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A257989
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The crank of the partition having Heinz number n.
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10
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-1, 2, -2, 3, 0, 4, -3, 2, 0, 5, -2, 6, 0, 3, -4, 7, 1, 8, -1, 4, 0, 9, -3, 3, 0, 2, -1, 10, 1, 11, -5, 5, 0, 4, -2, 12, 0, 6, -3, 13, 1, 14, -1, 3, 0, 15, -4, 4, 1, 7, -1, 16, 2, 5, -2, 8, 0, 17, -1, 18, 0, 4, -6, 6, 1, 19, -1, 9, 1, 20, -3, 21, 0, 3, -1, 5, 1, 22, -4, 2, 0, 23, -1, 7, 0, 10, -2, 24, 2, 6, -1
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OFFSET
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2,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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)
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MAPLE
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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);
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MATHEMATICA
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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]];
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 *)
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CROSSREFS
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A001522 counts partitions of crank 0.
A003242 counts anti-run compositions.
A064391 counts partitions by crank.
A064428 counts partitions of nonnegative crank.
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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