A181819 Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 11, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 14, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 13, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 14, 7, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 22, 2, 6, 6, 9, 2, 8, 2, 10, 8

Offset: 1

Author

Matthew Vandermast, Dec 07 2010

Keywords

• nonn
• easy
• mult

Comments

a(n) depends only on prime signature of n (cf. A025487). a(m) = a(n) iff m and n have the same prime signature, i.e., iff A046523(m) = A046523(n).

Because A046523 (the smallest representative of prime signature of n) and this sequence are functions of each other as A046523(n) = A181821(a(n)) and a(n) = a(A046523(n)), it implies that for all i, j: a(i) = a(j) <=> A046523(i) = A046523(j) <=> A101296(i) = A101296(j), i.e., that equivalence-class-wise this is equal to A101296, and furthermore, applying any function f on this sequence gives us a sequence b(n) = f(a(n)) whose equivalence class partitioning is equal to or coarser than that of A101296, i.e., b is then a sequence that depends only on the prime signature of n (the multiset of exponents of its prime factors), although not necessarily in a very intuitive way. - Antti Karttunen, Apr 28 2022

Examples

			20 = 2^2*5 has the exponents (2,1) in its prime factorization. Accordingly, a(20) = prime(2)*prime(1) = A000040(2)*A000040(1) = 3*2 = 6.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Column 1 of A353510 (see also A325239 and A325277).

Cf. A000040, A000265, A001511, A001222, A003963, A005361, A007814, A008578, A028234, A046523, A056239, A064553, A064989, A067029, A101296 (restricted growth sequence transform), A108951, A122111, A124010, A124859, A156552, A181820, A181821, A182850, A182855, A182857 (also A323014), A115621, A101296, A238690, A238745, A238747, A238748, A246029, A304465, A304647, A305732, A305733, A320118, A323022, A325501, A325502, A325507, A325508, A325755 (A353566), A325756, A328830 [= a(a(n))], A328835, A351564 (characteristic function of A130091), A351944, A351946, A353379.

Left inverse of A304660.

Programs

Haskell

a181819 = product . map a000040 . a124010_row
-- Reinhard Zumkeller, Mar 26 2012

Maple

A181819 := proc(n)
    local a;
    a := 1;
    for pf in ifactors(n)[2] do
        a := a*ithprime(pf[2]) ;
    end do:
    a ;
end proc:
seq(A181819(n),n=1..80) ; # R. J. Mathar, Jan 09 2019
# second Maple program:
a:= n-> mul(ithprime(i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..105);  # Alois P. Heinz, Nov 26 2024

Mathematica

{1}~Join~Table[Times @@ Prime@ Map[Last, FactorInteger@ n], {n, 2, 120}] (* Michael De Vlieger, Feb 07 2016 *)

PARI

a(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k,2]));} \\ Michel Marcus, Nov 16 2015

Scheme

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) (else (* (A000040 (A067029 n)) (A181819 (A028234 n))))))
;; Antti Karttunen, Feb 05 2016

Scheme

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) ((even? n) (* (A000040 (A007814 n)) (A181819 (A000265 n)))) (else (A181819 (A064989 n)))))
;; Antti Karttunen, Feb 07 2016

Formula

From Antti Karttunen, Feb 07 2016: (Start)

a(1) = 1; for n > 1, a(n) = A000040(A067029(n)) * a(A028234(n)).

a(1) = 1; for n > 1, a(n) = A008578(A001511(n)) * a(A064989(n)).

Other identities. For all n >= 1:

a(A124859(n)) = A122111(a(n)) = A238745(n). - from Matthew Vandermast's formulas for the latter sequence.

(End)

a(n) = A246029(A156552(n)). - Antti Karttunen, Oct 15 2016

From Antti Karttunen, Apr 28 & Apr 30 2022: (Start)

A181821(a(n)) = A046523(n) and a(A046523(n)) = a(n). [See comments]

a(n) = A329900(A124859(n)) = A319626(A124859(n)).

a(n) = A246029(A156552(n)).

a(a(n)) = A328830(n).

a(A304660(n)) = n.

a(A108951(n)) = A122111(n).

a(A185633(n)) = A322312(n).

a(A025487(n)) = A181820(n).

a(A276076(n)) = A275735(n) and a(A276086(n)) = A328835(n).

As the sequence converts prime exponents to prime indices, it effects the following mappings:

A001221(a(n)) = A071625(n). [Number of distinct indices --> Number of distinct exponents]

A001222(a(n)) = A001221(n). [Number of indices (i.e., the number of prime factors with multiplicity) --> Number of exponents (i.e., the number of distinct prime factors)]

A056239(a(n)) = A001222(n). [Sum of indices --> Sum of exponents]

A066328(a(n)) = A136565(n). [Sum of distinct indices --> Sum of distinct exponents]

A003963(a(n)) = A005361(n). [Product of indices --> Product of exponents]

A290103(a(n)) = A072411(n). [LCM of indices --> LCM of exponents]

A156061(a(n)) = A290107(n). [Product of distinct indices --> Product of distinct exponents]

A257993(a(n)) = A134193(n). [Index of the least prime not dividing n --> The least number not among the exponents]

A055396(a(n)) = A051904(n). [Index of the least prime dividing n --> Minimal exponent]

A061395(a(n)) = A051903(n). [Index of the greatest prime dividing n --> Maximal exponent]

A008966(a(n)) = A351564(n). [All indices are distinct (i.e., n is squarefree) --> All exponents are distinct]

A007814(a(n)) = A056169(n). [Number of occurrences of index 1 (i.e., the 2-adic valuation of n) --> Number of occurrences of exponent 1]

A056169(a(n)) = A136567(n). [Number of unitary prime divisors --> Number of exponents occurring only once]

A064989(a(n)) = a(A003557(n)) = A295879(n). [Indices decremented after <--> Exponents decremented before]

Other mappings:

A007947(a(n)) = a(A328400(n)) = A329601(n).

A181821(A007947(a(n))) = A328400(n).

A064553(a(n)) = A000005(n) and A000005(a(n)) = A182860(n).

A051903(a(n)) = A351946(n).

A003557(a(n)) = A351944(n).

A258851(a(n)) = A353379(n).

A008480(a(n)) = A309004(n).

a(A325501(n)) = A325507(n) and a(A325502(n)) = A038754(n+1).

a(n!) = A325508(n).

(End)

Extensions

Name "Prime shadow" (coined by Gus Wiseman in A325755) prefixed to the definition by Antti Karttunen, Apr 27 2022