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A328166
Heinz number of the run-lengths of the divisors of n.
21
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).
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.
Sequence in context: A366577 A102284 A273098 * A266447 A100700 A361478
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 07 2019
STATUS
approved