cp's OEIS Frontend

A304203 If n = Product (p_j^k_j) then a(n) = Product (p_j^prime(k_j)).

%I A304203 #38 Jan 20 2024 03:11:18
%S A304203 1,4,9,8,25,36,49,32,27,100,121,72,169,196,225,128,289,108,361,200,
%T A304203 441,484,529,288,125,676,243,392,841,900,961,2048,1089,1156,1225,216,
%U A304203 1369,1444,1521,800,1681,1764,1849,968,675,2116,2209,1152,343,500,2601,1352,2809,972,3025
%N A304203 If n = Product (p_j^k_j) then a(n) = Product (p_j^prime(k_j)).
%H A304203 Andrew Howroyd, <a href="/A304203/b304203.txt">Table of n, a(n) for n = 1..1000</a>
%H A304203 Ilya Gutkovskiy, <a href="/A304203/a304203.jpg">Scatter plot of a(n) up to n=50000</a>.
%H A304203 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A304203 a(prime(i)^k) = prime(i)^prime(k).
%F A304203 a(A000040(k)) = A001248(k).
%F A304203 a(A001248(k)) = A030078(k).
%F A304203 a(A030078(k)) = A050997(k).
%F A304203 a(A002110(k)) = A061742(k).
%F A304203 Multiplicative with a(p^e) = p^prime(e). - _M. F. Hasler_, Nov 20 2018
%F A304203 Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^prime(k)) = 1.80728269690724154161... . - _Amiram Eldar_, Jan 20 2024
%e A304203 a(12) = a(2^2*3^1) = 2^prime(2)*3^prime(1) = 2^3*3^2 = 72.
%p A304203 a:= n-> mul(i[1]^ithprime(i[2]), i=ifactors(n)[2]):
%p A304203 seq(a(n), n=1..55);  # _Alois P. Heinz_, Jan 20 2021
%t A304203 a[n_] := Times @@ (#[[1]]^Prime[#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 55}]
%o A304203 (PARI) a(n) = my(f=factor(n)); prod(k=1, #f~, f[k,1]^prime(f[k,2])); \\ _Michel Marcus_, May 09 2018
%o A304203 (PARI) apply( A304203(n)=factorback((n=factor(n))[,1],apply(prime,n[,2])), [1..50]) \\ _M. F. Hasler_, Nov 20 2018
%Y A304203 Cf. A000040, A001248, A002110, A003963, A008477, A030078, A048767, A050997, A056166 (records), A061742, A181819.
%Y A304203 Cf. A064988 (apply prime to p), A321874 (apply prime to both p & e).
%K A304203 nonn,mult
%O A304203 1,2
%A A304203 _Ilya Gutkovskiy_, May 09 2018