Displaying 51-60 of 106 results found.
page
1
2
3
4
5
6
7
8
9
10
11
Product of primes at even positions in the weakly increasing list (with multiplicity) of prime factors of n.
+10
17
1, 1, 1, 2, 1, 3, 1, 2, 3, 5, 1, 2, 1, 7, 5, 4, 1, 3, 1, 2, 7, 11, 1, 6, 5, 13, 3, 2, 1, 3, 1, 4, 11, 17, 7, 6, 1, 19, 13, 10, 1, 3, 1, 2, 3, 23, 1, 4, 7, 5, 17, 2, 1, 9, 11, 14, 19, 29, 1, 10, 1, 31, 3, 8, 13, 3, 1, 2, 23, 5, 1, 6, 1, 37, 5, 2, 11, 3, 1, 4, 9
EXAMPLE
The prime factors of 108 are (2,2,3,3,3), with even bisection (2,3), with product 6, so a(108) = 6.
The prime factors of 720 are (2,2,2,2,3,3,5), with even bisection (2,2,3), with product 12, so a(720) = 12.
MAPLE
f:= proc(n) local F, i;
F:= ifactors(n)[2];
F:= sort(map(t -> t[1]$t[2], F));
mul(F[i], i=2..nops(F), 2)
end proc:
MATHEMATICA
Table[Times@@Last/@Partition[Flatten[Apply[ConstantArray, FactorInteger[n], {1}]], 2], {n, 100}]
CROSSREFS
Positions of first appearances are A129597. The sum of prime indices of a(n) is A346698(n). A001221 counts distinct prime factors.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344606 counts alternating permutations of prime indices.
A344617 gives the sign of the alternating sum of prime indices.
A346633 adds up the even bisection of standard compositions.
Cf. A026424, A035363, A209281, A236913, A342768, A344653, A345957, A345958, A345960, A345961, A345962.
Numbers whose multiset of prime indices has alternating product <= 1.
+10
17
1, 2, 4, 6, 8, 9, 10, 14, 15, 16, 18, 21, 22, 24, 25, 26, 32, 33, 34, 35, 36, 38, 39, 40, 46, 49, 50, 51, 54, 55, 56, 57, 58, 60, 62, 64, 65, 69, 72, 74, 77, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 98, 100, 104, 106, 111, 115, 118, 119, 121, 122
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 alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
Also Heinz numbers integer partitions with reverse-alternating product <= 1, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Also numbers whose multiset of prime indices has alternating sum <= 1.
EXAMPLE
The initial terms and their prime indices:
1: {} 26: {1,6} 56: {1,1,1,4}
2: {1} 32: {1,1,1,1,1} 57: {2,8}
4: {1,1} 33: {2,5} 58: {1,10}
6: {1,2} 34: {1,7} 60: {1,1,2,3}
8: {1,1,1} 35: {3,4} 62: {1,11}
9: {2,2} 36: {1,1,2,2} 64: {1,1,1,1,1,1}
10: {1,3} 38: {1,8} 65: {3,6}
14: {1,4} 39: {2,6} 69: {2,9}
15: {2,3} 40: {1,1,1,3} 72: {1,1,1,2,2}
16: {1,1,1,1} 46: {1,9} 74: {1,12}
18: {1,2,2} 49: {4,4} 77: {4,5}
21: {2,4} 50: {1,3,3} 81: {2,2,2,2}
22: {1,5} 51: {2,7} 82: {1,13}
24: {1,1,1,2} 54: {1,2,2,2} 84: {1,1,2,4}
25: {3,3} 55: {3,5} 85: {3,7}
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], altprod[primeMS[#]]<=1&]
CROSSREFS
The additive version (alternating sum <= 0) is A028260. Allowing any alternating product < 1 gives A119899. Factorizations of this type are counted by A339846, complement A339890. Allowing any alternating product >= 1 gives A344609, multiplicative A347456. Partitions of this type are counted by A347443. Allowing any integer alternating product gives A347454, reciprocal A347451. A236913 counts partitions of 2n with reverse-alternating sum <= 0.
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 lists Heinz numbers of partitions with integer alternating product.
Numbers whose multiset of prime indices has integer alternating product.
+10
17
1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 25, 27, 28, 29, 31, 32, 36, 37, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 108, 109, 112, 113
COMMENTS
First differs from A265640 in having 42. 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)).
Also Heinz numbers of partitions with integer reverse-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: {} 20: {1,1,3} 47: {15}
2: {1} 23: {9} 48: {1,1,1,1,2}
3: {2} 25: {3,3} 49: {4,4}
4: {1,1} 27: {2,2,2} 50: {1,3,3}
5: {3} 28: {1,1,4} 52: {1,1,6}
7: {4} 29: {10} 53: {16}
8: {1,1,1} 31: {11} 59: {17}
9: {2,2} 32: {1,1,1,1,1} 61: {18}
11: {5} 36: {1,1,2,2} 63: {2,2,4}
12: {1,1,2} 37: {12} 64: {1,1,1,1,1,1}
13: {6} 41: {13} 67: {19}
16: {1,1,1,1} 42: {1,2,4} 68: {1,1,7}
17: {7} 43: {14} 71: {20}
18: {1,2,2} 44: {1,1,5} 72: {1,1,1,2,2}
19: {8} 45: {2,2,3} 73: {21}
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[altprod[primeMS[#]]]&]
CROSSREFS
Allowing any alternating product >= 1 gives A344609, multiplicative A347456. Factorizations of this type are counted by A347437. Allowing any alternating product <= 1 gives A347450. The version for reversed prime indices is A347457, complement A347455. 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.
A347461 counts possible alternating products of partitions.
A347462 counts possible reverse-alternating products of partitions.
Cf. A001105, A001222, A028982, A119620, A236913, A316523, A344653, A346703, A346704, A347443, A347439.
Number of distinct possible reverse-alternating products of integer partitions of n.
+10
17
1, 1, 2, 3, 4, 6, 8, 11, 13, 17, 22, 28, 33, 42, 51, 59, 69, 84, 100, 117, 137, 163, 191, 222, 256, 290, 332, 378, 429, 489, 564, 643, 729, 819, 929, 1040, 1167, 1313, 1473, 1647, 1845, 2045, 2272, 2521, 2785, 3076, 3398, 3744, 4115, 4548, 5010, 5524, 6086
COMMENTS
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.
EXAMPLE
Partitions representing each of the a(7) = 11 reverse-alternating products:
(7) -> 7
(61) -> 1/6
(52) -> 2/5
(511) -> 5
(43) -> 3/4
(421) -> 2
(4111) -> 1/4
(331) -> 1
(322) -> 3
(3211) -> 2/3
(2221) -> 1/2
MATHEMATICA
revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Union[revaltprod/@IntegerPartitions[n]]], {n, 0, 30}]
CROSSREFS
The version for non-reverse alternating sum instead of product is A004526. The non-reverse version is A347461. A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A122768 counts distinct submultisets of partitions.
A126796 counts complete partitions.
A293627 counts knapsack factorizations by sum.
A301957 counts distinct subset-products of prime indices.
A304793 counts distinct positive subset-sums of prime indices.
Cf. A000070, A001055, A002033, A002219, A028983, A119620, A325768, A345926, A347443, A347444, A347445, A347446.
Number of permutations of the prime indices of n with all equal parts contiguous and none appearing more than twice.
+10
16
1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 2, 1, 2, 2, 0, 1, 2, 1, 2, 2, 2, 1, 0, 1, 2, 0, 2, 1, 6, 1, 0, 2, 2, 2, 2, 1, 2, 2, 0, 1, 6, 1, 2, 2, 2, 1, 0, 1, 2, 2, 2, 1, 0, 2, 0, 2, 2, 1, 6, 1, 2, 2, 0, 2, 6, 1, 2, 2, 6, 1, 0, 1, 2, 2, 2, 2, 6, 1, 0, 0, 2, 1, 6, 2, 2, 2
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.
FORMULA
a(n) = A001221(n)! if n is cubefree, otherwise 0.
EXAMPLE
The a(90) = 6 permutations are (1,2,2,3), (1,3,2,2), (2,2,1,3), (2,2,3,1), (3,1,2,2), (3,2,2,1).
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}]
CROSSREFS
Permutations of prime indices are counted by A008480. Unsorted prime signature is A124010. Sorted prime signature is A118914. Permutations of prime indices with equal parts contiguous are A333175. STC-numbers of permutations of prime indices are A333221. (1,2,1) and (2,1,2)-avoiding permutations of prime indices are A333175. Numbers whose prime indices are inseparable are A335448. (1,2,1) or (2,1,2)-matching permutations of prime indices are A335460. (1,2,1) and (2,1,2)-matching permutations of prime indices are A335462. Strict permutations of prime indices are counted by A335489.
Number of integer partitions of n with no alternating permutation covering an initial interval of positive integers.
+10
16
0, 0, 1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 8, 10, 11, 15, 16, 18, 23, 27, 30, 35, 41, 47, 54, 62, 71, 82, 92, 103, 121, 137, 151, 173, 195, 220, 248, 277, 311, 350, 393, 435, 488, 546, 605, 678, 754, 835, 928, 1029, 1141, 1267, 1400, 1544, 1712, 1891, 2081, 2298, 2533, 2785, 3068
COMMENTS
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,3,2,2,2,2,1) has no alternating permutations, even though it has anti-run permutations (2,3,2,3,2,1,2), (2,3,2,1,2,3,2), and (2,1,2,3,2,3,2).
Sequences covering an initial interval (patterns) are counted by A000670 and ranked by A333217.
EXAMPLE
The a(2) = 1 through a(10) = 6 partitions:
11 111 1111 2111 21111 2221 221111 22221 32221
11111 111111 211111 2111111 321111 222211
1111111 11111111 2211111 3211111
21111111 22111111
111111111 211111111
1111111111
MATHEMATICA
normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[IntegerPartitions[n], normQ[#]&&Select[Permutations[#], wigQ[#]&]=={}&]], {n, 0, 15}]
PROG
(PARI) P(n, m)={Vec(1/prod(k=1, m, 1-y*x^k, 1+O(x*x^n)))}
a(n) = {(n >= 2) + sum(k=2, (sqrtint(8*n+1)-1)\2, my(r=n-binomial(k+1, 2), v=P(r, k)); sum(i=1, min(k, 2*r\k), sum(j=k-1, (2*r-(k-1)*(i-1))\(i+1), my(p=(j+k+(i==1||i==k))\2); if(p*i<=r, polcoef(v[r-p*i+1], j-p)) )))} \\ Andrew Howroyd, Jan 31 2024
CROSSREFS
The complement in covering partitions is counted by A345163. A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A344604 counts alternating compositions with twins.
A344605 counts alternating patterns with twins.
A345164 counts alternating permutations of prime indices.
A345170 counts partitions with a alternating permutation, ranked by A345172.
Cf. A000070, A103919, A335126, A344614, A344615, A344653, A344654, A344740, A344742, A345167, A345168, A345192, A348609.
Number of strict integer partitions of n with no difference -2.
+10
16
1, 1, 1, 2, 1, 3, 3, 4, 4, 7, 7, 8, 11, 12, 15, 18, 21, 23, 31, 32, 40, 45, 54, 59, 73, 78, 94, 106, 122, 136, 161, 177, 203, 231, 259, 293, 334, 372, 417, 476, 525, 592, 663, 742, 821, 931, 1020, 1147, 1271, 1416, 1558, 1752, 1916, 2137, 2357, 2613, 2867
EXAMPLE
The a(1) = 1 through a(12) = 11 partitions (A..C = 10..12):
1 2 3 4 5 6 7 8 9 A B C
21 32 51 43 62 54 73 65 84
41 321 52 71 63 82 74 93
61 521 72 91 83 A2
81 541 92 B1
432 721 A1 543
621 4321 632 651
821 732
741
921
6321
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], FreeQ[Differences[#], 0|-2]&]], {n, 0, 30}]
CROSSREFS
The version for no difference 0 is A000009. The version for subsets of prescribed maximum is A005314. A027187 counts partitions of even length.
Cf. A000929, A003000, A018819, A040039, A045690, A045691, A154402, A303362, A323094, A342095, A342097.
Product of primes at odd positions in the weakly increasing list (with multiplicity) of prime factors of n.
+10
15
1, 2, 3, 2, 5, 2, 7, 4, 3, 2, 11, 6, 13, 2, 3, 4, 17, 6, 19, 10, 3, 2, 23, 4, 5, 2, 9, 14, 29, 10, 31, 8, 3, 2, 5, 6, 37, 2, 3, 4, 41, 14, 43, 22, 15, 2, 47, 12, 7, 10, 3, 26, 53, 6, 5, 4, 3, 2, 59, 6, 61, 2, 21, 8, 5, 22, 67, 34, 3, 14, 71, 12, 73, 2, 15, 38
EXAMPLE
The prime factors of 108 are (2,2,3,3,3), with odd bisection (2,3,3), with product 18, so a(108) = 18.
The prime factors of 720 are (2,2,2,2,3,3,5), with odd bisection (2,2,3,5), with product 60, so a(720) = 60.
MATHEMATICA
Table[Times@@First/@Partition[Append[Flatten[Apply[ConstantArray, FactorInteger[n], {1}]], 0], 2], {n, 100}]
CROSSREFS
The even reverse version appears to be A329888. Positions of first appearances are A342768. A001221 counts distinct prime factors.
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).
A344606 counts alternating permutations of prime indices.
A344617 gives the sign of the alternating sum of prime indices.
A346633 adds up the even bisection of standard compositions.
A346698 gives the sum of the even bisection of prime indices.
A346700 gives the sum of the even bisection of reversed prime indices.
Cf. A025047, A027187, A027193, A053738, A097805, A106356, A341446, A344653, A345957, A345958, A345959.
Number of non-alternating compositions of n, excluding twins (x,x).
+10
15
0, 0, 0, 1, 3, 9, 19, 45, 98, 208, 436, 906, 1861, 3803, 7731, 15659, 31628, 63747, 128257, 257722, 517338, 1037652, 2079983, 4167325, 8346203, 16710572, 33449694, 66944254, 133959020, 268028868, 536231902, 1072737537, 2145905284, 4292486690, 8586035992
COMMENTS
First differs from A348382 at a(6) = 19, A348382(6) = 17. The two non-alternating non-twin compositions of 6 that are not an anti-run are (1,2,3) and (3,2,1). A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). Alternating permutations of multisets are a generalization of alternating or up-down permutations of {1..n}.
EXAMPLE
The a(3) = 1 through a(6) = 19 compositions:
(1,1,1) (1,1,2) (1,1,3) (1,1,4)
(2,1,1) (1,2,2) (1,2,3)
(1,1,1,1) (2,2,1) (2,2,2)
(3,1,1) (3,2,1)
(1,1,1,2) (4,1,1)
(1,1,2,1) (1,1,1,3)
(1,2,1,1) (1,1,2,2)
(2,1,1,1) (1,1,3,1)
(1,1,1,1,1) (1,2,2,1)
(1,3,1,1)
(2,1,1,2)
(2,2,1,1)
(3,1,1,1)
(1,1,1,1,2)
(1,1,1,2,1)
(1,1,2,1,1)
(1,2,1,1,1)
(2,1,1,1,1)
(1,1,1,1,1,1)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MatchQ[#, {___, x_, y_, z_, ___}/; x<=y<=z||x>=y>=z]&]], {n, 0, 15}]
CROSSREFS
Non-twin compositions are counted by A051049. The complement is counted by A344604. A001250 counts alternating permutations.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A344614 counts compositions avoiding (1,2,3) and (3,2,1) adjacent.
A345165 = partitions with no alternating permutations, ranked by A345171.
A345170 = partitions with an alternating permutation, ranked by A345172.
Cf. A005649, A178470, A238279, A333755, A335126, A344653, A344740, A345166, A345169, A345173, A348379, A348381.
Number of ways to choose a divisor of each prime index of n (taken in weakly increasing order) such that the result is weakly increasing.
+10
15
1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2, 4, 3, 3, 1, 2, 3, 4, 2, 5, 2, 3, 2, 3, 4, 4, 3, 4, 3, 2, 1, 3, 2, 4, 3, 6, 4, 7, 2, 2, 5, 4, 2, 4, 3, 4, 2, 6, 3, 3, 4, 5, 4, 3, 3, 7, 4, 2, 3, 6, 2, 7, 1, 6, 3, 2, 2, 5, 4, 6, 3, 4, 6, 4, 4, 4, 7, 4, 2, 5, 2, 2, 5, 3, 4, 7
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.
EXAMPLE
The a(15) = 3 ways are: (1,1), (1,3), (2,3).
The a(18) = 3 ways are: (1,1,1), (1,1,2), (1,2,2).
The a(2) = 1 through a(19) = 4 ways:
1 1 11 1 11 1 111 11 11 1 111 1 11 11 1111 1 111 1
2 3 12 2 12 13 5 112 2 12 13 7 112 2
4 22 3 14 23 122 4
6 8
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Select[Tuples[Divisors/@primeMS[n]], LessEqual@@#&]], {n, 100}]
CROSSREFS
Choosing a multiset instead of sequence gives A355733, firsts A355734. Positions of first appearances are A355736. A003963 multiplies together the prime indices of n.
A061395 selects the maximum prime index.
A120383 lists numbers divisible by all of their prime indices.
A324850 lists numbers divisible by the product of their prime indices.
Cf. A000720, A076610, A316524, A335433, A335448, A340827, A340852, A344616, A355737, A355739, A355740, A355742.
Search completed in 0.065 seconds
|