A346702 -id:A346702 - cp's OEIS frontend

A344651 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with alternating sum k, with k ranging from n mod 2 to n in steps of 2.

1, 1, 1, 1, 2, 1, 2, 2, 1, 4, 2, 1, 3, 5, 2, 1, 7, 5, 2, 1, 5, 9, 5, 2, 1, 12, 10, 5, 2, 1, 7, 17, 10, 5, 2, 1, 19, 19, 10, 5, 2, 1, 11, 28, 20, 10, 5, 2, 1, 30, 33, 20, 10, 5, 2, 1, 15, 47, 35, 20, 10, 5, 2, 1, 45, 57, 36, 20, 10, 5, 2, 1, 22, 73, 62, 36, 20, 10, 5, 2, 1

Offset: 0

Gus Wiseman, Jun 05 2021

The alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. This is equal to the number of odd parts in the conjugate partition, so T(n,k) is the number of integer partitions of n with k odd parts in the conjugate partition, which is also the number of partitions of n with k odd parts.

Also the number of integer partitions of n with odd-indexed parts (odd bisection) summing to k, ceiling(n/2) <= k <= n. The even-indexed version is A346633. - Gus Wiseman, Nov 29 2021

			Triangle begins:
   1
   1
   1   1
   2   1
   2   2   1
   4   2   1
   3   5   2   1
   7   5   2   1
   5   9   5   2   1
  12  10   5   2   1
   7  17  10   5   2   1
  19  19  10   5   2   1
  11  28  20  10   5   2   1
  30  33  20  10   5   2   1
  15  47  35  20  10   5   2   1
  45  57  36  20  10   5   2   1
  22  73  62  36  20  10   5   2   1
  67  92  64  36  20  10   5   2   1
  30 114 102  65  36  20  10   5   2   1
  97 147 107  65  36  20  10   5   2   1
Row n = 10 counts the following partitions (A = 10):
  (55)          (64)         (73)       (82)     (91)   (A)
  (3322)        (442)        (433)      (622)    (811)
  (4411)        (541)        (532)      (721)
  (222211)      (3331)       (631)      (7111)
  (331111)      (4222)       (5221)     (61111)
  (22111111)    (4321)       (6211)
  (1111111111)  (5311)       (42211)
                (22222)      (52111)
                (32221)      (511111)
                (33211)      (4111111)
                (43111)
                (322111)
                (421111)
                (2221111)
                (3211111)
                (31111111)
                (211111111)
The conjugate version is:
  (A)      (55)      (3331)     (331111)    (31111111)   (1111111111)
  (64)     (73)      (5311)     (511111)    (211111111)
  (82)     (91)      (7111)     (3211111)
  (442)    (433)     (33211)    (4111111)
  (622)    (532)     (43111)    (22111111)
  (4222)   (541)     (52111)
  (22222)  (631)     (61111)
           (721)     (322111)
           (811)     (421111)
           (3322)    (2221111)
           (4321)
           (4411)
           (5221)
           (6211)
           (32221)
           (42211)
           (222211)

This is A103919 with all zeros removed.
The strict version is A152146 interleaved with A152157.
The rows are those of A239830 interleaved with those of A239829.
The reverse version is the right half of A344612.
The strict reverse version is the right half of A344739.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A027187 counts partitions with rev-alternating sum <= 0, ranked by A028260.
A124754 lists alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A325534/A325535 count separable/inseparable partitions.
A344607 counts partitions with rev-alternating sum >= 0, ranked by A344609.
A344608 counts partitions with rev-alternating sum < 0, ranked by A119899.
A344610 counts partitions of n by positive rev-alternating sum.
A344611 counts partitions of 2n with rev-alternating sum >= 0.
A345197 counts compositions by sum, length, and alternating sum.
A346697 gives the sum of odd-indexed prime indices (reverse: A346699).
A346702 represents the odd bisection of compositions, sums A209281.
Cf. A000070, A000097, A003242, A006330, A025047, A097805, A114121, A116406, A131577, A344617, A344649, A344650, A344654, A346633.

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],ats[#]==k&]],{n,0,15},{k,Mod[n,2],n,2}]

A209281 Start with first run [0,1] then, for n >= 2, the n-th run has length 2^n and is the concatenation of [a(1),a(2),...,a(2^n/2)] and [n-a(1),n-a(2),...,n-a(2^n/2)].

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 1, 2, 4, 3, 2, 3, 1, 2, 3, 2, 5, 4, 3, 4, 2, 3, 4, 3, 1, 2, 3, 2, 4, 3, 2, 3, 6, 5, 4, 5, 3, 4, 5, 4, 2, 3, 4, 3, 5, 4, 3, 4, 1, 2, 3, 2, 4, 3, 2, 3, 5, 4, 3, 4, 2, 3, 4, 3, 7, 6, 5, 6, 4, 5, 6, 5, 3, 4, 5, 4, 6, 5, 4, 5, 2, 3, 4, 3, 5, 4, 3

Offset: 1

Benoit Cloitre, Jan 16 2013

Also the sum of the odd bisection (odd-indexed parts) of the (n-1)-th composition in standard order, where the k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. Note that this sequence counts {} as composition number 1 (instead of the usual 0). For example, composition number 741 in standard order is (2,1,1,3,2,1), with odd bisection (2,1,2), so a(742) = 2 + 1 + 2 = 5. - Gus Wiseman, Aug 24 2021

			[0,1] -> [0,1] U [2-0,2-1] =
[0,1,2,1] -> [0,1,2,1] U [3-0,3-1,3-2,3-1] =
[0,1,2,1,3,2,1,2] etc.
From _Gus Wiseman_, Aug 08 2021: (Start)
As a triangle without the initial 0, row-lengths A000079:
  1
  2 1
  3 2 1 2
  4 3 2 3 1 2 3 2
  5 4 3 4 2 3 4 3 1 2 3 2 4 3 2 3
  6 5 4 5 3 4 5 4 2 3 4 3 5 4 3 4 1 2 3 2 4 3 2 3 5 4 3 4 2 3 4 3
(End)

Cf. A010060 (Thue-Morse), A103204 (indices of 1's).
Cf. A029837 (binary order), A000120 (binary weight), A006068 (inverse Gray), A272020 (bit positions).
Cf. A089215.
As a triangle: A000079 (row lengths), A001792 (row sums).
Other composition part sums: A124754. A346633.
Also the sum of row A346702(n-1) of A066099.
Cf. A346697 (on prime indices).

Table[Total[First/@Partition[Append[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse,0],2]],{n,0,100}] (* Gus Wiseman, Aug 08 2021 *)

/* compute 2^15 terms */ v=[0,1];for(n=2,15,v=concat(v,vector(2^n/2,i,n-v[i]));a(n)=v[n];)

a(n) = n--; my(s=1,ns); while((ns=n>>s), n=bitxor(n,ns); s<<=1); hammingweight(n); \\ Kevin Ryde, May 14 2022

Let T(n)=A010060(n) then for n>=1 a(2n)=a(n)+1-T(n-1) and a(2n+1)=a(n+1)+T(n).
For n>=2 a(n) = a(ceiling(n/2))+T(n-1) hence:
a(n) = Sum_{k=0..ceiling(log(n-1)/log(2))} T(floor((n-1)/2^k)).
For k>=0 a(3*2^k+1)=1 (more precisely a(n)=1 iff n is in A103204), a(2^k+1)=k+1, a(5*2^k+1)=2, a(7*2^k+1)=k+2 etc.
From Gus Wiseman, Aug 18 2021: (Start)
a(n + 1) = (A029837(n) + A124754(n))/2.
a(n + 1) = A029837(n) - A346633(n).
a(n + 1) = A346633(n) - A124754(n).
a(n + 1) = A029837(A346702(n)).
(End)
From Kevin Ryde, May 14 2022: (Start)
a(n) = A000120(A006068(n-1)), binary weight of inverse binary Gray code.
a(n) = Sum_{k=1..A000120(n-1)} (-1)^(k-1) * A272020(n-1,k), alternating sum of 1-bit positions.
a(n) = A089215(n) - 1.
(End)

A346700 Sum of the even bisection (even-indexed parts) of the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 2, 0, 2, 0, 1, 2, 1, 0, 2, 3, 1, 2, 1, 0, 2, 0, 2, 2, 1, 3, 3, 0, 1, 2, 2, 0, 2, 0, 1, 2, 1, 0, 2, 4, 3, 2, 1, 0, 3, 3, 2, 2, 1, 0, 3, 0, 1, 2, 3, 3, 2, 0, 1, 2, 3, 0, 3, 0, 1, 3, 1, 4, 2, 0, 2, 4, 1, 0, 3, 3, 1, 2, 2, 0, 3, 4, 1, 2, 1, 3, 3, 0, 4, 2, 4, 0, 2, 0, 2, 3

Offset: 1

Gus Wiseman, Aug 03 2021

nonn

First differs from A334107 at a(64) = 3, A334107(64) = 2.

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 partition with Heinz number 1100 is (5,3,3,1,1), so a(1100) = 3 + 1 = 4.
The partition with Heinz number 2100 is (4,3,3,2,1,1), so a(2100) = 3 + 2 + 1 = 6.

Antti Karttunen, Table of n, a(n) for n = 1..10201
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to Heinz numbers

Sum of prime indices of A329888(n).
Subtracting from the odd version gives A344616 (non-reverse: A316524).
The unreversed version for standard compositions is A346633.
The odd non-reverse version is A346697.
The non-reverse version (multisets instead of partitions) is A346698.
The odd version is A346699.
A001414 adds up prime factors, row sums of A027746.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
Cf. A000070, A025047, A124754, A209281, A329888, A341446, A344617, A344653, A345957, A345958, A346701, A346702, A346704.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Last/@Partition[Reverse[primeMS[n]],2]],{n,100}]

A346700(n) = if(1==n,0,my(f=factor(n),s=0,p=0); forstep(k=#f~,1,-1,while(f[k,2], s += (p%2)*primepi(f[k,1]); f[k,2]--; p++)); (s)); \\ Antti Karttunen, Sep 21 2021

a(n) = A056239(n) - A346699(n).
a(n) = A346699(n) - A344616(n).
a(n even omega) = A346697(n).
a(n odd omega) = A346698(n).
A316524(n) = A346697(n) - A346698(n).
a(n) = A056239(A329888(n)). - Gus Wiseman and Antti Karttunen, Oct 13 2021

Data section extended up to 105 terms by Antti Karttunen, Sep 21 2021

A329888 a(n) = A329900(A329602(n)); Heinz number of the even bisection (even-indexed parts) of the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 1, 3, 1, 2, 3, 2, 1, 4, 5, 2, 3, 2, 1, 3, 1, 4, 3, 2, 5, 6, 1, 2, 3, 4, 1, 3, 1, 2, 3, 2, 1, 4, 7, 5, 3, 2, 1, 6, 5, 4, 3, 2, 1, 6, 1, 2, 3, 8, 5, 3, 1, 2, 3, 5, 1, 6, 1, 2, 5, 2, 7, 3, 1, 4, 9, 2, 1, 6, 5, 2, 3, 4, 1, 6, 7, 2, 3, 2, 5, 8, 1, 7, 3, 10, 1, 3, 1, 4, 5

Offset: 1

Antti Karttunen, Dec 22 2019

nonn

From Gus Wiseman, Aug 05 2021 and Antti Karttunen, Oct 13 2021: (Start)
Also the product of primes at even positions in the weakly decreasing list (with multiplicity) of prime factors of n. For example, the prime factors of 108 are (3,3,3,2,2), with even bisection (3,2), with product 6, so a(108) = 6.
Proof: A108951(n) gives a number with the same largest prime factor (A006530) and its exponent (A071178) as in n, and with each smaller prime p = 2, 3, 5, 7, ... < A006530(n) having as its exponent the partial sum of the exponents of all prime factors >= p present in n (with primes not present in n having the exponent 0). Then applying A000188 replaces each such "partial sum exponent" k with floor(k/2). Finally, A319626 replaces those halved exponents with their first differences (here the exponent of the largest prime present stays intact, because the next larger prime's exponent is 0 in n). It should be easy to see that if prime q is not present in n (i.e., does not divide it), then neither it is present in a(n). Moreover, if the partial sum exponent of q is odd and only one larger than the partial sum exponent of the next larger prime factor of n, then q will not be present in a(n), while in all other cases q is present in a(n). See also the last example.
(End)

			From _Gus Wiseman_, Aug 15 2021: (Start)
The list of all numbers with image 12 and their corresponding prime factors begins:
  144: (3,3,2,2,2,2)
  216: (3,3,3,2,2,2)
  240: (5,3,2,2,2,2)
  288: (3,3,2,2,2,2,2)
  336: (7,3,2,2,2,2)
  360: (5,3,3,2,2,2)
(End)
The positions from the left are indexed as 1, 2, 3, ..., etc, so e.g., for 240 we pick the second, the fourth and the sixth prime factor, 3, 2 and 2, to obtain a(240) = 3*2*2 = 12. For 288, we similarly pick the second (3), the fourth (2) and the sixth (2) to obtain a(288) = 3*2*2 = 12. - _Antti Karttunen_, Oct 13 2021
Consider n = 11945934 = 2*3*3*3*7*11*13*13*17. Its primorial inflation is A108951(11945934) = 96478365991115908800000 = 2^9 * 3^8 * 5^5 * 7^5 * 11^4 * 13^3 * 17^1. Applying A000188 to this halves each exponent (floored down if the exponent is odd), leaving the factors 2^4 * 3^4 * 5^2 * 7^2 * 11^2 * 13^1 = 2497294800. Then applying A319626 to this number retains the largest prime factor (and its exponent), and subtracts from the exponent of each of the rest of primes the exponent of the next larger prime, so from 2^4 * 3^4 * 5^2 * 7^2 * 11^2 * 13^1 we get 2^(4-4) * 3^(4-2) * 5^(2-2) * 7^(2-2) * 11^(2-1) * 13^1 = 3^2 * 11^1 * 13^1 = 1287 = a(11945934), which is obtained also by selecting every second prime from the list [17, 13, 13, 11, 7, 3, 3, 3, 2] and taking their product. - _Antti Karttunen_, Oct 15 2021

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to Heinz numbers
Index entries for sequences related to primorial numbers

Cf. A000188, A108951, A319626, A329602, A329900.
A left inverse of A000290.
Positions of 1's are A008578.
Positions of primes are A168645.
The sum of prime indices of a(n) is A346700(n).
The odd version is A346701.
The odd non-reverse version is A346703.
The non-reverse version is A346704.
The version for standard compositions is A346705, odd A346702.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A001414 adds up prime factors, row sums of A027746.
A027187 counts partitions of even length, ranked by A028260.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A346633 adds up the even bisection of standard compositions.
A346698 adds up the even bisection of prime indices.
Cf. A000097, A035363, A102750, A236913, A253553, A344606, A344617, A344653, A345957, A345958, A345959.

Table[Times@@Last/@Partition[Reverse[Flatten[Apply[ConstantArray,FactorInteger[n],{1}]]],2],{n,100}] (* Gus Wiseman, Oct 13 2021 *)

PARI
```
A329888(n) = A329900(A329602(n));
```

A329888(n) = if(1==n,n,my(f=factor(n),m=1,p=0); forstep(k=#f~,1,-1,while(f[k,2], m *= f[k,1]^(p%2); f[k,2]--; p++)); (m)); \\ (After Wiseman's new interpretation) - Antti Karttunen, Sep 21 2021

a(n) = A329900(A329602(n)) = A329900(A000188(A108951(n))).
A108951(a(n)) = A329602(n).
a(n^2) = n for all n >= 1.
a(n) * A346701(n) = n. - Gus Wiseman, Aug 07 2021
A056239(a(n)) = A346700(n). - Gus Wiseman, Aug 07 2021
Antti Karttunen, Sep 21 2021
From Antti Karttunen, Oct 13 2021: (Start)
a(n) = A319626(A329602(n)) = A319626(A000188(A108951(n))).
For all x in A102750, a(x) = a(A253553(x)). (End)

Name amended with Gus Wiseman's new interpretation - Antti Karttunen, Oct 13 2021

A346701 Heinz number of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 6, 13, 7, 5, 4, 17, 6, 19, 10, 7, 11, 23, 6, 5, 13, 9, 14, 29, 10, 31, 8, 11, 17, 7, 6, 37, 19, 13, 10, 41, 14, 43, 22, 15, 23, 47, 12, 7, 10, 17, 26, 53, 9, 11, 14, 19, 29, 59, 10, 61, 31, 21, 8, 13, 22, 67, 34, 23, 14, 71

Offset: 1

Gus Wiseman, Aug 03 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.

			The partition (2,2,2,1,1) has Heinz number 108 and odd bisection (2,2,1) with Heinz number 18, so a(108) = 18.
The partitions (3,2,2,1,1), (3,2,2,2,1), (3,3,2,1,1) have Heinz numbers 180, 270, 300 and all have odd bisection (3,2,1) with Heinz number 30, so a(180) = a(270) = a(300) = 30.

Positions of last appearances are A000290 without the first term 0.
Positions of primes are A037143 (complement: A033942).
The even version is A329888.
Positions of first appearances are A342768.
The sum of prime indices of a(n) is A346699(n), non-reverse: A346697.
The non-reverse version is A346703.
The even non-reverse version is A346704.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum, reverse A344612.
A209281 (shifted) adds up the odd bisection of standard compositions.
A316524 gives the alternating sum of prime indices, reverse A344616.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344606 counts alternating permutations of prime indices.
A344617 gives the sign of the alternating sum of prime indices.
A346700 gives the sum of the even bisection of reversed prime indices.
Cf. A025047, A027187, A027193, A053738, A106356, A277103, A341446, A344653, A345957, A345958, A345959, A346698, A346702.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@First/@Partition[Append[Reverse[primeMS[n]],0],2],{n,100}]

a(n) * A329888(n) = n.

A056239(a(n)) = A346699(n).

A346705 The a(n)-th composition in standard order is the even bisection of the n-th composition in standard order.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 4, 2, 1, 3, 0, 1, 2, 1, 4, 2, 1, 3, 8, 4, 2, 5, 1, 3, 6, 3, 0, 1, 2, 1, 4, 2, 1, 3, 8, 4, 2, 5, 1, 3, 6, 3, 16, 8, 4, 9, 2, 5, 10, 5, 1, 3, 6, 3, 12, 6, 3, 7, 0, 1, 2, 1, 4, 2, 1, 3, 8, 4, 2, 5, 1, 3, 6, 3, 16, 8, 4, 9, 2, 5

Offset: 0

Gus Wiseman, Aug 19 2021

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.

a(n) is the row number in A066099 of the even bisection (even-indexed terms) of the n-th row of A066099.

			Composition number 741 in standard order is (2,1,1,3,2,1), with even bisection (1,3,1), which is composition number 25 in standard order, so a(741) = 25.

Length of the a(n)-th standard composition is A000120(n)/2 rounded down.
Positions of first appearances appear to be A088698, sorted: A277335.
The version for reversed prime indices appears to be A329888, sums A346700.
Sum of the a(n)-th standard composition is A346633.
An unordered reverse version for odd bisection is A346701, sums A346699.
The version for odd bisection is A346702, sums A209281(n+1).
An unordered version for odd bisection is A346703, sums A346697.
An unordered version is A346704, sums A346698.
A011782 counts compositions.
A029837 gives length of binary expansion, or sometimes A070939.
A066099 lists compositions in standard order.
A097805 counts compositions by alternating sum.
Cf. A000302, A025047, A088218, A124754, A124767, A228351, A290258, A290259, A344618, A345197.

Table[Total[2^Accumulate[Reverse[Last/@Partition[ Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse,2]]]]/2,{n,0,100}]

A029837(a(n)) = A346633(n).

cp's OEIS Frontend

A344651 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with alternating sum k, with k ranging from n mod 2 to n in steps of 2.

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A209281 Start with first run [0,1] then, for n >= 2, the n-th run has length 2^n and is the concatenation of [a(1),a(2),...,a(2^n/2)] and [n-a(1),n-a(2),...,n-a(2^n/2)].

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A346700 Sum of the even bisection (even-indexed parts) of the integer partition with Heinz number n.

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A329888 a(n) = A329900(A329602(n)); Heinz number of the even bisection (even-indexed parts) of the integer partition with Heinz number n.

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A346701 Heinz number of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

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A346705 The a(n)-th composition in standard order is the even bisection of the n-th composition in standard order.

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