A078311 -id:A078311 - 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.

A078312 Greatest prime factor of n*rad(n)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 5, 3, 13, 37, 5, 17, 7, 101, 61, 73, 17, 197, 113, 11, 29, 109, 181, 67, 17, 97, 53, 29, 7, 677, 41, 131, 421, 53, 37, 13, 109, 89, 613, 31, 137, 17, 761, 401, 29, 353, 37, 19, 13, 73, 17, 17, 43, 167, 1301, 41, 281, 13, 89, 157, 13, 673, 1741, 1801, 1861, 769

Offset: 1

Reinhard Zumkeller, Nov 23 2002

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

Cf. A006530, A007947, A078310, A078311.

a078312 = a006530 . a078310  -- Reinhard Zumkeller, Jul 23 2013

a[n_] := FactorInteger[1 + n * Times @@ FactorInteger[n][[;;, 1]]][[-1, 1]]; Array[a, 100] (* Amiram Eldar, Apr 10 2025 *)

a(n) = my(f = factor(1 + n * vecprod(factor(n)[, 1]))); f[#f~, 1]; \\ Amiram Eldar, Apr 10 2025

a(n) = A006530(A078310(n)).

cp's OEIS Frontend

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

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A078312 Greatest prime factor of n*rad(n)+1, where rad = A007947 (squarefree kernel).

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Haskell

Mathematica

PARI

Formula