Displaying 61-70 of 106 results found.
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Number of inseparable factorizations of n into factors > 1.
+10
14
0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0
COMMENTS
A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
EXAMPLE
The a(n) factorizations for n = 4, 16, 96, 144, 64, 192:
2*2 4*4 2*2*2*12 12*12 8*8 3*4*4*4
2*2*2*2 2*2*2*2*6 2*2*2*18 4*4*4 2*2*2*24
2*2*2*2*2*3 2*2*2*2*9 2*2*2*8 2*2*2*2*12
2*2*2*2*3*3 2*2*2*2*4 2*2*2*2*2*6
2*2*2*2*2*2 2*2*2*2*3*4
2*2*2*2*2*2*3
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], Select[Permutations[#], !MatchQ[#, {___, x_, x_, ___}]&]=={}&]], {n, 100}]
CROSSREFS
The version for partitions is A325535. The version for multisets with prescribed multiplicities is A335126. Separable partitions are ranked by A335433. Inseparable partitions are ranked by A335448. Anti-run permutations of prime indices are A335452. Patterns contiguously matched by compositions are A335457. Cf. A106351, A292884, A295370, A333628, A333755, A335463, A335125, A335127, A335407, A335474, A335516, A335838.
Sum of even-indexed parts (even bisection) of the n-th composition in standard order.
+10
14
0, 0, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 3, 2, 1, 2, 0, 1, 2, 1, 3, 2, 1, 2, 4, 3, 2, 3, 1, 2, 3, 2, 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, 0, 1, 2, 1, 3, 2, 1, 2, 4, 3, 2, 3, 1, 2, 3, 2, 5, 4, 3, 4, 2, 3, 4
COMMENTS
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. This gives a bijective correspondence between nonnegative integers and integer compositions.
EXAMPLE
Composition number 741 in standard order is (2,1,1,3,2,1), so a(741) = 1 + 3 + 1 = 5.
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Total[Last/@Partition[Append[stc[n], 0], 2]], {n, 0, 100}]
CROSSREFS
Including odd-indexed parts gives A029837. Subtracting from the odd version gives A124754. The odd-indexed version is A209281(n+1). A097805 counts compositions by alternating (or reverse-alternating) sum.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
Sum of the even bisection (even-indexed parts) of the integer partition with Heinz number n.
+10
14
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
COMMENTS
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.
EXAMPLE
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.
MATHEMATICA
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}]
PROG
(PARI) 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
CROSSREFS
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. A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344606 counts alternating permutations of prime indices.
Cf. A000070, A025047, A124754, A209281, A329888, A341446, A344617, A344653, A345957, A345958, A346701, A346702, A346704.
Heinz numbers of odd-length integer partitions with integer alternating (or reverse-alternating) product.
+10
14
2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, 27, 28, 29, 31, 32, 37, 41, 42, 43, 44, 45, 47, 48, 50, 52, 53, 59, 61, 63, 67, 68, 71, 72, 73, 75, 76, 78, 79, 80, 83, 89, 92, 97, 98, 99, 101, 103, 107, 108, 109, 112, 113, 114, 116, 117, 124, 125, 127, 128, 130
COMMENTS
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 odd length and integer alternating product, where a prime index of n is a number m such that prime(m) divides n.
EXAMPLE
The terms and their prime indices begin:
2: {1} 29: {10} 61: {18}
3: {2} 31: {11} 63: {2,2,4}
5: {3} 32: {1,1,1,1,1} 67: {19}
7: {4} 37: {12} 68: {1,1,7}
8: {1,1,1} 41: {13} 71: {20}
11: {5} 42: {1,2,4} 72: {1,1,1,2,2}
12: {1,1,2} 43: {14} 73: {21}
13: {6} 44: {1,1,5} 75: {2,3,3}
17: {7} 45: {2,2,3} 76: {1,1,8}
18: {1,2,2} 47: {15} 78: {1,2,6}
19: {8} 48: {1,1,1,1,2} 79: {22}
20: {1,1,3} 50: {1,3,3} 80: {1,1,1,1,3}
23: {9} 52: {1,1,6} 83: {23}
27: {2,2,2} 53: {16} 89: {24}
28: {1,1,4} 59: {17} 92: {1,1,9}
MATHEMATICA
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], OddQ[PrimeOmega[#]]&&IntegerQ[altprod[primeMS[#]]]&]
CROSSREFS
Allowing any alternating product <= 1 gives A001105. Allowing any alternating product gives A026424. Factorizations of this type are counted by A347441. These partitions are counted by A347444. Allowing any alternating product > 1 gives A347465. A027193 counts odd-length partitions.
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.
A347446 counts partitions with integer alternating product.
A347457 ranks partitions with integer alt product, complement A347455.
A347461 counts possible alternating products of partitions.
A347462 counts possible reverse-alternating products of partitions.
Cf. A001222, A028260, A028982, A028983, A339890, A344617, A344653, A345958, A346703, A346704, A347437, A347443, A347450, A347451.
Number of separable multisets of size n covering an initial interval of positive integers.
+10
13
1, 1, 1, 3, 5, 13, 24, 56, 108, 236, 464, 976, 1936, 3984, 7936, 16128, 32192, 64960, 129792, 260864, 521472, 1045760, 2091008, 4188160, 8375296, 16763904, 33525760, 67080192, 134156288, 268374016, 536739840, 1073610752, 2147205120, 4294688768, 8589344768, 17179279360, 34358493184
COMMENTS
A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
Alternatively, a multiset is separable if its greatest multiplicity is greater than the sum of its remaining multiplicities plus one. Hence a(n) is the number of compositions of n whose greatest part is at most one more than the sum of its other parts. For example, the a(1) = 1 through a(5) = 13 compositions are:
(1) (11) (12) (22) (23)
(21) (112) (32)
(111) (121) (113)
(211) (122)
(1111) (131)
(212)
(221)
(311)
(1112)
(1121)
(1211)
(2111)
(11111)
FORMULA
a(n) = 2^(n-1) - (floor(n/2)+1) * 2^(floor(n/2)-2) for n >= 2. - David A. Corneth, Jul 09 2020 a(n) = 2*a(n-1) + 4*a(n-2) - 8*a(n-3) - 4*a(n-4) + 8*a(n-5) for n > 6.
G.f.: (x - 1)*(2*x^5 + 7*x^4 - 5*x^2 + 1)/((2*x - 1)*(2*x^2 - 1)^2). (End)
EXAMPLE
The a(1) = 1 through a(5) = 13 separable multisets:
{1} {1,2} {1,1,2} {1,1,2,2} {1,1,1,2,2}
{1,2,2} {1,1,2,3} {1,1,1,2,3}
{1,2,3} {1,2,2,3} {1,1,2,2,2}
{1,2,3,3} {1,1,2,2,3}
{1,2,3,4} {1,1,2,3,3}
{1,1,2,3,4}
{1,2,2,2,3}
{1,2,2,3,3}
{1,2,2,3,4}
{1,2,3,3,3}
{1,2,3,3,4}
{1,2,3,4,4}
{1,2,3,4,5}
MATHEMATICA
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
sepQ[m_]:=Select[Permutations[m], !MatchQ[#, {___, x_, x_, ___}]&]!={};
Table[Length[Select[allnorm[n], sepQ]], {n, 0, 5}]
(* or *)
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n], With[{mx=Max@@#}, mx<=1+Total[DeleteCases[#, mx, {1}, 1]]]&]], {n, 0, 15}] (* or *)
CoefficientList[Series[(x - 1) (2 x^5 + 7 x^4 - 5 x^2 + 1)/((2 x - 1) (2 x^2 - 1)^2), {x, 0, 36}], x] (* Michael De Vlieger, Apr 07 2021 *)
CROSSREFS
The inseparable version is A336102. The strong (weakly decreasing multiplicities) case is A336106. Sequences covering an initial interval are A000670. Inseparable partitions are A325535. Inseparable factorizations are A333487. Anti-run compositions are ranked by A333489. Heinz numbers of inseparable partitions are A335448. Cf. A001792, A019472, A025065, A049610, A052841, A106351, A269134, A292884, A335126, A335433, A335452.
Sum of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.
+10
13
0, 1, 2, 1, 3, 2, 4, 2, 2, 3, 5, 3, 6, 4, 3, 2, 7, 3, 8, 4, 4, 5, 9, 3, 3, 6, 4, 5, 10, 4, 11, 3, 5, 7, 4, 3, 12, 8, 6, 4, 13, 5, 14, 6, 5, 9, 15, 4, 4, 4, 7, 7, 16, 4, 5, 5, 8, 10, 17, 4, 18, 11, 6, 3, 6, 6, 19, 8, 9, 5, 20, 4, 21, 12, 5, 9, 5, 7, 22, 5, 4
COMMENTS
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.
EXAMPLE
The partition with Heinz number 1100 is (5,3,3,1,1), so a(1100) = 5 + 3 + 1 = 9.
The partition with Heinz number 2100 is (4,3,3,2,1,1), so a(2100) = 4 + 3 + 1 = 8.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Total[First/@Partition[Append[Reverse[primeMS[n]], 0], 2]], {n, 100}]
CROSSREFS
The version for standard compositions is A209281(n+1) (even: A346633). The non-reverse version (multisets instead of partitions) is A346697. The even non-reverse version is A346698. A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344606 counts alternating permutations of prime indices.
Cf. A000070, A025047, A120452, A124754, A329888, A341446, A344617, A345957, A345958, A346701, A346703.
Number of odd-length integer partitions of n with integer alternating product.
+10
13
0, 1, 1, 2, 2, 4, 4, 8, 7, 14, 13, 24, 21, 40, 35, 62, 55, 99, 85, 151, 128, 224, 195, 331, 283, 481, 416, 690, 593, 980, 844, 1379, 1189, 1918, 1665, 2643, 2292, 3630, 3161, 4920, 4299, 6659, 5833, 8931, 7851, 11905, 10526, 15805, 13987, 20872, 18560, 27398
COMMENTS
We define the alternating product of a sequence (y_1, ... ,y_k) to be the Product_i y_i^((-1)^(i-1)).
The reverse version (integer reverse-alternating product) is the same.
EXAMPLE
The a(1) = 1 through a(9) = 14 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(111) (211) (221) (222) (322) (332) (333)
(311) (411) (331) (422) (441)
(11111) (21111) (421) (611) (522)
(511) (22211) (621)
(22111) (41111) (711)
(31111) (2111111) (22221)
(1111111) (32211)
(33111)
(42111)
(51111)
(2211111)
(3111111)
(111111111)
MATHEMATICA
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&IntegerQ[altprod[#]]&]], {n, 0, 30}]
CROSSREFS
Allowing any alternating product gives A027193. The multiplicative version (factorizations) is A347441. Allowing any length and alternating product > 1 gives A347448. Allowing any reverse-alternating product > 1 gives A347449. The even-length instead of odd-length version is A347704. A025047 counts wiggly compositions.
A026424 lists numbers with odd bigomega.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1, ranked by A028982.
A339890 counts odd-length factorizations.
A347437 counts factorizations with integer alternating product.
A347461 counts possible alternating products of partitions.
Cf. A000070, A236559, A236913, A236914, A304620, A344654, A347439, A347442, A347456, A347457, A347460, A347462, A347463.
Numbers whose multiset of prime indices has integer reciprocal alternating product.
+10
13
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
COMMENTS
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).
EXAMPLE
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}
MATHEMATICA
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[#]]]&]
CROSSREFS
The version for reversed prime indices is A028982, counted by A119620. 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. 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.
Number of non-weakly alternating permutations of the multiset of prime factors of n.
+10
13
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0
COMMENTS
The first odd term is a(144) = 7, whose non-weakly alternating permutations are shown in the example below.
We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. Then a sequence is alternating in the sense of A025047 iff it is a weakly alternating anti-run. For n > 1, the multiset of prime factors of n is row n of A027746. The prime indices A112798 can also be used.
EXAMPLE
The following are the weakly alternating permutations for selected n.
n = 30 60 72 120 144 180
---------------------------------------------
235 2235 22332 22235 222332 22353
532 2352 23223 22352 223223 23235
2532 23322 22532 223322 23325
3225 32232 23225 232232 23523
5223 23522 233222 23532
5322 25223 322223 25323
25322 322322 32235
32252 32253
52232 32352
53222 32532
33225
35223
35322
52233
52332
53223
53232
MATHEMATICA
whkQ[y_]:=And@@Table[If[EvenQ[m], y[[m]]<=y[[m+1]], y[[m]]>=y[[m+1]]], {m, 1, Length[y]-1}];
Table[Length[Select[Permutations[Flatten[ConstantArray@@@ FactorInteger[n]]], !whkQ[#]&&!whkQ[-#]&]], {n, 100}]
CROSSREFS
Counting all permutations of prime factors gives A008480. The complement is counted by A349056. The version counting ordered factorizations is A350139, complement A349059. Positions of nonzero terms are A350353. A335452 counts anti-run permutations of prime factors, complement A336107.
A345165 counts partitions w/o an alternating permutation, ranked by A345171.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A348379 counts factorizations with an alternating permutation.
Cf. A003242, A335433, A335448, A344614, A344652, A344653, A345173, A348613, A349798, A350252, A349800.
Number of permutations of the prime indices of n with at least one non-singleton run, or non-separations.
+10
12
0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 0, 4, 1, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 5, 1, 2, 0, 2, 0, 4, 0, 4, 0, 0, 0, 6, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 0, 9, 0, 0, 2, 2, 0, 0, 0, 5, 1, 0, 0, 6, 0, 0, 0
COMMENTS
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. A separation (or Carlitz composition) of a multiset is a permutation with no adjacent equal parts.
EXAMPLE
The a(n) non-separations for n = 12, 36, 60, 72, 180, 420:
(11) (112) (1122) (1123) (11122) (11223) (11234)
(211) (1221) (1132) (11212) (11232) (11243)
(2112) (2113) (11221) (11322) (11324)
(2211) (2311) (12112) (12213) (11342)
(3112) (12211) (12231) (11423)
(3211) (21112) (13122) (11432)
(21121) (13221) (21134)
(21211) (21123) (21143)
(22111) (21132) (23114)
(22113) (23411)
(22131) (24113)
(22311) (24311)
(23112) (31124)
(23211) (31142)
(31122) (32114)
(31221) (32411)
(32112) (34112)
(32211) (34211)
(41123)
(41132)
(42113)
(42311)
(43112)
(43211)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Select[Permutations[primeMS[n]], MatchQ[#, {___, x_, x_, ___}]&]], {n, 100}]
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