OFFSET
2,2
COMMENTS
The crank of a partition p is defined to be (i) the largest part of p if there is no 1 in p and (ii) (the number of parts larger than the number of 1's) minus (the number of 1's).
We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.
In the Maple program the subprogram B yields the partition with Heinz number n, the subprogram b yields the number of 1's in the partition with Heinz number n and the subprogram c yields the number of parts that are larger than the number of 1's in the partition with the Heinz number n.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 2..10000
G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167-171.
B. C. Berndt, H. H. Chan, S. H. Chan, W.-C. Liaw, Cranks and dissections in Ramanujan's lost notebook, J. Comb. Theory, Ser. A, 109, 2005, 91-120.
B. C. Berndt, H. H. Chan, S. H. Chan, W.-C. Liaw, Cranks - really the final problem, Ramanujan J., 23, 2010, 3-15.
G. E. Andrews, K. Ono, Ramanujan's congruences and Dyson's crank, Proc. Natl. Acad. Sci. USA, 102, 2005, 15277.
FindStat, Dyson's crank of a partition.
K. Mahlburg, Partition congruences and the Andrews-Garvan-Dyson crank, Proc. Natl. Acad. Sci. USA, 102, 2005, 15373-15376.
Wikipedia, Crank of a partition
EXAMPLE
a(12) = - 2 because the partition with Heinz number 12 = 2*2*3 is [1,1,2], the number of parts larger than the number of 1's is 0 and the number of 1's is 2; 0 - 2 = -2.
a(945) = 4 because the partition with Heinz number 945 = 3^3 * 5 * 7 is [2,2,2,3,4] which has no part 1; the largest part is 4.
From Gus Wiseman, Apr 05 2021: (Start)
The partitions (center) with each Heinz number (left), and the corresponding terms (right):
2: (1) -> -1
3: (2) -> 2
4: (1,1) -> -2
5: (3) -> 3
6: (2,1) -> 0
7: (4) -> 4
8: (1,1,1) -> -3
9: (2,2) -> 2
10: (3,1) -> 0
11: (5) -> 5
12: (2,1,1) -> -2
13: (6) -> 6
14: (4,1) -> 0
15: (3,2) -> 3
16: (1,1,1,1) -> -4
(End)
MAPLE
with(numtheory): a := proc (n) local B, b, c: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do; [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: b := proc (n) if `mod`(n, 2) = 1 then 0 else 1+b((1/2)*n) end if end proc: c := proc (n) local b, B, ct, i: b := proc (n) if `mod`(n, 2) = 1 then 0 else 1+b((1/2)*n) end if end proc: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: ct := 0: for i to bigomega(n) do if b(n) < B(n)[i] then ct := ct+1 else end if end do: ct end proc: if b(n) = 0 then max(B(n)) else c(n)-b(n) end if end proc: seq(a(n), n = 2 .. 150);
MATHEMATICA
B[n_] := Module[{nn, j, m}, nn = FactorInteger[n]; For[j = 1, j <= Length[nn], j++, m[j] = nn[[j]]]; Flatten[Table[Table[PrimePi[m[i][[1]]], {q, 1, m[i][[2]]}], {i, 1, Length[nn]}]]];
b[n_] := b[n] = If[OddQ[n], 0, 1 + b[n/2]];
c[n_] := Module[{ct, i}, ct = 0; For[i = 1, i <= PrimeOmega[n], i++, If[ b[n] < B[n][[i]], ct++]]; ct];
a[n_] := If[b[n] == 0, Max[B[n]], c[n] - b[n]];
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Apr 25 2017, after Emeric Deutsch *)
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
ck[y_]:=With[{w=Count[y, 1]}, If[w==0, Max@@y, Count[y, _?(#>w&)]-w]];
Table[ck[primeMS[n]], {n, 2, 30}] (* Gus Wiseman, Apr 05 2021 *)
CROSSREFS
KEYWORD
sign
AUTHOR
Emeric Deutsch, May 18 2015
STATUS
approved