A347457 -id:A347457 - cp's OEIS frontend

A347049 Number of odd-length ordered factorizations of n with integer alternating product.

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 3, 1, 1, 1, 7, 1, 1, 1, 11, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 14, 1, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 7, 1, 1, 3, 15, 1, 1, 1, 3, 1, 1, 1, 24, 1, 1, 3, 3, 1, 1, 1, 14, 4, 1, 1, 7, 1, 1, 1, 5, 1, 7, 1, 3, 1, 1, 1, 24, 1, 3, 3, 11

Offset: 1

Gus Wiseman, Oct 10 2021

nonn

An ordered factorization of n is a sequence of positive integers > 1 with product n.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

			The a(n) ordered factorizations for n = 2, 8, 12, 16, 24, 32, 36, 48:
  2   8       12      16      24      32          36      48
      2*2*2   2*2*3   2*2*4   2*2*6   2*2*8       2*2*9   2*4*6
              3*2*2   2*4*2   3*2*4   2*4*4       2*3*6   3*2*8
                      4*2*2   4*2*3   4*2*4       2*6*3   3*4*4
                              6*2*2   4*4*2       3*2*6   4*2*6
                                      8*2*2       3*3*4   4*4*3
                                      2*2*2*2*2   3*6*2   6*2*4
                                                  4*3*3   6*4*2
                                                  6*2*3   8*2*3
                                                  6*3*2   12*2*2
                                                  9*2*2   2*2*12
                                                          2*2*2*2*3
                                                          2*2*3*2*2
                                                          3*2*2*2*2

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Positions of 2's appear to be A030078.
Positions of 3's appear to be A054753.
Positions of 1's appear to be A167207.
Allowing non-integer alternating product gives A174726, unordered A339890.
The even-length version is A347048.
The unordered version is A347441, with same reverse version.
The case of partitions is A347444, ranked by A347453.
Allowing any length gives A347463.
A001055 counts factorizations (strict A045778, ordered A074206).
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1, ranked by A028982.
A339846 counts even-length factorizations, ordered A174725.
A347050 = factorizations with alternating permutation, complement A347706.
A347437 = factorizations with integer alternating product, reverse A347442.
A347438 = factorizations with alternating product 1, on squares A273013.
A347439 = factorizations with integer reciprocal alternating product.
A347446 = partitions with integer alternating product, reverse A347445.
A347457 lists Heinz numbers of partitions with integer alternating product.
A347460 counts possible alternating products of factorizations.
A347708 counts possible alternating products of odd-length factorizations.
Cf. A025047, A035363, A038548, A116406, A347440, A347454, A347456, A347458, A347459, A347464, A347704.

ordfacs[n_]:=If[n<=1,{{}},Join@@Table[Prepend[#,d]&/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[ordfacs[n],OddQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,100}]

A347049(n, m=n, ap=1, e=0) = if(1==n,(e%2) && 1==denominator(ap), sumdiv(n, d, if(d>1, A347049(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024

a(n) = A347463(n) - A347048(n).

Data section extended up to a(100) by Antti Karttunen, Jul 28 2024

A348384 Heinz numbers of integer partitions whose length is 2/3 their sum.

Original entry on oeis.org

1, 6, 36, 40, 216, 224, 240, 1296, 1344, 1408, 1440, 1600, 6656, 7776, 8064, 8448, 8640, 8960, 9600, 34816, 39936, 46656, 48384, 50176, 50688, 51840, 53760, 56320, 57600, 64000, 155648, 208896, 239616, 266240, 279936, 290304, 301056, 304128, 311040, 315392

Offset: 1

Gus Wiseman, Nov 13 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose sum of prime indices is 3/2 their number. Counting the partitions with these Heinz numbers gives A035377(n) = A000041(n/3) if n is a multiple of 3, otherwise 0.

			The terms and their prime indices begin:
     1: {}
     6: {1,2}
    36: {1,1,2,2}
    40: {1,1,1,3}
   216: {1,1,1,2,2,2}
   224: {1,1,1,1,1,4}
   240: {1,1,1,1,2,3}
  1296: {1,1,1,1,2,2,2,2}
  1344: {1,1,1,1,1,1,2,4}
  1408: {1,1,1,1,1,1,1,5}
  1440: {1,1,1,1,1,2,2,3}
  1600: {1,1,1,1,1,1,3,3}
  6656: {1,1,1,1,1,1,1,1,1,6}
  7776: {1,1,1,1,1,2,2,2,2,2}

David A. Corneth, Table of n, a(n) for n = 1..10000

These partitions are counted by A035377.
Rounding down gives A348550 or A347452, counted by A108711 or A119620.
A000041 counts integer partitions.
A001222 counts prime factors with multiplicity.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344606 counts alternating permutations of prime factors.
Cf. A000070, A000097, A028260, A028982, A032766, A236914, A316413, A347457, A348551.

Select[Range[1000],2*Total[Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]]==3*PrimeOmega[#]&]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
isA348384(n) = (A056239(n)==(3/2)*bigomega(n)); \\ Antti Karttunen, Nov 22 2021

The sequence contains n iff A056239(n) = 3*A001222(n)/2. Here, A056239 adds up prime indices, while A001222 counts them with multiplicity.

Intersection of A028260 and A347452.

cp's OEIS Frontend

A347049 Number of odd-length ordered factorizations of n with integer alternating product.

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