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)
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 07 2019
STATUS
approved