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:
select(filter, [$1..200]); # Robert Israel, Sep 10 2020
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 11 2018
EXTENSIONS
Clarification (with repetition) added to the definition by Antti Karttunen, Dec 06 2021
STATUS
approved