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A324515
Numbers > 1 where the maximum prime index minus the minimum prime index equals the number of prime factors minus the number of distinct prime factors.
11
2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 23, 29, 31, 37, 40, 41, 43, 45, 47, 53, 59, 61, 67, 71, 73, 75, 79, 83, 89, 97, 100, 101, 103, 107, 109, 112, 113, 120, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 175, 179, 180, 181, 189, 191, 193, 197, 199, 211, 223
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.
Also Heinz numbers of the integer partitions enumerated by A324516. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
LINKS
FORMULA
A243055(a(n)) = A061395(a(n)) - A055396(a(n)) = A001222(a(n)) - A001221(a(n)) = A046660(a(n)).
EXAMPLE
The sequence of terms together with their prime indices begins:
2: {1}
3: {2}
5: {3}
7: {4}
11: {5}
12: {1,1,2}
13: {6}
17: {7}
18: {1,2,2}
19: {8}
23: {9}
29: {10}
31: {11}
37: {12}
40: {1,1,1,3}
41: {13}
43: {14}
45: {2,2,3}
MAPLE
filter:= proc(n) local F, Inds, t;
if isprime(n) then return true fi;
F:= ifactors(n)[2];
Inds:= map(numtheory:-pi, F[.., 1]);
max(Inds) - min(Inds) = add(t[2], t=F) - nops(F)
end proc:
select(filter, [$2..300]); # Robert Israel, Nov 19 2023
MATHEMATICA
Select[Range[2, 100], With[{f=FactorInteger[#]}, PrimePi[f[[-1, 1]]]-PrimePi[f[[1, 1]]]==Total[Last/@f]-Length[f]]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 06 2019
STATUS
approved