|
|
A302696
|
|
Numbers whose prime indices (with repetition) are pairwise coprime. Nonprime Heinz numbers of integer partitions with pairwise coprime parts.
|
|
78
|
|
|
1, 2, 4, 6, 8, 10, 12, 14, 15, 16, 20, 22, 24, 26, 28, 30, 32, 33, 34, 35, 38, 40, 44, 46, 48, 51, 52, 55, 56, 58, 60, 62, 64, 66, 68, 69, 70, 74, 76, 77, 80, 82, 85, 86, 88, 92, 93, 94, 95, 96, 102, 104, 106, 110, 112, 116, 118, 119, 120, 122, 123, 124, 128, 132
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
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 has a common divisor other than 1. A single number is not considered coprime unless it is equal to 1.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Number 36 = prime(1)*prime(1)*prime(2)*prime(2) is not included in the sequence, because the pair of prime indices {2,2} is not coprime. - Gus Wiseman, Dec 06 2021
|
|
LINKS
|
|
|
EXAMPLE
|
Sequence of integer partitions with pairwise coprime parts begins: (), (1), (11), (21), (111), (31), (211), (41), (32), (1111), (311), (51), (2111), (61), (411), (321).
Missing from this list are: (2), (3), (4), (22), (5), (6), (7), (221), (8), (42), (9), (33), (222).
|
|
MAPLE
|
filter:= proc(n) local F;
F:= ifactors(n)[2];
if nops(F)=1 then if F[1][1] = 2 then return true else return false fi fi;
if ormap(t -> t[2]>1 and t[1] <> 2, F) then return false fi;
F:= map(t -> numtheory:-pi(t[1]), F);
ilcm(op(F))=convert(F, `*`)
end proc:
|
|
MATHEMATICA
|
primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[200], Or[#===1, CoprimeQ@@primeMS[#]]&]
|
|
PROG
|
(PARI) isA302696(n) = if(isprimepower(n), !(n%2), if(!issquarefree(n>>valuation(n, 2)), 0, my(pis=apply(primepi, factor(n)[, 1])); (lcm(pis)==factorback(pis)))); \\ Antti Karttunen, Dec 06 2021
|
|
CROSSREFS
|
Cf. A000837, A000961, A001222, A005117, A007359, A051424, A275024, A289508, A289509, A298748, A302568, A302569, A302697, A302698, A327512, A327513.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Clarification (with repetition) added to the definition by Antti Karttunen, Dec 06 2021
|
|
STATUS
|
approved
|
|
|
|