Displaying 1-10 of 32 results found.
Numbers that are not a product of elements of A304711.
+20
1
3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 42, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 78, 79, 81, 83, 87, 89, 91, 97, 101, 103, 105, 107, 109, 111, 113, 114, 115, 117, 121, 125, 126, 127, 129, 130, 131, 133, 137, 139, 147, 149
COMMENTS
A304711 lists numbers whose distinct prime indices are pairwise coprime.
The first term divisible by 4 is a(421) = 1092.
EXAMPLE
The sequence of terms together with their prime indices begins:
3: {2} 39: {2,6} 78: {1,2,6}
5: {3} 41: {13} 79: {22}
7: {4} 42: {1,2,4} 81: {2,2,2,2}
9: {2,2} 43: {14} 83: {23}
11: {5} 47: {15} 87: {2,10}
13: {6} 49: {4,4} 89: {24}
17: {7} 53: {16} 91: {4,6}
19: {8} 57: {2,8} 97: {25}
21: {2,4} 59: {17} 101: {26}
23: {9} 61: {18} 103: {27}
25: {3,3} 63: {2,2,4} 105: {2,3,4}
27: {2,2,2} 65: {3,6} 107: {28}
29: {10} 67: {19} 109: {29}
31: {11} 71: {20} 111: {2,12}
37: {12} 73: {21} 113: {30}
MATHEMATICA
nn=100;
dat=Select[Range[nn], CoprimeQ@@PrimePi/@First/@FactorInteger[#]&];
facsusing[s_, n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facsusing[Select[s, Divisible[n/d, #]&], n/d], Min@@#>=d&]], {d, Select[s, Divisible[n, #]&]}]];
Select[Range[nn], facsusing[dat, #]=={}&]
CROSSREFS
A302696 lists numbers with coprime prime indices.
A304711 lists numbers with coprime distinct prime indices.
Cf. A007916, A112798, A302569, A327516, A333228, A335238, A335239, A335240, A335241, A336424, A336568, A336736.
1, 2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 34, 35, 36, 38, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 77, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 106
COMMENTS
A304711 lists numbers whose distinct prime indices are pairwise coprime.
First differs from A304711 in having 84.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {} 28: {1,1,4} 52: {1,1,6}
2: {1} 30: {1,2,3} 54: {1,2,2,2}
4: {1,1} 32: {1,1,1,1,1} 55: {3,5}
6: {1,2} 33: {2,5} 56: {1,1,1,4}
8: {1,1,1} 34: {1,7} 58: {1,10}
10: {1,3} 35: {3,4} 60: {1,1,2,3}
12: {1,1,2} 36: {1,1,2,2} 62: {1,11}
14: {1,4} 38: {1,8} 64: {1,1,1,1,1,1}
15: {2,3} 40: {1,1,1,3} 66: {1,2,5}
16: {1,1,1,1} 44: {1,1,5} 68: {1,1,7}
18: {1,2,2} 45: {2,2,3} 69: {2,9}
20: {1,1,3} 46: {1,9} 70: {1,3,4}
22: {1,5} 48: {1,1,1,1,2} 72: {1,1,1,2,2}
24: {1,1,1,2} 50: {1,3,3} 74: {1,12}
26: {1,6} 51: {2,7} 75: {2,3,3}
MATHEMATICA
nn=100;
dat=Select[Range[nn], CoprimeQ@@PrimePi/@First/@FactorInteger[#]&];
facsusing[s_, n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facsusing[Select[s, Divisible[n/d, #]&], n/d], Min@@#>=d&]], {d, Select[s, Divisible[n, #]&]}]];
Select[Range[nn], facsusing[dat, #]!={}&]
CROSSREFS
A302696 lists numbers with coprime prime indices.
A304711 lists numbers with coprime distinct prime indices.
Cf. A001221, A007360, A007916, A056239, A112798, A302569, A327516, A333228, A335238, A335239, A335240, A336424, A336497, A336736.
Number of integer partitions of n that are empty, (1), or have at least two parts and these parts are pairwise coprime.
+10
53
1, 1, 1, 2, 3, 5, 6, 9, 11, 14, 17, 22, 26, 32, 37, 42, 50, 59, 69, 80, 91, 101, 115, 133, 152, 170, 190, 210, 235, 265, 300, 334, 366, 398, 441, 484, 541, 597, 648, 703, 770, 848, 935, 1022, 1102, 1184, 1281, 1406, 1534, 1661, 1789, 1916, 2062, 2244, 2435
COMMENTS
The Heinz numbers of these partitions are given by A302696. Note that the definition excludes partitions with repeated parts other than 1 (cf. A038348, A304709).
EXAMPLE
The a(1) = 1 through a(8) = 11 partitions:
(1) (11) (21) (31) (32) (51) (43) (53)
(111) (211) (41) (321) (52) (71)
(1111) (311) (411) (61) (431)
(2111) (3111) (511) (521)
(11111) (21111) (3211) (611)
(111111) (4111) (5111)
(31111) (32111)
(211111) (41111)
(1111111) (311111)
(2111111)
(11111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], #=={}||CoprimeQ@@#&]], {n, 0, 30}]
CROSSREFS
A000837 is the relatively prime instead of pairwise coprime version.
A101268 is the ordered version (with singletons).
A307719 counts these partitions of length 3.
A018783 counts partitions with a common divisor.
A328673 counts pairwise non-coprime partitions.
Squarefree numbers whose prime indices are pairwise indivisible. Heinz numbers of strict integer partitions with pairwise indivisible parts.
+10
33
1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 83, 85, 89, 91, 93, 95, 97, 101, 103, 107, 109, 113, 119, 123, 127, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163, 165, 167, 173
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
EXAMPLE
Sequence of entries together with their corresponding multiset multisystems (see A302242) begins: 1: {}
2: {{}}
3: {{1}}
5: {{2}}
7: {{1,1}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
19: {{1,1,1}}
23: {{2,2}}
29: {{1,3}}
31: {{5}}
33: {{1},{3}}
35: {{2},{1,1}}
MATHEMATICA
Select[Range[300], SquareFreeQ[#]&&Select[Tuples[PrimePi/@First/@FactorInteger[#], 2], UnsameQ@@#&&Divisible@@#&]==={}&]
CROSSREFS
Cf. A000009, A005117, A006126, A056239, A073576, A285572, A285573, A293606, A293993, A302696, A302796, A303362, A303365, A304711.
Number of ways to express the integer n as a product of its unitary divisors ( A034444).
+10
24
1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 5, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 2, 1, 2, 2, 2, 2, 5, 1, 2, 1, 2, 1, 5, 2, 2, 2, 2, 1, 5, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 5, 1, 2, 5
COMMENTS
Equivalently, a(n) is the number of ways to express the cyclic group Z_n as a direct sum of its Hall subgroups. A Hall subgroup of a finite group G is a subgroup whose order is coprime to its index.
a(n) is the number of ways to partition the set of distinct prime factors of n.
Also the number of singleton or pairwise coprime factorizations of n. - Gus Wiseman, Sep 24 2019
EXAMPLE
a(60) = 5 because we have: 60 = 4*3*5 = 4*15 = 3*20 = 5*12.
For n = 36, its unitary divisors are 1, 4, 9, 36. From these we obtain 36 either as 1*36 or 4*9, thus a(36) = 2. - Antti Karttunen, Oct 21 2017
MAPLE
map(combinat:-bell @ nops @ numtheory:-factorset, [$1..100]); # Robert Israel, Jul 09 2015
MATHEMATICA
Table[BellB[PrimeNu[n]], {n, 1, 75}]
(* second program *)
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], Length[#]==1||CoprimeQ@@#&]], {n, 100}] (* Gus Wiseman, Sep 24 2019 *)
PROG
(PARI) a(n) = my(t=omega(n), x='x, m=contfracpnqn(matrix(2, t\2, y, z, if( y==1, -z*x^2, 1 - (z+1)*x)))); polcoeff(1/(1 - x + m[2, 1]/m[1, 1]) + O(x^(t+1)), t) \\ Charles R Greathouse IV, Jun 30 2017
CROSSREFS
Differs from A050320 for the first time at n=36. Differs from A354870 for the first time at n=210, where a(210) = 15, while A354870(210) = 12. Related classes of factorizations:
- Constant or distinct factors coprime: A327399- Constant or relatively prime: A327400- Distinct factors coprime: A327695
Number of integer partitions of n whose distinct parts are pairwise coprime.
+10
21
1, 1, 2, 3, 6, 7, 13, 16, 23, 29, 42, 49, 69, 83, 102, 126, 161, 191, 239, 281, 336, 402, 484, 566, 672, 787, 919, 1067, 1251, 1449, 1684, 1934, 2223, 2554, 2920, 3341, 3821, 4344, 4928, 5586, 6334, 7163, 8091, 9100, 10228, 11492, 12902, 14449, 16167, 18058
COMMENTS
Two parts are coprime if they have no common divisor greater than 1. For partitions of length 1 note that (1) is coprime but (x) is not coprime for x > 1.
EXAMPLE
The a(6) = 7 integer partitions of 6 whose distinct parts are pairwise coprime are (51), (411), (321), (3111), (2211), (21111), (111111).
MATHEMATICA
Table[Select[IntegerPartitions[n], CoprimeQ@@Union[#]&]//Length, {n, 20}]
PROG
(PARI)
lista(nn)={local(Cache=Map());
my(excl=vector(nn, n, sum(i=1, n-1, if(gcd(i, n)>1, 2^(n-i)))));
my(c(n, m, b)=
if(n==0, 1,
while(m>n || bittest(b, 0), m--; b>>=1);
my(hk=[n, m, b], z);
if(!mapisdefined(Cache, hk, &z),
z = if(m, self()(n, m-1, b>>1) + self()(n-m, m, bitor(b, excl[m])), 0);
mapput(Cache, hk, z)); z));
my(a(n)=c(n, n, 0) + 1 - numdiv(n));
for(n=1, nn, print1(a(n), ", "))
CROSSREFS
Cf. A000005, A007359, A007360, A018783, A051424, A078374, A101268, A289508, A289509, A302569, A302696, A302698, A302796, A302797, A304711, A304712.
Number of ordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is not considered coprime unless it is (1).
+10
16
0, 0, 0, 1, 3, 6, 9, 9, 18, 15, 24, 21, 42, 24, 51, 30, 54, 42, 93, 45, 102, 54, 99, 69, 162, 66, 150, 87, 168, 96, 264, 93, 228, 120, 246, 126, 336, 132, 315, 168, 342, 162, 486, 165, 420, 216, 411, 213, 618, 207, 558, 258, 540, 258, 783, 264, 654, 324, 660
EXAMPLE
The a(3) = 1 through a(8) = 18 triples:
(1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5) (1,1,6)
(1,2,1) (1,2,2) (1,2,3) (1,3,3) (1,2,5)
(2,1,1) (1,3,1) (1,3,2) (1,5,1) (1,3,4)
(2,1,2) (1,4,1) (2,2,3) (1,4,3)
(2,2,1) (2,1,3) (2,3,2) (1,5,2)
(3,1,1) (2,3,1) (3,1,3) (1,6,1)
(3,1,2) (3,2,2) (2,1,5)
(3,2,1) (3,3,1) (2,3,3)
(4,1,1) (5,1,1) (2,5,1)
(3,1,4)
(3,2,3)
(3,3,2)
(3,4,1)
(4,1,3)
(4,3,1)
(5,1,2)
(5,2,1)
(6,1,1)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n, {3}], CoprimeQ@@Union[#]&]], {n, 0, 100}]
CROSSREFS
A337461 is the strict case except for any number of 1's.
A337602 considers all singletons to be coprime.
A000217(n - 2) counts 3-part compositions.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A304711 ranks partitions whose distinct parts are pairwise coprime.
A305713 counts strict pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
Cf. A000740, A001840, A007359, A087087, A178472, A284825, A302696, A307719, A335235, A337561, A337695.
Number of unordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is not considered coprime unless it is (1).
+10
15
0, 0, 0, 1, 1, 2, 2, 3, 4, 4, 5, 6, 8, 7, 10, 7, 11, 11, 17, 12, 19, 12, 19, 17, 29, 16, 28, 19, 31, 23, 46, 23, 42, 25, 45, 27, 59, 31, 57, 34, 61, 37, 84, 38, 75, 42, 74, 47, 107, 45, 98, 51, 96, 56, 135, 54, 115, 63, 117, 67, 174, 65, 139, 75, 144, 75, 194
EXAMPLE
The a(3) = 1 through a(14) = 10 partitions (A = 10, B = 11, C = 12):
111 211 221 321 322 332 441 433 443 543 544 554
311 411 331 431 522 532 533 552 553 743
511 521 531 541 551 651 661 752
611 711 721 722 732 733 761
811 731 741 751 833
911 831 922 851
921 B11 941
A11 A31
B21
C11
MATHEMATICA
Table[Length[Select[IntegerPartitions[n, {3}], CoprimeQ@@Union[#]&]], {n, 0, 100}]
CROSSREFS
A304709 counts these partitions of any length.
A307719 is the strict case except for any number of 1's.
A337600 considers singletons to be coprime.
A000217 counts 3-part compositions.
A000837 counts relatively prime partitions.
A023023 counts relatively prime 3-part partitions.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A337461 counts pairwise coprime 3-part compositions.
Odd numbers that are either prime or whose prime indices are pairwise coprime.
+10
14
3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 83, 85, 89, 93, 95, 97, 101, 103, 107, 109, 113, 119, 123, 127, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163, 165, 167, 173, 177, 179
COMMENTS
Also Heinz numbers of partitions with pairwise coprime parts all greater than 1 ( A007359), where singletons are considered coprime. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
EXAMPLE
The sequence of terms together with their prime indices begins:
3: {2} 43: {14} 89: {24} 141: {2,15}
5: {3} 47: {15} 93: {2,11} 143: {5,6}
7: {4} 51: {2,7} 95: {3,8} 145: {3,10}
11: {5} 53: {16} 97: {25} 149: {35}
13: {6} 55: {3,5} 101: {26} 151: {36}
15: {2,3} 59: {17} 103: {27} 155: {3,11}
17: {7} 61: {18} 107: {28} 157: {37}
19: {8} 67: {19} 109: {29} 161: {4,9}
23: {9} 69: {2,9} 113: {30} 163: {38}
29: {10} 71: {20} 119: {4,7} 165: {2,3,5}
31: {11} 73: {21} 123: {2,13} 167: {39}
33: {2,5} 77: {4,5} 127: {31} 173: {40}
35: {3,4} 79: {22} 131: {32} 177: {2,17}
37: {12} 83: {23} 137: {33} 179: {41}
41: {13} 85: {3,7} 139: {34} 181: {42}
Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of multiset systems. 03: {{1}}
05: {{2}}
07: {{1,1}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
19: {{1,1,1}}
23: {{2,2}}
29: {{1,3}}
31: {{5}}
33: {{1},{3}}
35: {{2},{1,1}}
37: {{1,1,2}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
51: {{1},{4}}
53: {{1,1,1,1}}
MATHEMATICA
primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1, 400, 2], Or[PrimeQ[#], CoprimeQ@@primeMS[#]]&]
CROSSREFS
A007359 counts partitions with these Heinz numbers.
A337694 is the pairwise non-coprime instead of pairwise coprime version.
A337984 does not include the primes.
A305713 counts pairwise coprime strict partitions.
A337561 counts pairwise coprime strict compositions.
A337697 counts pairwise coprime compositions with no 1's.
Cf. A005408, A051424, A056239, A087087, A112798, A200976, A302797, A303282, A304711, A335235, A338468.
Number of integer partitions of n whose parts are all equal or whose distinct parts are pairwise coprime.
+10
12
1, 1, 2, 3, 5, 7, 10, 14, 19, 25, 32, 43, 54, 70, 86, 105, 130, 162, 196, 240, 286, 339, 405, 485, 573, 674, 790, 922, 1072, 1252, 1456, 1685, 1939, 2226, 2557, 2923, 3349, 3822, 4347, 4931, 5593, 6335, 7170, 8092, 9105, 10233, 11495, 12903, 14458, 16169, 18063
COMMENTS
Two parts are coprime if they have no common divisor greater than 1.
EXAMPLE
The a(6) = 10 partitions whose parts are all equal or whose distinct parts are pairwise coprime are (6), (51), (411), (33), (321), (3111), (222), (2211), (21111), (111111).
MAPLE
g:= proc(n, i, s) `if`(n=0, 1, `if`(i<1, 0,
b(n, i, select(x-> x<=i, s))))
end:
b:= proc(n, i, s) option remember; g(n, i-1, s)+(f->
`if`(f intersect s={}, add(g(n-i*j, i-1, s union f)
, j=1..n/i), 0))(numtheory[factorset](i))
end:
a:= n-> g(n$2, {}):
MATHEMATICA
Table[Select[IntegerPartitions[n], Or[SameQ@@#, CoprimeQ@@Union[#]]&]//Length, {n, 20}]
(* Second program: *)
g[n_, i_, s_] := If[n == 0, 1, If[i < 1, 0, b[n, i, Select[s, # <= i &]]]];
b[n_, i_, s_] := b[n, i, s] = g[n, i - 1, s] + Function[f,
If[f ~Intersection~ s == {}, Sum[g[n - i*j, i - 1, s ~Union~ f],
{j, 1, n/i}], 0]][FactorInteger[i][[All, 1]]];
a[n_] := g[n, n, {}];
CROSSREFS
Cf. A000837, A007359, A018783, A051424, A056239, A078374, A101268, A289508, A289509, A298748, A300486, A302569, A302696, A302698, A302796, A302797, A304709, A304711.
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