A078310 - OEIS

%I #31 Jun 24 2020 06:17:00

%S 2,5,10,9,26,37,50,17,28,101,122,73,170,197,226,33,290,109,362,201,

%T 442,485,530,145,126,677,82,393,842,901,962,65,1090,1157,1226,217,

%U 1370,1445,1522,401,1682,1765,1850,969,676,2117,2210,289,344,501,2602,1353

%N a(n) = n*rad(n) + 1, where rad = A007947 (squarefree kernel).

%C A112526(a(n) - 1) = 1, see also A224866. - _Reinhard Zumkeller_, Jul 23 2013

%C Increase each exponent in the prime factorization by one, then add 1 to the new product. - _M. F. Hasler_, Jan 22 2017

%H Charles R Greathouse IV, <a href="/A078310/b078310.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A064549(n)+1.

%p a:= n-> 1+n*mul(i[1], i=ifactors(n)[2]):

%p seq(a(n), n=1..60);  # _Alois P. Heinz_, Jan 22 2017

%t A078310[n_] := n*Times @@ FactorInteger[n][[All, 1]] + 1; Array[A078310, 50] (* _G. C. Greubel_, Apr 25 2017 *)

%o (Haskell)

%o a078310 n = n * a007947 n + 1

%o -- _Reinhard Zumkeller_, Jul 23 2013, Oct 19 2011

%o (PARI) rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])

%o a(n)=n*rad(n)+1 \\ _Charles R Greathouse IV_, Jul 09 2013

%o (PARI) a(n)={n=factor(n);n[,2]+=vectorv(matsize(n)[1],i,1);factorback(n)+1} \\ _M. F. Hasler_, Jan 22 2017

%o (PARI) a(n)=prod(k=1,matsize(n=factor(n))[1],n[k,1]^(n[k,2]+1))+1 \\ _M. F. Hasler_, Jan 22 2017

%Y Smallest, greatest factor: A078311, A078312, number of factors: A078313, A078314, min, max exponent: A078315, A078316, number, sum of divisors: A078317, A078318, sum of prime factors: A078319, A078320, Euler's totient: A078321, squarefree kernel: A078322, arithmetic derivative: A078323.

%Y Cf. primes: A078324, squarefree: A078325, squareful: A078326.

%K nonn

%O 1,1

%A _Reinhard Zumkeller_, Nov 23 2002