A078315 -id:A078315 - 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.

A078313 Number of distinct prime factors of n*rad(n)+1, where rad=A007947 (squarefree kernel).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 2, 3, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 1, 2, 2, 3, 3, 2, 2, 4, 1, 2, 2, 2, 3, 3, 2, 3, 2, 3, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 4, 1, 2, 2, 3, 2, 2, 2, 2, 2, 4, 1, 2, 2, 3, 2, 3, 2, 3, 3, 3, 1, 2, 1, 3, 1, 3, 3, 2, 2, 3, 2, 3

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

a(n)=A001221(A078310(n)).

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

Cf. A078314, A078315.

a078313 = a001221 . a078310  -- Reinhard Zumkeller, Jul 23 2013

A078313[n_] := PrimeNu[n*Times @@ FactorInteger[n][[All, 1]] + 1]; Array[A078313, 50] (* G. C. Greubel, Apr 25 2017 *)

a(n)=omega(n*vecprod(factor(n)[,1])+1) \\ Charles R Greathouse IV, Jul 09 2013

A078314 Total number of prime factors of n*rad(n)+1 counted with multiplicity, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

1, 1, 2, 2, 2, 1, 3, 1, 3, 1, 2, 1, 3, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 2, 4, 1, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 3, 2, 1, 3, 2, 4, 3, 4, 2, 4, 2, 4, 2, 2, 3, 3, 3, 3, 2, 5, 2, 2, 1, 2, 2, 3, 2, 2, 1, 3, 3, 2, 3, 2, 1, 4, 1, 2, 4, 3, 2, 2, 3, 3, 3, 4, 1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 1, 2, 1, 3, 1, 4, 3, 2, 2, 3, 2, 3

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

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

Cf. A001222, A007947, A078310, A078313, A078315.

a078314 = a001222 . a078310  -- Reinhard Zumkeller, Jul 23 2013

a[n_] := PrimeOmega[1 + n * Times @@ FactorInteger[n][[;;, 1]]]; Array[a, 100] (* Amiram Eldar, Sep 08 2024 *)

a(n)=bigomega(n*vecprod(factor(n)[,1])+1) \\ Charles R Greathouse IV, Jul 09 2013

a(n) = A001222(A078310(n)).

A078316 Maximum exponent in the prime factorization of n*rad(n)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

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

Cf. A007947, A051903, A078310, A078313, A078315.

a078316 = a051903 . a078310  -- Reinhard Zumkeller, Jul 23 2013

a[n_] := Max[FactorInteger[1 + n * Times @@ FactorInteger[n][[;;, 1]]][[;;, 2]]]; Array[a, 100] (* Amiram Eldar, Sep 07 2024 *)

a(n) = vecmax(factor(1 + n * vecprod(factorint(n)[, 1]))[, 2]); \\ Amiram Eldar, Sep 07 2024

a(n) = A051903(A078310(n)).

A078326 Numbers n such that n-1 and n are a pair of consecutive powerful numbers.

Original entry on oeis.org

9, 289, 676, 9801, 12168, 235225, 332929, 465125, 1825201, 11309769, 384199201, 592192225, 4931691076, 5425069448, 13051463049, 221322261601, 443365544449, 865363202001, 8192480787001, 11968683934832, 13325427460801, 15061377048201, 28821995554248

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

a(n) = u*rad(u) = v*rad(v)+1 for appropriate u, v, where rad(n) = A007947(n) is the squarefree kernel.

Also numbers n such that n(n-1) is a powerful number. - Charles R Greathouse IV, Aug 08 2013

Donovan Johnson, Table of n, a(n) for n = 1..39 (terms < 10^22)
Jérôme Germoni, Nombres puissants au bac S, Images des Mathématiques, CNRS, 2021 (in French).
Eric Weisstein's World of Mathematics, Powerful Number

Cf. A078310, A064549, A078315, A078325, A060355, A078326.

is(n)=ispowerful(n-1)&&ispowerful(n) \\ Charles R Greathouse IV, Aug 08 2013

a(n) = A060355(n)+1.

a(22)-a(23) from Donovan Johnson, Jul 29 2011

cp's OEIS Frontend

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

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A078313 Number of distinct prime factors of n*rad(n)+1, where rad=A007947 (squarefree kernel).

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A078314 Total number of prime factors of n*rad(n)+1 counted with multiplicity, where rad = A007947 (squarefree kernel).

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A078316 Maximum exponent in the prime factorization of n*rad(n)+1, where rad = A007947 (squarefree kernel).

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A078326 Numbers n such that n-1 and n are a pair of consecutive powerful numbers.

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