%I #13 May 06 2022 20:32:14

%S 0,1,1,0,1,-1,1,1,0,-1,1,1,1,-1,-1,0,1,1,1,1,-1,-1,1,-1,0,-1,1,1,1,1,

%T 1,1,-1,-1,-1,0,1,-1,-1,-1,1,1,1,1,1,-1,1,1,0,1,-1,1,1,-1,-1,-1,-1,-1,

%U 1,-1,1,-1,1,0,-1,1,1,1,-1,1,1,1,1,-1,1,1,-1,1,1,1,0,-1,1,-1,-1,-1,-1,-1,1,-1,-1,1,-1

%N Sign of the alternating sum of the prime indices of n.

%C Also the sign of the reverse-alternating sum of the partition with Heinz number n.

%C The alternating sum of a reversed partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(k-1) times the number of odd parts in the conjugate partition. The alternating sum of the prime indices of n is given by A316524(n).

%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

%H Antti Karttunen, <a href="/A344617/b344617.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(n) = 0 if n is a square (A000290); otherwise a(n) = (-1)^(k-1), where k = A001222(n).

%F a(n) = a(A046523(n)). - _Antti Karttunen_, May 06 2022

%e The pre-images of -1, 0, and 1, together with their prime indices, begin:

%e 6: {1,2} 1: {} 2: {1}

%e 10: {1,3} 4: {1,1} 3: {2}

%e 14: {1,4} 9: {2,2} 5: {3}

%e 15: {2,3} 16: {1,1,1,1} 7: {4}

%e 21: {2,4} 25: {3,3} 8: {1,1,1}

%e 22: {1,5} 11: {5}

%e 24: {1,1,1,2} 12: {1,1,2}

%e 26: {1,6} 13: {6}

%e 17: {7}

%e 18: {1,2,2}

%e 19: {8}

%e 20: {1,1,3}

%e 23: {9}

%e 27: {2,2,2}

%e 28: {1,1,4}

%e 29: {10}

%e 30: {1,2,3}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];

%t Sign[Table[ats[primeMS[n]],{n,100}]]

%o (PARI) A344617(n) = ((!issquare(n)) * ((-1)^(1+bigomega(n)))); \\ _Antti Karttunen_, May 06 2022

%Y Positions of nonzeros are A000037.

%Y Positions of 0's are A000290.

%Y Positions of 1's are A026424.

%Y The absolute value is A049240.

%Y Positions of -1's are A119899.

%Y a(n) is the sign of A316524(n).

%Y A000041 counts partitions of 2n with alternating sum 0.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A071321 is the alternating sum of prime factors.

%Y A071322 is the reverse-alternating sum of prime factors.

%Y A103919 counts partitions by sum and alternating sum.

%Y A316523 is the alternating sum of prime multiplicities.

%Y A335433 ranks separable partitions.

%Y A335448 ranks inseparable partitions.

%Y A344606 counts wiggly permutations of prime indices with twins.

%Y A344610 counts partitions by sum and positive reverse-alternating sum.

%Y A344612 counts partitions by sum and reverse-alternating sum.

%Y A344616 is the alternating sum of the reversed prime indices of n.

%Y A344618 gives reverse-alternating sum of standard compositions.

%Y Cf. A000070, A001222, A046523, A028260, A116406, A124754, A239829, A343938, A344607, A344608, A344609, A344653, A344739.

%K sign

%O 1

%A _Gus Wiseman_, Jun 03 2021

%E More terms from _Antti Karttunen_, May 06 2022