A347457 -id:A347457 - cp's OEIS frontend

A347451 Numbers whose multiset of prime indices has integer reciprocal alternating product.

1, 2, 4, 6, 8, 9, 10, 14, 16, 18, 21, 22, 24, 25, 26, 32, 34, 36, 38, 39, 40, 46, 49, 50, 54, 56, 57, 58, 62, 64, 65, 72, 74, 81, 82, 84, 86, 87, 88, 90, 94, 96, 98, 100, 104, 106, 111, 115, 118, 121, 122, 126, 128, 129, 133, 134, 136, 142, 144, 146, 150, 152

Offset: 1

Gus Wiseman, Sep 24 2021

nonn

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.
We define the reciprocal alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^i).
Also Heinz numbers integer partitions with integer reverse-reciprocal alternating product, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The terms and their prime indices begin:
      1: {}            32: {1,1,1,1,1}       65: {3,6}
      2: {1}           34: {1,7}             72: {1,1,1,2,2}
      4: {1,1}         36: {1,1,2,2}         74: {1,12}
      6: {1,2}         38: {1,8}             81: {2,2,2,2}
      8: {1,1,1}       39: {2,6}             82: {1,13}
      9: {2,2}         40: {1,1,1,3}         84: {1,1,2,4}
     10: {1,3}         46: {1,9}             86: {1,14}
     14: {1,4}         49: {4,4}             87: {2,10}
     16: {1,1,1,1}     50: {1,3,3}           88: {1,1,1,5}
     18: {1,2,2}       54: {1,2,2,2}         90: {1,2,2,3}
     21: {2,4}         56: {1,1,1,4}         94: {1,15}
     22: {1,5}         57: {2,8}             96: {1,1,1,1,1,2}
     24: {1,1,1,2}     58: {1,10}            98: {1,4,4}
     25: {3,3}         62: {1,11}           100: {1,1,3,3}
     26: {1,6}         64: {1,1,1,1,1,1}    104: {1,1,1,6}

The version for reversed prime indices is A028982, counted by A119620.
The additive version is A119899, strict A028260.
Allowing any alternating product >= 1 gives A344609.
Factorizations of this type are counted by A347439.
Allowing any alternating product <= 1 gives A347450.
The non-reciprocal version is A347454.
Allowing any alternating product > 1 gives A347465, reverse A028983.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433 lists numbers whose prime indices are separable, complement A335448.
A344606 counts alternating permutations of prime indices.
A347457 ranks partitions with integer alternating product.
Cf. A001222, A236913, A316523, A344617, A345958, A345959, A346703, A346704, A347437, A347446, A347455.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],IntegerQ[1/altprod[primeMS[#]]]&]

A347458 Number of factorizations of n^2 with integer alternating product.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 17, 2, 6, 6, 15, 2, 17, 2, 16, 6, 6, 2, 41, 4, 6, 8, 16, 2, 31, 2, 27, 6, 6, 6, 56, 2, 6, 6, 39, 2, 31, 2, 17, 17, 6, 2, 90, 4, 17, 6, 17, 2, 41, 6, 39, 6, 6, 2, 105, 2, 6, 17, 48, 6, 31, 2, 17, 6, 31, 2, 148, 2, 6, 17, 17, 6, 32, 2, 86, 15, 6, 2, 107, 6, 6, 6, 40, 2, 109, 6, 17

Offset: 1

Gus Wiseman, Sep 21 2021

nonn

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
The even-length case, the case of alternating product 1, and the case of alternating sum 0 are all counted by A001055.

			The a(2) = 2 through a(8) = 8 factorizations:
  4     9     16        25    36        49    64
  2*2   3*3   4*4       5*5   6*6       7*7   8*8
              2*2*4           2*2*9           2*4*8
              2*2*2*2         2*3*6           4*4*4
                              3*3*4           2*2*16
                              2*2*3*3         2*2*4*4
                                              2*2*2*2*4
                                              2*2*2*2*2*2

Antti Karttunen, Table of n, a(n) for n = 1..16415

Positions of 2's are A000040, squares A001248.
The restriction to powers of 2 is A344611.
This is the restriction to perfect squares of A347437.
The nonsquared even-length version is A347438.
The reciprocal version is A347459, non-squared A347439.
The additive version (partitions) is the even bisection of A347446.
The nonsquared ordered version is A347463.
The case of alternating product 1 in the ordered version is A347464.
Allowing any alternating product gives A347466.
A000290 lists squares, complement A000037.
A001055 counts factorizations.
A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors of n (reverse: A071322).
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347460 counts possible alternating products of factorizations.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A347457 ranks partitions with integer alternating product.
Cf. A062312, A119620, A330972, A346635, A347440, A347441, A347442, A347445, A347451, A347456, A347704, A347705.
Apparently, A006881 gives the positions of 6's. - Antti Karttunen, Oct 22 2023

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[facs[n^2],IntegerQ[altprod[#]]&]],{n,100}]

A347437(n, m=n, ap=1, e=0) = if(1==n, if(e%2, 1==denominator(ap), 1==numerator(ap)), sumdiv(n, d, if((d>1)&&(d<=m), A347437(n/d, d, ap * d^((-1)^e), 1-e))));
A347458(n) = A347437(n*n); \\ Antti Karttunen, Oct 22 2023

a(2^n) = A344611(n).

a(n) = A347437(n^2).

Data section extended up to a(92) by Antti Karttunen, Oct 22 2023

A347459 Number of factorizations of n^2 with integer reciprocal alternating product.

Original entry on oeis.org

1, 1, 1, 3, 1, 4, 1, 6, 3, 4, 1, 11, 1, 4, 4, 12, 1, 11, 1, 12, 4, 4, 1, 28, 3, 4, 6, 12, 1, 19, 1, 22, 4, 4, 4, 38, 1, 4, 4, 29, 1, 21, 1, 12, 11, 4, 1, 65, 3, 11, 4, 12, 1, 29, 4, 29, 4, 4, 1, 71, 1, 4, 11, 40, 4, 22, 1, 12, 4, 18, 1, 107, 1, 4, 11, 12, 4, 22, 1, 66, 12, 4, 1, 76, 4, 4, 4, 30, 1, 71, 4, 12, 4, 4, 4, 141

Offset: 1

Gus Wiseman, Sep 22 2021

nonn

We define the reciprocal alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^i).
A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
All such factorizations have even length.
Image appears to be 1, 3, 4, 6, 11, ... , missing some numbers such as 2, 5, 7, 8, 9, ...
The case of alternating product 1, the case of alternating sum 0, and the reverse version are all counted by A001055.

			The a(2) = 1 through a(10) = 4 factorizations:
    2*2  3*3  2*8      5*5  6*6      7*7  8*8          9*9      2*50
              4*4           2*18          2*32         3*27     5*20
              2*2*2*2       3*12          4*16         3*3*3*3  10*10
                            2*2*3*3       2*2*2*8               2*2*5*5
                                          2*2*4*4
                                          2*2*2*2*2*2

Antti Karttunen, Table of n, a(n) for n = 1..16384

Positions of 1's are 1 and A000040, squares A001248.
The additive version (partitions) is A000041, the even bisection of A119620.
Partitions of this type are ranked by A028982 and A347451.
The restriction to powers of 2 is A236913, the even bisection of A027187.
The nonsquared nonreciprocal even-length version is A347438.
This is the restriction to perfect squares of A347439.
The nonreciprocal version is A347458, non-squared A347437.
A000290 lists squares, complement A000037.
A001055 counts factorizations.
A046099 counts factorizations with no alternating permutations.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A347460 counts possible alternating products of factorizations.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A347457 ranks partitions with integer alternating product.
A347466 counts factorizations of n^2.
Cf. A062312, A316523, A330972, A344653, A346635, A347440, A347441, A347442, A347453, A347461, A347463, A347464, A347705.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
recaltprod[q_]:=Product[q[[i]]^(-1)^i,{i,Length[q]}];
Table[Length[Select[facs[n^2],IntegerQ[recaltprod[#]]&]],{n,100}]

A347439(n, m=n, ap=1, e=0) = if(1==n, !(e%2) && 1==denominator(ap), sumdiv(n, d, if(d>1 && d<=m, A347439(n/d, d, ap * d^((-1)^e), 1-e))));
A347459(n) = A347439(n^2); \\ Antti Karttunen, Jul 28 2024

a(2^n) = A236913(n).

a(n) = A347439(n^2).

Data section extended up to a(96) by Antti Karttunen, Jul 28 2024

A347465 Numbers whose multiset of prime indices has alternating product > 1.

Original entry on oeis.org

3, 5, 7, 11, 12, 13, 17, 19, 20, 23, 27, 28, 29, 30, 31, 37, 41, 42, 43, 44, 45, 47, 48, 52, 53, 59, 61, 63, 66, 67, 68, 70, 71, 73, 75, 76, 78, 79, 80, 83, 89, 92, 97, 99, 101, 102, 103, 105, 107, 108, 109, 110, 112, 113, 114, 116, 117, 120, 124, 125, 127

Offset: 1

Gus Wiseman, Sep 27 2021

nonn

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.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
All terms have odd bigomega (A001222).
Also Heinz numbers integer partitions with reverse-alternating product > 1.

			The terms and their prime indices begin:
      3: {2}         37: {12}            68: {1,1,7}
      5: {3}         41: {13}            70: {1,3,4}
      7: {4}         42: {1,2,4}         71: {20}
     11: {5}         43: {14}            73: {21}
     12: {1,1,2}     44: {1,1,5}         75: {2,3,3}
     13: {6}         45: {2,2,3}         76: {1,1,8}
     17: {7}         47: {15}            78: {1,2,6}
     19: {8}         48: {1,1,1,1,2}     79: {22}
     20: {1,1,3}     52: {1,1,6}         80: {1,1,1,1,3}
     23: {9}         53: {16}            83: {23}
     27: {2,2,2}     59: {17}            89: {24}
     28: {1,1,4}     61: {18}            92: {1,1,9}
     29: {10}        63: {2,2,4}         97: {25}
     30: {1,2,3}     66: {1,2,5}         99: {2,2,5}
     31: {11}        67: {19}           101: {26}

The squarefree case is A030059 without 2.
The reverse version is A028983, counted by A119620.
The opposite version (< 1 instead of > 1) is A119899.
Factorizations of this type are counted by A339890, reverse A347705.
The weak version (>= 1 instead of > 1) is A344609.
Partitions of this type are counted by A347449, reverse A347448.
The complement is A347450, counted by A339846 or A347443.
Allowing any integer reverse-alternating product gives A347454.
Allowing any integer alternating product gives A347457.
A335433 ranks inseparable partitions, complement A335448.
A347446 counts partitions with integer alternating product, reverse A347445.
Cf. A008549, A344607, A344608, A344611, A347442, A347444, A347447, A347453, A347456, A347461, A347462.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],altprod[primeMS[#]]>1&]

A347464 Number of even-length ordered factorizations of n^2 into factors > 1 with alternating product 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 5, 1, 6, 2, 5, 1, 26, 1, 5, 5, 20, 1, 26, 1, 26, 5, 5, 1, 134, 2, 5, 6, 26, 1, 73, 1, 70, 5, 5, 5, 230, 1, 5, 5, 134, 1, 73, 1, 26, 26, 5, 1, 670, 2, 26, 5, 26, 1, 134, 5, 134, 5, 5, 1, 686, 1, 5, 26, 252, 5, 73, 1, 26, 5, 73, 1, 1714, 1, 5, 26

Offset: 1

Gus Wiseman, Sep 23 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)).
Also the number of ordered pairs of ordered factorizations of n, both of the same length.
Note that the version for all n (not just squares) is 0 except at perfect squares.

			The a(12) = 26 ordered factorizations:
  (2*2*6*6)      (3*2*4*6)      (6*2*2*6)  (4*2*3*6)  (12*12)
  (2*3*6*4)      (3*3*4*4)      (6*3*2*4)  (4*3*3*4)
  (2*4*6*3)      (3*4*4*3)      (6*4*2*3)  (4*4*3*3)
  (2*6*6*2)      (3*6*4*2)      (6*6*2*2)  (4*6*3*2)
  (2*2*2*2*3*3)  (3*2*2*2*2*3)
  (2*2*2*3*3*2)  (3*2*2*3*2*2)
  (2*2*3*2*2*3)  (3*3*2*2*2*2)
  (2*2*3*3*2*2)
  (2*3*2*2*3*2)
  (2*3*3*2*2*2)
For example, the ordered factorization 6*3*2*4 = 144 has alternating product 6/3*2/4 = 1, so is counted under a(12).

Index entries for sequences computed from exponents in factorization of n

Positions of 1's are A008578 (1 and A000040).
The restriction to powers of 2 is A000984.
Positions of 2's are A001248.
The not necessarily even-length version is A273013.
A000290 lists squares, complement A000037.
A001055 counts factorizations.
A027187 counts even-length partitions.
A074206 counts ordered factorizations.
A119620 counts partitions with alternating product 1, ranked by A028982.
A339846 counts even-length factorizations, ordered A347706.
A347438 counts factorizations with alternating product 1.
A347457 ranks partitions with integer alternating product.
A347460 counts possible alternating products of factorizations.
A347466 counts factorizations of n^2.
Cf. A062312, A339890, A347437, A347439, A347440, A347442, A347456, A347459, A347463, A347705.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[Join@@Permutations/@facs[n^2],EvenQ[Length[#]]&&altprod[#]==1&]],{n,100}]

A347464aux(n, k=0, t=1) = if(1==n, (0==k)&&(1==t), my(s=0); fordiv(n, d, if((d>1), s += A347464aux(n/d, 1-k, t*(d^((-1)^k))))); (s));
A347464(n) = A347464aux(n^2); \\ Antti Karttunen, Oct 30 2021

A347466 Number of factorizations of n^2.

Original entry on oeis.org

1, 2, 2, 5, 2, 9, 2, 11, 5, 9, 2, 29, 2, 9, 9, 22, 2, 29, 2, 29, 9, 9, 2, 77, 5, 9, 11, 29, 2, 66, 2, 42, 9, 9, 9, 109, 2, 9, 9, 77, 2, 66, 2, 29, 29, 9, 2, 181, 5, 29, 9, 29, 2, 77, 9, 77, 9, 9, 2, 269, 2, 9, 29, 77, 9, 66, 2, 29, 9, 66, 2, 323, 2, 9, 29, 29

Offset: 1

Gus Wiseman, Sep 23 2021

nonn

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

			The a(1) = 1 through a(8) = 11 factorizations:
  ()  (4)    (9)    (16)       (25)   (36)       (49)   (64)
      (2*2)  (3*3)  (2*8)      (5*5)  (4*9)      (7*7)  (8*8)
                    (4*4)             (6*6)             (2*32)
                    (2*2*4)           (2*18)            (4*16)
                    (2*2*2*2)         (3*12)            (2*4*8)
                                      (2*2*9)           (4*4*4)
                                      (2*3*6)           (2*2*16)
                                      (3*3*4)           (2*2*2*8)
                                      (2*2*3*3)         (2*2*4*4)
                                                        (2*2*2*2*4)
                                                        (2*2*2*2*2*2)

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Positions of 2's are the primes (A000040), which have squares A001248.
The restriction to powers of 2 is A058696.
The additive version (partitions) is A072213.
The case of integer alternating product is A347459, nonsquared A347439.
A000290 lists squares, complement A000037.
A001055 counts factorizations.
A339846 counts even-length factorizations.
A339890 counts odd-length factorizations.
A347050 = factorizations with alternating permutation, complement A347706.
Cf. A000041, A062312, A120452, A144338, A273013, A330972, A345957, A346635, A347437, A347438, A347457, A347460, A347464.

b:= proc(n, k) option remember; `if`(n>k, 0, 1)+`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=numtheory[divisors](n) minus {1, n}))
    end:
a:= proc(n) option remember; b((l-> mul(ithprime(i)^l[i], i=1..nops(l)))(
      sort(map(i-> i[2], ifactors(n^2)[2]), `>`))$2)
    end:
seq(a(n), n=1..76);  # Alois P. Heinz, Oct 14 2021

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[facs[n^2]],{n,25}]

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
A347466(n) = A001055(n^2); \\ Antti Karttunen, Oct 13 2021

a(n) = A001055(A000290(n)).

A347704 Number of even-length integer partitions of n with integer alternating product.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 6, 4, 11, 8, 18, 13, 33, 22, 49, 38, 79, 58, 122, 90, 186, 139, 268, 206, 402, 304, 569, 448, 817, 636, 1152, 907, 1612, 1283, 2220, 1791, 3071, 2468, 4162, 3409, 5655, 4634, 7597, 6283, 10171, 8478, 13491, 11336, 17906, 15088, 23513, 20012

Offset: 0

Gus Wiseman, Sep 17 2021

nonn

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

			The a(2) = 1 through a(9) = 8 partitions:
  (11)  (21)  (22)    (41)    (33)      (61)      (44)        (63)
              (31)    (2111)  (42)      (2221)    (62)        (81)
              (1111)          (51)      (4111)    (71)        (3321)
                              (2211)    (211111)  (2222)      (4221)
                              (3111)              (3221)      (6111)
                              (111111)            (3311)      (222111)
                                                  (4211)      (411111)
                                                  (5111)      (21111111)
                                                  (221111)
                                                  (311111)
                                                  (11111111)

Allowing any alternating product >= 1 gives A000041, reverse A344607.
Allowing any alternating product gives A027187, odd bisection A236914.
The Heinz numbers of these partitions are given by A028260 /\ A347457.
The reverse and reciprocal versions are both A035363.
The multiplicative version (factorizations) is A347438, reverse A347439.
The odd-length instead of even-length version is A347444.
Allowing any length gives A347446.
A034008 counts even-length compositions, ranked by A053754.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Cf. A000070, A067661, A236913, A304620, A339846, A347437, A347441, A347442, A347445, A347448, A347449, A347454, A347462.

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,0,30}]

A347452 Heinz numbers of integer partitions whose sum is 3/2 their length, rounded down.

Original entry on oeis.org

1, 2, 6, 12, 36, 40, 72, 80, 216, 224, 240, 432, 448, 480, 1296, 1344, 1408, 1440, 1600, 2592, 2688, 2816, 2880, 3200, 6656, 7776, 8064, 8448, 8640, 8960, 9600, 13312, 15552, 16128, 16896, 17280, 17920, 19200, 34816, 39936, 46656, 48384, 50176, 50688, 51840

Offset: 1

Gus Wiseman, Oct 28 2021

nonn

Also numbers whose sum of prime indices is 3/2 their number, rounded down, where a prime index of n is a number m such that prime(m) divides n.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The sequence contains n iff A056239(n) = floor(3*A001222(n)/2). Here, A056239 adds up prime indices, and A001222 counts them with multiplicity.
Counting the partitions with these Heinz numbers gives A119620 with zeros interspersed every three terms.

			The initial terms and their prime indices:
      1: {}
      2: {1}
      6: {1,2}
     12: {1,1,2}
     36: {1,1,2,2}
     40: {1,1,1,3}
     72: {1,1,1,2,2}
     80: {1,1,1,1,3}
    216: {1,1,1,2,2,2}
    224: {1,1,1,1,1,4}
    240: {1,1,1,1,2,3}
    432: {1,1,1,1,2,2,2}
    448: {1,1,1,1,1,1,4}
    480: {1,1,1,1,1,2,3}
   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}

Counting terms by Heinz weight (in A032766) gives A119620.
An adjoint version is A348550, counted by A108711.
A000041 counts 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 wiggly permutations of prime factors.
Cf. A000070, A000097, A028982, A236914, A316413, A347457, A348551.

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

A347455 Heinz numbers of integer partitions with non-integer alternating product.

Original entry on oeis.org

15, 30, 33, 35, 51, 55, 60, 66, 69, 70, 77, 85, 91, 93, 95, 102, 105, 110, 119, 120, 123, 132, 135, 138, 140, 141, 143, 145, 154, 155, 161, 165, 170, 177, 182, 186, 187, 190, 201, 203, 204, 205, 209, 210, 215, 217, 219, 220, 221, 231, 238, 240, 246, 247, 249

Offset: 1

Gus Wiseman, Oct 04 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
Also numbers whose multiset of prime indices has non-integer reverse-alternating product.

			The terms and their reversed prime indices begin:
     15: (3,2)        102: (7,2,1)        161: (9,4)
     30: (3,2,1)      105: (4,3,2)        165: (5,3,2)
     33: (5,2)        110: (5,3,1)        170: (7,3,1)
     35: (4,3)        119: (7,4)          177: (17,2)
     51: (7,2)        120: (3,2,1,1,1)    182: (6,4,1)
     55: (5,3)        123: (13,2)         186: (11,2,1)
     60: (3,2,1,1)    132: (5,2,1,1)      187: (7,5)
     66: (5,2,1)      135: (3,2,2,2)      190: (8,3,1)
     69: (9,2)        138: (9,2,1)        201: (19,2)
     70: (4,3,1)      140: (4,3,1,1)      203: (10,4)
     77: (5,4)        141: (15,2)         204: (7,2,1,1)
     85: (7,3)        143: (6,5)          205: (13,3)
     91: (6,4)        145: (10,3)         209: (8,5)
     93: (11,2)       154: (5,4,1)        210: (4,3,2,1)
     95: (8,3)        155: (11,3)         215: (14,3)
For example, (4,3,2,1) has alternating product 4/3*2/1 = 8/3, so the Heinz number 210 is in the sequence.

The reciprocal version is A028983, complement A028982.
Factorizations not of this type are counted by A347437.
Partitions not of this type are counted by A347446.
The complement of the reverse reciprocal version is A347451.
The complement in the odd-length case is A347453.
The complement of the reverse version is A347454.
The complement is A347457.
A056239 adds up prime indices, row sums of A112798.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433 lists numbers whose prime indices are separable, complement A335448.
A347461 counts possible alternating products of partitions, reverse A347462.
Cf. A001105, A001222, A028260, A119620, A119899, A316523, A344606, A344617, A346703, A346704, A347448, A347450, A347465.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Select[Range[100],!IntegerQ[altprod[Reverse[primeMS[#]]]]&]

A347048 Number of even-length ordered factorizations of n with integer alternating product.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 4, 0, 0, 0, 7, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 6, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 11, 0, 0, 0, 1, 0, 0, 0, 11, 0, 0, 1, 1, 0, 0, 0, 6, 3, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 8, 0, 1, 1, 7, 0, 0, 0, 1, 0

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 = 16, 32, 36, 48, 64, 96:
  4*4       8*4       6*6       12*4      8*8           24*4
  8*2       16*2      12*3      24*2      16*4          48*2
  2*2*2*2   2*2*4*2   18*2      2*2*6*2   32*2          3*2*8*2
            4*2*2*2   2*2*3*3   3*2*4*2   2*2*4*4       4*2*6*2
                      2*3*3*2   4*2*3*2   2*2*8*2       6*2*4*2
                      3*2*2*3   6*2*2*2   2*4*4*2       8*2*3*2
                      3*3*2*2             4*2*2*4       12*2*2*2
                                          4*2*4*2       2*2*12*2
                                          4*4*2*2
                                          8*2*2*2
                                          2*2*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 0's are A005117 \ {2}.
The restriction to powers of 2 is A027306.
Heinz numbers of partitions of this type are A028260 /\ A347457.
Positions of 3's appear to be A030514.
Positions of 1's are 1 and A082293.
Allowing non-integer alternating product gives A174725, unordered A339846.
The odd-length version is A347049.
The unordered version is A347438, reverse A347439.
Allowing any length gives A347463.
Partitions of this type are counted by A347704, reverse A035363.
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.
A273013 counts ordered factorizations of n^2 with alternating product 1.
A339890 counts odd-length factorizations, ordered A174726.
A347050 = factorizations with alternating permutation, complement A347706.
A347437 = factorizations with integer alternating product, reverse A347442.
A347446 = partitions with integer alternating product, reverse A347445.
A347460 counts possible alternating products of factorizations.
Cf. A025047, A038548, A116406, A138364, A347440, A347441, A347454, A347456, A347458, A347459, A347464.

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],EvenQ[Length[#]]&&IntegerQ[altprod[#]]&]],{n,100}]

A347048(n, m=n, ap=1, e=0) = if(1==n,!(e%2) && 1==numerator(ap), sumdiv(n, d, if(d>1, A347048(n/d, d, ap * d^((-1)^e), 1-e)))); \\ Antti Karttunen, Jul 28 2024

a(n) = A347463(n) - A347049(n).

Data section extended up to a(105) by Antti Karttunen, Jul 28 2024

cp's OEIS Frontend

A347451 Numbers whose multiset of prime indices has integer reciprocal alternating product.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A347458 Number of factorizations of n^2 with integer alternating product.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A347459 Number of factorizations of n^2 with integer reciprocal alternating product.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A347465 Numbers whose multiset of prime indices has alternating product > 1.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A347464 Number of even-length ordered factorizations of n^2 into factors > 1 with alternating product 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A347466 Number of factorizations of n^2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A347704 Number of even-length integer partitions of n with integer alternating product.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A347452 Heinz numbers of integer partitions whose sum is 3/2 their length, rounded down.

Views