A078322 -id:A078322 - cp's OEIS frontend

A078310 a(n) = n*rad(n) + 1, where rad = A007947 (squarefree kernel).

2, 5, 10, 9, 26, 37, 50, 17, 28, 101, 122, 73, 170, 197, 226, 33, 290, 109, 362, 201, 442, 485, 530, 145, 126, 677, 82, 393, 842, 901, 962, 65, 1090, 1157, 1226, 217, 1370, 1445, 1522, 401, 1682, 1765, 1850, 969, 676, 2117, 2210, 289, 344, 501, 2602, 1353

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

A112526(a(n) - 1) = 1, see also A224866. - Reinhard Zumkeller, Jul 23 2013

Increase each exponent in the prime factorization by one, then add 1 to the new product. - M. F. Hasler, Jan 22 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

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.

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

a078310 n = n * a007947 n + 1
-- Reinhard Zumkeller, Jul 23 2013, Oct 19 2011

a:= n-> 1+n*mul(i[1], i=ifactors(n)[2]):
seq(a(n), n=1..60);  # Alois P. Heinz, Jan 22 2017

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

rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])
a(n)=n*rad(n)+1 \\ Charles R Greathouse IV, Jul 09 2013

a(n)={n=factor(n);n[,2]+=vectorv(matsize(n)[1],i,1);factorback(n)+1} \\ M. F. Hasler, Jan 22 2017

a(n)=prod(k=1,matsize(n=factor(n))[1],n[k,1]^(n[k,2]+1))+1 \\ M. F. Hasler, Jan 22 2017

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

A078323 Arithmetic derivative of n*rad(n)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

1, 1, 7, 6, 15, 1, 45, 1, 32, 1, 63, 1, 129, 1, 115, 14, 213, 1, 183, 70, 281, 102, 381, 34, 165, 1, 43, 134, 423, 70, 581, 18, 773, 102, 615, 38, 969, 459, 763, 1, 957, 358, 1715, 431, 780, 102, 1847, 34, 524, 170, 1303, 607, 1977, 155, 1725, 162, 3825, 678, 1743

Offset: 1

Reinhard Zumkeller, Nov 23 2002

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A003415, A007947 (rad), A078310, A078322.

a078323 = a003415 . a078310  -- Reinhard Zumkeller, Jul 23 2013

ad[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := ad[1 + n * Times @@ FactorInteger[n][[;;, 1]]]; Array[a, 100] (* Amiram Eldar, Apr 10 2025 *)

a(n) = A003415(A078310(n)).

A188525 a(n) = rad(rad(n)^2+1), where rad = A007947.

Original entry on oeis.org

2, 5, 10, 5, 26, 37, 10, 5, 10, 101, 122, 37, 170, 197, 226, 5, 290, 37, 362, 101, 442, 485, 530, 37, 26, 677, 10, 197, 842, 901, 962, 5, 1090, 1157, 1226, 37, 1370, 85, 1522, 101, 58, 1765, 370, 485, 226, 2117, 2210, 37, 10, 101, 2602, 677, 2810, 37

Offset: 1

Jonathan Vos Post, Apr 03 2011

			a(7) = rad(rad(7)^2 + 1) = rad(7^2 + 1) = rad(50) = 10.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A007947, A078322.

[ &*PrimeDivisors((&*PrimeDivisors(n))^2+1): n in [1..51] ]; // Bruno Berselli, Apr 04 2011

with(numtheory):
rad:= n-> mul(i, i=factorset(n)):
a:= n-> rad(rad(n)^2+1):
seq(a(n), n=1..70);  # Alois P. Heinz, Apr 03 2011

rad[n_] := Times @@ FactorInteger[n][[All, 1]];
a[n_] := rad[rad[n]^2 + 1];
Array[a, 70] (* Jean-François Alcover, Mar 27 2017 *)

rad(n)=my(f=factor(n)[,1]);prod(i=1,#f,f[i])
a(n)=rad(rad(n)^2+1) \\ Charles R Greathouse IV, Aug 08 2013

a(n) = A007947(A078615(n)+1). - R. J. Mathar, Apr 04 2011

cp's OEIS Frontend

A078310 a(n) = n*rad(n) + 1, where rad = A007947 (squarefree kernel).

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A078323 Arithmetic derivative of n*rad(n)+1, where rad = A007947 (squarefree kernel).

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A188525 a(n) = rad(rad(n)^2+1), where rad = A007947.

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