cp's OEIS Frontend

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)).

%I A181819 #132 Apr 21 2025 13:23:05
%S A181819 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,
%T A181819 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,
%U A181819 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
%N 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)).
%C A181819 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).
%C A181819 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
%H A181819 Reinhard Zumkeller, <a href="/A181819/b181819.txt">Table of n, a(n) for n = 1..10000</a>
%H A181819 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A181819 From _Antti Karttunen_, Feb 07 2016: (Start)
%F A181819 a(1) = 1; for n > 1, a(n) = A000040(A067029(n)) * a(A028234(n)).
%F A181819 a(1) = 1; for n > 1, a(n) = A008578(A001511(n)) * a(A064989(n)).
%F A181819 Other identities. For all n >= 1:
%F A181819 a(A124859(n)) = A122111(a(n)) = A238745(n). - from _Matthew Vandermast_'s formulas for the latter sequence.
%F A181819 (End)
%F A181819 a(n) = A246029(A156552(n)). - _Antti Karttunen_, Oct 15 2016
%F A181819 From _Antti Karttunen_, Apr 28 & Apr 30 2022: (Start)
%F A181819 A181821(a(n)) = A046523(n) and a(A046523(n)) = a(n). [See comments]
%F A181819 a(n) = A329900(A124859(n)) = A319626(A124859(n)).
%F A181819 a(n) = A246029(A156552(n)).
%F A181819 a(a(n)) = A328830(n).
%F A181819 a(A304660(n)) = n.
%F A181819 a(A108951(n)) = A122111(n).
%F A181819 a(A185633(n)) = A322312(n).
%F A181819 a(A025487(n)) = A181820(n).
%F A181819 a(A276076(n)) = A275735(n) and a(A276086(n)) = A328835(n).
%F A181819 As the sequence converts prime exponents to prime indices, it effects the following mappings:
%F A181819 A001221(a(n)) = A071625(n). [Number of distinct indices --> Number of distinct exponents]
%F A181819 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)]
%F A181819 A056239(a(n)) = A001222(n). [Sum of indices --> Sum of exponents]
%F A181819 A066328(a(n)) = A136565(n). [Sum of distinct indices --> Sum of distinct exponents]
%F A181819 A003963(a(n)) = A005361(n). [Product of indices --> Product of exponents]
%F A181819 A290103(a(n)) = A072411(n). [LCM of indices --> LCM of exponents]
%F A181819 A156061(a(n)) = A290107(n). [Product of distinct indices --> Product of distinct exponents]
%F A181819 A257993(a(n)) = A134193(n). [Index of the least prime not dividing n --> The least number not among the exponents]
%F A181819 A055396(a(n)) = A051904(n). [Index of the least prime dividing n --> Minimal exponent]
%F A181819 A061395(a(n)) = A051903(n). [Index of the greatest prime dividing n --> Maximal exponent]
%F A181819 A008966(a(n)) = A351564(n). [All indices are distinct (i.e., n is squarefree) --> All exponents are distinct]
%F A181819 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]
%F A181819 A056169(a(n)) = A136567(n). [Number of unitary prime divisors --> Number of exponents occurring only once]
%F A181819 A064989(a(n)) = a(A003557(n)) = A295879(n). [Indices decremented after <--> Exponents decremented before]
%F A181819 Other mappings:
%F A181819 A007947(a(n)) = a(A328400(n)) = A329601(n).
%F A181819 A181821(A007947(a(n))) = A328400(n).
%F A181819 A064553(a(n)) = A000005(n) and A000005(a(n)) = A182860(n).
%F A181819 A051903(a(n)) = A351946(n).
%F A181819 A003557(a(n)) = A351944(n).
%F A181819 A258851(a(n)) = A353379(n).
%F A181819 A008480(a(n)) = A309004(n).
%F A181819 a(A325501(n)) = A325507(n) and a(A325502(n)) = A038754(n+1).
%F A181819 a(n!) = A325508(n).
%F A181819 (End)
%e A181819 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.
%p A181819 A181819 := proc(n)
%p A181819     local a;
%p A181819     a := 1;
%p A181819     for pf in ifactors(n)[2] do
%p A181819         a := a*ithprime(pf[2]) ;
%p A181819     end do:
%p A181819     a ;
%p A181819 end proc:
%p A181819 seq(A181819(n),n=1..80) ; # _R. J. Mathar_, Jan 09 2019
%p A181819 # second Maple program:
%p A181819 a:= n-> mul(ithprime(i[2]), i=ifactors(n)[2]):
%p A181819 seq(a(n), n=1..105);  # _Alois P. Heinz_, Nov 26 2024
%t A181819 {1}~Join~Table[Times @@ Prime@ Map[Last, FactorInteger@ n], {n, 2, 120}] (* _Michael De Vlieger_, Feb 07 2016 *)
%o A181819 (Haskell)
%o A181819 a181819 = product . map a000040 . a124010_row
%o A181819 -- _Reinhard Zumkeller_, Mar 26 2012
%o A181819 (PARI) a(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k,2]));} \\ _Michel Marcus_, Nov 16 2015
%o A181819 (Scheme)
%o A181819 ;; With memoization-macro definec.
%o A181819 (definec (A181819 n) (cond ((= 1 n) 1) (else (* (A000040 (A067029 n)) (A181819 (A028234 n))))))
%o A181819 ;; _Antti Karttunen_, Feb 05 2016
%o A181819 (Scheme)
%o A181819 ;; With memoization-macro definec.
%o A181819 (definec (A181819 n) (cond ((= 1 n) 1) ((even? n) (* (A000040 (A007814 n)) (A181819 (A000265 n)))) (else (A181819 (A064989 n)))))
%o A181819 ;; _Antti Karttunen_, Feb 07 2016
%Y A181819 Column 1 of A353510 (see also A325239 and A325277).
%Y A181819 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.
%Y A181819 Left inverse of A304660.
%K A181819 nonn,easy,mult
%O A181819 1,2
%A A181819 _Matthew Vandermast_, Dec 07 2010
%E A181819 Name "Prime shadow" (coined by _Gus Wiseman_ in A325755) prefixed to the definition by _Antti Karttunen_, Apr 27 2022