A328166 Heinz number of the run-lengths of the divisors of n.

2, 3, 4, 6, 4, 10, 4, 12, 8, 12, 4, 28, 4, 12, 16, 24, 4, 40, 4, 36, 16, 12, 4, 112, 8, 12, 16, 48, 4, 120, 4, 48, 16, 12, 16, 224, 4, 12, 16, 144, 4, 120, 4, 48, 64, 12, 4, 448, 8, 48, 16, 48, 4, 160, 16, 144, 16, 12, 4, 832, 4, 12, 64, 96, 16, 160, 4, 48, 16

Offset

1,1

Comments

The Heinz number of an integer partition or multiset {y_1,...,y_k} is prime(y_1)*...*prime(y_k).

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Wikipedia, Run (cards)

Index entries for sequences related to Heinz numbers

Formula

A001222(a(n)) = A137921(n).

A056239(a(n)) = A000005(n).

Example

Splitting the divisors of 30 into runs gives {{1, 2, 3}, {5, 6}, {10}, {15}, {30}}, and the Heinz number of {1, 1, 1, 2, 3} is 120, so a(30) = 120.
More examples from Antti Karttunen, Dec 09 2021: (Start)
Splitting the divisors of 1 into runs gives {1}, and the Heinz number of that is 2.
Splitting the divisors of 2 into runs gives {1, 2}, and the Heinz number of that is 3. [one run of length 2, therefore a(2) = prime(2)^1].
Splitting the divisors of 3 into runs gives {1} and {3}, and the Heinz number of that is 4. [two runs of length 1, therefore a(3) = prime(1)^2].
Splitting the divisors of 4 into runs gives {1, 2} and {4}, and the Heinz number of that is 6. [one run of length 1, and other run of length 2, therefore a(4) = prime(1)*prime(2)].
Splitting the divisors of 5 into runs gives {1} and {5}, and the Heinz number of that is 4. [two runs of length 1, therefore a(5) = prime(1)^2].
(End)

Mathematica

Table[Times@@Prime/@Length/@Split[Divisors[n], #2==#1+1&], {n, 30}]

Prog

(PARI) A328166(n) = { my(rl=0, pd=0, v=vector(numdiv(n)), m=1); fordiv(n, d, if(d>(1+pd), v[rl]++; rl=0); pd=d; rl++); v[rl]++; for(i=1, #v, m *= prime(i)^v[i]); (m); }; \\ Antti Karttunen, Dec 09 2021

Crossrefs

The longest run of divisors of n has length A055874(n).

Numbers whose divisors > 1 have no non-singleton runs are A088725.

The number of successive pairs of divisors of n is A129308(n).

The Heinz number of the set of divisors of n is A275700(n).

Numbers whose divisors do not have weakly decreasing run-lengths are A328165.

Cf. A000005, A056239, A060680, A060775, A119313, A137921, A181063, A199970.

Sequence in context: A366577 A102284 A273098 * A266447 A100700 A361478

Adjacent sequences: A328163 A328164 A328165 * A328167 A328168 A328169

Keyword

nonn

Author

Gus Wiseman, Oct 07 2019

Status

approved