prifacs[n]]], {n, 100}]

Prog

(PARI) A363131(n) = if(1==n, 0, my(f=factorint(n), minexp=vecmin(f[, 2])); sum(i=1, #f~, (f[i, 2]>minexp))); \\ Antti Karttunen, Oct 23 2025

Crossrefs

Positions of terms > 0 are A059404.

Positions of first appearances appear to converge to A228593.

For modes instead of non-co-modes we have A362611, triangle A362614.

For co-modes instead of non-co-modes we have A362613, triangle A362615.

For parts instead of multiplicities we have A362983.

For non-modes instead of non-co-modes we have A363127, triangle A363126.

The triangle for this rank statistic (number of non-co-modes) is A363130.

A027746 lists prime factors, A112798 indices, A124010 exponents.

A047966 counts uniform partitions, ranks A072774.

A363128 counts partitions with more than one non-mode, complement A363129.

Cf. A001221, A001222, A002865, A051903, A051904, A056239, A098859, A237984, A327472, A353863, A362616.

Sequence in context: A391436 A294937 A389905 * A355447 A045701 A277156

Adjacent sequences: A363128 A363129 A363130 * A363132 A363133 A363134

Keyword

nonn

Author

Gus Wiseman, May 18 2023

Extensions

More terms and secondary offset added by Antti Karttunen, Oct 23 2025

Status

approved