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Search: a289509 -id:a289509
Displaying 61-70 of 208 results found. page 1 2 3 4 5 6 7 8 9 10 ... 21
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A353394 Product of prime shadows of prime indices of n (with multiplicity). +10
14
1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 2, 2, 4, 3, 4, 1, 2, 4, 5, 2, 6, 2, 3, 2, 4, 4, 8, 3, 4, 4, 2, 1, 4, 2, 6, 4, 6, 5, 8, 2, 2, 6, 4, 2, 8, 3, 4, 2, 9, 4, 4, 4, 7, 8, 4, 3, 10, 4, 2, 4, 6, 2, 12, 1, 8, 4, 2, 2, 6, 6, 6, 4, 4, 6, 8, 5, 6, 8, 4, 2, 16, 2, 2, 6, 4, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
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 prime shadow A181819(n) to be the product of primes indexed by the exponents in the prime factorization of n. For example, 90 = prime(1)*prime(2)^2*prime(3) has prime shadow prime(1)*prime(2)*prime(1) = 12.
LINKS
Table of n, a(n) for n=1..86.
FORMULA
a(n) = Product_i A181819(A112798(n,i)).
Positions where a(n) = A003963(n) are A003586.
Positions where a(n) = A005361(n) are A353399, counted by A353398.
Positions where a(n) = A181819(n) are A353395, counted by A353396.
EXAMPLE
We have 42 = prime(1)*prime(2)*prime(4), so a(42) = 1*2*3 = 6.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
red[n_]:=If[n==1, 1, Times@@Prime/@Last/@FactorInteger[n]];
Table[Times@@red/@primeMS[n], {n, 100}]
CROSSREFS
Positions of first appearances are A353397.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914, product A005361.
A181819 gives prime shadow, with an inverse A181821.
A324850 lists numbers divisible by the product of their prime indices.
A325131 lists numbers relatively prime to their prime shadow.
A325755 lists numbers divisible by their prime shadow, quotient also A325756, with recursion A353393.
Cf. A003586, A005117, A008480, A182850, A266477, A289509, A316428, A320325, A325702, A353389, A353395, A353399.
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 17 2022
STATUS
approved
A324966 Number of distinct odd prime indices of n. +10
13
0, 1, 0, 1, 1, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 1, 1, 1, 0, 2, 0, 2, 1, 1, 1, 1, 0, 1, 0, 2, 1, 1, 1, 2, 1, 1, 0, 1, 0, 2, 1, 1, 0, 2, 1, 2, 1, 1, 0, 2, 1, 1, 0, 1, 2, 1, 0, 1, 1, 2, 0, 2, 0, 1, 1, 2, 1, 2, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 0, 2, 0, 2, 1, 1, 2, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,10
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.
If x and y are coprime then a(x*y) = a(x)+a(y). - Robert Israel, Mar 24 2019
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = A001221(n) - A324967(n). - Robert Israel, Mar 24 2019
G.f.: Sum_{k>=1} x^prime(2*k-1) / (1 - x^prime(2*k-1)). - Ilya Gutkovskiy, Feb 12 2020
Additive with a(p^e) = 1 if primepi(p) is odd and 0 otherwise. - Amiram Eldar, Oct 06 2023
EXAMPLE
180180 has prime indices {1,1,2,2,3,4,5,6}, so a(180180) = 3.
MAPLE
f:= proc(n) nops(select(type, map(numtheory:-pi, numtheory:-factorset(n)), odd)) end proc:
map(f, [$1..100]); # Robert Israel, Mar 24 2019
MATHEMATICA
Table[Count[If[n==1, {}, FactorInteger[n]], {_?(OddQ[PrimePi[#]]&), _}], {n, 100}]
PROG
(PARI) a(n) = my(f=factor(n)[, 1]); sum(k=1, #f, primepi(f[k]) % 2); \\ Michel Marcus, Mar 22 2019
CROSSREFS
Cf. A000720, A001221, A003963, A005087, A066208, A112798, A257991, A257992, A289509, A324929, A324967.
KEYWORD
nonn,easy
AUTHOR
Gus Wiseman, Mar 21 2019
STATUS
approved
A328220 Number of strict integer partitions of n with no pair of consecutive parts relatively prime. +10
13
1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 5, 1, 5, 4, 6, 3, 10, 3, 11, 7, 12, 3, 19, 5, 18, 12, 23, 9, 36, 11, 33, 21, 40, 20, 58, 19, 58, 35, 70, 31, 98, 36, 101, 65, 112, 56, 155, 64, 164, 97, 188, 88, 250, 112, 256, 157, 293, 145, 392, 163, 399, 241, 461, 242 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500
EXAMPLE
The a(2) = 1 through a(20) = 11 partitions (A..K = 10..20):
2 3 4 5 6 7 8 9 A B C D E F G H I J K
42 62 63 64 84 86 96 A6 863 A8 964 C8
82 93 A4 A5 C4 962 C6 A63 E6
A2 C2 C3 E2 E4 F5
642 842 862 F3 G4
A42 G2 I2
864 A64
963 A82
A62 C62
C42 E42
8642
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&!MatchQ[#, {___, x_, y_, ___}/; GCD[x, y]==1]&]], {n, 0, 30}]
CROSSREFS
The non-strict case is A328187.
Partitions with all consecutive parts relatively prime are A328172, with strict case A328188.
Strict partitions with relatively prime parts are A078374.
Partitions with no consecutive divisibilities are A328171.
Cf. A000837, A018783, A070211, A289509, A318980, A325545, A328170, A328221, A328335, A328336.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 14 2019
STATUS
approved
A355535 Odd numbers of which it is not possible to choose a different prime factor of each prime index. +10
13
9, 21, 25, 27, 45, 49, 57, 63, 75, 81, 99, 105, 115, 117, 121, 125, 133, 135, 147, 153, 159, 171, 175, 189, 195, 207, 225, 231, 243, 245, 261, 273, 275, 279, 285, 289, 297, 315, 325, 333, 343, 345, 351, 357, 361, 363, 369, 371, 375, 387, 393, 399, 405, 423 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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.
LINKS
Table of n, a(n) for n=1..54.
EXAMPLE
The terms together with their prime indices begin:
9: {2,2}
21: {2,4}
25: {3,3}
27: {2,2,2}
45: {2,2,3}
49: {4,4}
57: {2,8}
63: {2,2,4}
75: {2,3,3}
81: {2,2,2,2}
99: {2,2,5}
105: {2,3,4}
For example, the prime indices of 897 are {2,6,9}, of which we can choose prime factors in two ways: (2,2,3) or (2,3,3); but neither of these has all distinct elements, so 897 is in the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], OddQ[#]&&Select[Tuples[primeMS/@primeMS[#]], UnsameQ@@#&]=={}&]
CROSSREFS
Including evens gives A355529.
The version for all divisors including evens is A355740, zeros of A355739.
Choices of a prime factor of each prime index: A355741, unordered A355744.
A001221 counts distinct prime factors, with sum A001414.
A001222 counts prime factors with multiplicity.
A003963 multiplies together the prime indices of n.
A056239 adds up prime indices, row sums of A112798.
A120383 lists numbers divisible by all of their prime indices.
Cf. A000720, A076610, A289509, A302796, A327486, A355731, A355733, A355742.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 22 2022
STATUS
approved
A303282 Numbers whose prime indices have no common divisor other than 1 but are not pairwise coprime. +10
12
18, 36, 42, 45, 50, 54, 72, 75, 78, 84, 90, 98, 99, 100, 105, 108, 114, 126, 130, 135, 144, 150, 153, 156, 162, 168, 174, 175, 180, 182, 195, 196, 198, 200, 207, 210, 216, 222, 225, 228, 230, 231, 234, 242, 245, 250, 252, 258, 260, 266, 270, 275, 279, 285, 288 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A prime index of n is a number m such that prime(m) divides n. Two or more numbers are coprime if no pair of them has a common divisor other than 1.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
LINKS
Table of n, a(n) for n=1..55.
EXAMPLE
The sequence of integer partitions whose Heinz numbers belong to this sequence begins (221), (2211), (421), (322), (331), (2221), (22111), (332), (621), (4211), (3221), (441), (522), (3311), (432), (22211).
MATHEMATICA
primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[400], !CoprimeQ@@primeMS[#]&&GCD@@primeMS[#]===1&]
CROSSREFS
Cf. A000837, A001222, A018783, A051424, A056239, A078374, A168532, A289508, A289509, A296150, A298748, A300486, A302569, A302696, A302796, A303138, A303139, A303140, A303283.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 20 2018
STATUS
approved
A304712 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Two parts are coprime if they have no common divisor greater than 1.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
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, {}):
seq(a(n), n=0..60); # Alois P. Heinz, May 17 2018
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, {}];
a /@ Range[0, 60] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)
CROSSREFS
Cf. A000837, A007359, A018783, A051424, A056239, A078374, A101268, A289508, A289509, A298748, A300486, A302569, A302696, A302698, A302796, A302797, A304709, A304711.
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 17 2018
STATUS
approved
A305732 Heinz numbers of reducible integer partitions. Numbers n > 1 that are prime or whose prime indices are relatively prime and such that A181819(n) is already in the sequence. +10
12
2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). A prime index of n is a number m such that prime(m) divides n. A multiset m whose distinct elements are m_1, m_2, ..., m_k with multiplicities y_1, y_2, ..., y_k is reducible if either m is of size 1 or gcd(m_1,...,m_k) = 1 and the multiset {y_1,...,y_k} is also reducible.
LINKS
Table of n, a(n) for n=1..67.
EXAMPLE
60 has relatively prime prime indices {1,1,2,3} with multiplicities {1,1,2} corresponding to A181819(90) = 12. 12 has relatively prime prime indices {1,1,2} with multiplicities {1,2} corresponding to A181819(12) = 6. 6 has relatively prime prime indices {1,2} with multiplicities {1,1} corresponding to A181819(6) = 4. 4 has relatively prime prime indices {1,1} with multiplicities {2} corresponding to A181819(4) = 3. 3 is prime, so we conclude that 60 belongs to the sequence.
MATHEMATICA
rdzQ[n_]:=And[n>1, Or[PrimeQ[n], And[rdzQ[Times@@Prime/@FactorInteger[n][[All, 2]]], GCD@@PrimePi/@FactorInteger[n][[All, 1]]==1]]];
Select[Range[50], rdzQ]
CROSSREFS
Cf. A000837, A007916, A056239, A181819, A182850, A289508, A289509, A298748, A304465, A304687, A304818, A305563, A305731, A305733.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 22 2018
STATUS
approved
A319055 Maximum product of an integer partition of n with relatively prime parts. +10
12
1, 1, 2, 3, 6, 6, 12, 18, 24, 36, 54, 72, 108, 162, 216, 324, 486, 648, 972, 1458, 1944, 2916, 4374, 5832, 8748, 13122, 17496, 26244, 39366, 52488, 78732, 118098, 157464, 236196, 354294, 472392, 708588, 1062882, 1417176, 2125764, 3188646, 4251528, 6377292 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
After a(7), this appears to be the same as A319054.
LINKS
Table of n, a(n) for n=1..43.
MATHEMATICA
Table[Max[Times@@@Select[IntegerPartitions[n], GCD@@#==1&]], {n, 20}]
CROSSREFS
Cf. A000740, A000792, A000793, A000837, A001414, A007916, A100953, A281116, A289508, A289509, A296302, A319054, A319057.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 09 2018
STATUS
approved
A328170 Number of integer partitions of n whose parts minus 1 are relatively prime. +10
12
0, 0, 1, 1, 2, 3, 5, 8, 12, 18, 27, 38, 53, 74, 102, 137, 184, 241, 317, 413, 536, 687, 880, 1112, 1405, 1765, 2215, 2755, 3424, 4229, 5216, 6402, 7847, 9572, 11662, 14148, 17139, 20688, 24940, 29971, 35969, 43044, 51438, 61311, 72985, 86678, 102807, 121675 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
A partition is relatively prime if the GCD of its parts is 1. Zeros are ignored when computing GCD, and the empty set has GCD 0.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: Sum_{d>=1} mu(d)*(-1/(1-x) + 1/(Prod_{k>=0} 1 - x^(k*d + 1))). - Andrew Howroyd, Oct 17 2019
EXAMPLE
The a(2) = 1 through a(9) = 18 partitions:
(2) (21) (22) (32) (42) (43) (62) (54)
(211) (221) (222) (52) (332) (63)
(2111) (321) (322) (422) (72)
(2211) (421) (431) (432)
(21111) (2221) (521) (522)
(3211) (2222) (621)
(22111) (3221) (3222)
(211111) (4211) (3321)
(22211) (4221)
(32111) (4311)
(221111) (5211)
(2111111) (22221)
(32211)
(42111)
(222111)
(321111)
(2211111)
(21111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], GCD@@(#-1)==1&]], {n, 0, 30}]
PROG
(PARI) seq(n)=Vec(sum(d=1, n-1, moebius(d)*(-1/(1-x) + 1/prod(k=0, n\d, 1 - x*x^(k*d) + O(x*x^n)))), -(n+1)) \\ Andrew Howroyd, Oct 17 2019
CROSSREFS
The Heinz numbers of these partitions are given by A328168.
Partitions whose parts are relatively prime are A000837.
Partitions whose parts plus 1 are relatively prime are A318980.
The GCD of the prime indices of n, all minus 1, is A328167(n).
Cf. A007359, A018783, A258409, A289509, A328163, A328164.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 09 2019
STATUS
approved
A337452 Number of relatively prime strict integer partitions of n with no 1's. +10
12
0, 0, 0, 0, 0, 1, 0, 2, 1, 3, 2, 6, 3, 9, 7, 11, 11, 20, 15, 28, 24, 35, 36, 55, 47, 73, 71, 95, 96, 136, 123, 180, 177, 226, 235, 305, 299, 403, 406, 503, 523, 668, 662, 852, 873, 1052, 1115, 1370, 1391, 1720, 1784, 2125, 2252, 2701, 2786, 3348, 3520, 4116 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..300
EXAMPLE
The a(5) = 1 through a(16) = 11 partitions (A = 10, B = 11, C = 12, D = 13):
32 43 53 54 73 65 75 76 95 87 97
52 72 532 74 543 85 B3 B4 B5
432 83 732 94 653 D2 D3
92 A3 743 654 754
542 B2 752 753 763
632 643 932 762 853
652 5432 843 943
742 852 952
832 942 B32
A32 6532
6432 7432
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&!MemberQ[#, 1]&&GCD@@#==1&]], {n, 0, 15}]
CROSSREFS
A078374 is the version allowing 1's.
A302698 is the non-strict version.
A332004 is the ordered version allowing 1's.
A337450 is the ordered non-strict version.
A337451 is the ordered version.
A337485 is the pairwise coprime version.
A000837 counts relatively prime partitions.
A078374 counts relatively prime strict partitions.
A002865 counts partitions with no 1's.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A337561 counts pairwise coprime strict compositions.
Cf. A007359, A101268, A289509, A337485, A337563.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 31 2020
STATUS
approved
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Last modified August 29 09:16 EDT 2024. Contains 375511 sequences. (Running on oeis4.)