A333235 a(n) is the product of indices of unitary prime power divisors of n.

1, 1, 2, 3, 4, 2, 5, 6, 7, 4, 8, 6, 9, 5, 8, 10, 11, 7, 12, 12, 10, 8, 13, 12, 14, 9, 15, 15, 16, 8, 17, 18, 16, 11, 20, 21, 19, 12, 18, 24, 20, 10, 21, 24, 28, 13, 22, 20, 23, 14, 22, 27, 24, 15, 32, 30, 24, 16, 25, 24, 26, 17, 35, 27, 36, 16, 28, 33, 26, 20

Offset

1,3

Comments

Equivalently: replace each prime power p^e in the prime factorisation of n by its index in A246655. - M. F. Hasler, Jun 16 2021

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

If n = Product (p_j^k_j) then a(n) = Product (A025528(p_j^k_j)).

a(prime(n)) = A027883(n).

a(2^n) = A182908(n).

a(A246655(n)) = n.

Example

a(600) = a(2^3 * 3 * 5^2) = a(A246655(6) * A246655(2) * A246655(14)) = 6 * 2 * 14 = 168.

Maple

N:= 1000: # for a(1)..a(N)
R:= NULL: p:= 2:
while p < N do
  R:= R,  seq(p^k, k=1..ilog[p](N));
  p:= nextprime(p);
od:
L:= sort([R]):
f:= proc(n) local F, t;
  F:= ifactors(n)[2];
  mul(ListTools:-BinarySearch(L, t[1]^t[2]), t=F)
end proc:
map(f, [$1..N]); # Robert Israel, Feb 11 2021

Mathematica

PrimePowerPi[n_] := Sum[Boole[PrimePowerQ[k]], {k, 1, n}]; a[1] = 1; a[n_] := Times @@ (PrimePowerPi[#[[1]]^#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 70}]

Prog

(PARI) apply( {A333235(n)=vecprod([A322981(f[1]^f[2])|f<-factor(n)~])}, [1..99]) \\ M. F. Hasler, Jun 16 2021

Crossrefs

Cf. A003963, A025528, A027883, A141128, A141809, A156061, A182908, A246655.

Cf. A322981 (the index of n = p^e in A246655).

Sequence in context: A369422 A361020 A368693 * A295887 A293450 A322990

Adjacent sequences: A333232 A333233 A333234 * A333236 A333237 A333238

Keyword

nonn,mult,look

Author

Ilya Gutkovskiy, Mar 12 2020

Status

approved