A078310 -id:A078310 - cp's OEIS frontend

A064549 a(n) = n * Product_{primes p|n} p.

1, 4, 9, 8, 25, 36, 49, 16, 27, 100, 121, 72, 169, 196, 225, 32, 289, 108, 361, 200, 441, 484, 529, 144, 125, 676, 81, 392, 841, 900, 961, 64, 1089, 1156, 1225, 216, 1369, 1444, 1521, 400, 1681, 1764, 1849, 968, 675, 2116, 2209, 288, 343, 500, 2601, 1352

Offset: 1

Henry Bottomley, Oct 16 2001

Index of first occurrence of n in A003557. - Franklin T. Adams-Watters, Jul 25 2014

			a(12) = 72 since 12 = 2^2*3 and 12*2*3 = 72.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Nadia Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, Vol. 113, No. 8 (2006), pp. 1732-1745; arXiv preprint, arXiv:math/0509316 [math.NT], 2005-2006.

A permutation of the powerful numbers A001694.

Cf. A003557 (a left inverse), A007947, A057521, A078310, A082695, A202535.

a064549 n = a007947 n * n  -- Reinhard Zumkeller, Jul 23 2013

[n^2/( (&+[Floor(k^n/n)-Floor((k^n - 1)/n) : k in [1..n]]) ): n in [1..50]]; // G. C. Greubel, Nov 02 2018

a:= n -> n * convert(numtheory:-factorset(n), `*`):
seq(a(n),n=1..100); # Robert Israel, Jul 25 2014

a[n_] := n * Times @@ FactorInteger[n][[All, 1]]; Array[a, 100] (* Jean-François Alcover, Feb 17 2017 *)
Table[n*Product[If[PrimeQ[d], d, 1], {d, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Jun 15 2019 *)

popf(n)= { local(f,p=1); f=factor(n); for(i=1, matsize(f)[1], p*=f[i, 1]); return(p) } { for (n=1, 1000, write("b064549.txt", n, " ", n*popf(n)) ) } \\ Harry J. Smith, Sep 18 2009

A064549(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]++); factorback(f); }; \\ Antti Karttunen, Aug 30 2018

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + p^2*X)/(1 - p*X))[n], ", ")) \\ Vaclav Kotesovec, Jun 24 2020

Multiplicative with a(p^k)=p^(k+1) when k>0.
a(n) = n*A007947(n) = n^2/A003557(n).
Dirichlet convolution of A000027 and A202535. - R. J. Mathar, Dec 20 2011
a(n) = A078310(n) - 1. - Reinhard Zumkeller, Jul 23 2013
A003557(a(n)) = n; a(A003557(n)) = A057521(n). - Antti Karttunen, Aug 30 2018
G.f.: Sum_{k>=1} mu(k)^2*phi(k)*k*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Nov 02 2018
From Vaclav Kotesovec, Jun 24 2020: (Start)
Dirichlet g.f.: zeta(s-2) * zeta(s-1) * Product_{primes p} (1 + p^(3-2*s) - p^(4-2*s) - p^(1-s)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = A065463/3 = A065464*Pi^2/18 = 0.234814...
(End)
Sum_{k>=1} 1/a(k) = zeta(2)*zeta(3)/zeta(6) = A082695. - Vaclav Kotesovec, Sep 19 2020
Sum_{k>=1} (-1)^(k+1)/a(k) = zeta(2)*zeta(3)/(3*zeta(6)) = (1/3) * A082695. - Amiram Eldar, Nov 18 2020

A073353 Sum of n and its squarefree kernel.

Original entry on oeis.org

2, 4, 6, 6, 10, 12, 14, 10, 12, 20, 22, 18, 26, 28, 30, 18, 34, 24, 38, 30, 42, 44, 46, 30, 30, 52, 30, 42, 58, 60, 62, 34, 66, 68, 70, 42, 74, 76, 78, 50, 82, 84, 86, 66, 60, 92, 94, 54, 56, 60, 102, 78, 106, 60, 110, 70, 114, 116, 118, 90, 122, 124, 84, 66, 130, 132

Offset: 1

Reinhard Zumkeller, Jul 29 2002

a(n) is even; a(n)=2*n iff n is squarefree.
Least k >n such that n divides k^n. - Benoit Cloitre, Oct 09 2002
a(n) is the smallest integer > n such that the positive integers coprime to a(n) are also coprime to n. - Leroy Quet, Dec 24 2006

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

Cf. A007947, A005117, A065463, A066503, A064549, A003557, A073354, A078310.

a073353 n = n + a007947 n  -- Reinhard Zumkeller, Jul 23 2013

a[n_] := n + Times @@ FactorInteger[n][[;;, 1]]; Array[a, 100] (* Amiram Eldar, Dec 07 2023 *)

a(n)=vecprod(factor(n)[,1])+n \\ Charles R Greathouse IV, Nov 05 2017

a(n) = n + A007947(n).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 + A065463 = 1.704442... . - Amiram Eldar, Dec 07 2023

A078325 Squarefree numbers of the form m*rad(m)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 10, 17, 26, 33, 37, 65, 73, 82, 101, 109, 122, 129, 145, 170, 197, 201, 217, 226, 257, 290, 362, 393, 401, 433, 442, 485, 501, 530, 577, 626, 649, 677, 730, 785, 842, 865, 901, 962, 969, 973, 1001, 1090, 1126, 1153, 1157, 1226, 1297, 1353, 1370, 1373

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

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

Intersection of A005117 and A224866.

Cf. A008966, A078310, A078324.

a078325 n = a078325_list !! (n-1)
a078325_list = filter ((== 1) . a008966) a224866_list
-- Reinhard Zumkeller, Jul 23 2013

powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Range[1400], SquareFreeQ[#] && powQ[# - 1] &] (* Amiram Eldar, Jul 31 2022 *)

is(n) = n>1 && issquarefree(n) && ispowerful(n-1); \\ Amiram Eldar, Jul 31 2022

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)).

A078315 Minimum exponent in prime factorization of n*rad(n)+1, where rad = A007947 (the radical or squarefree kernel).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

2 = a(4) = a(45) = a(48) = a(140) = a(529) = a(682) = a(3264) = a(3564) = a(4680) = a(4756) = a(166320) = a(194873) = a(330096) = a(364905) = a(2100332) = a(4160200) with all terms in between equal to 1. Is there an n with a(n) > 2? - Charles R Greathouse IV, May 20 2013

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

Cf. A051904, A078310, A078314, A078316.

a078315 = a051904 . a078310  -- Reinhard Zumkeller, Jul 23 2013

a[n_] := Min[FactorInteger[1 + n * Times @@ FactorInteger[n][[;;, 1]]][[;;, 2]]]; Array[a, 100] (* Amiram Eldar, Sep 08 2024 *)

a(n)=my(f=factor(n));f[,2]=apply(n->n+1,f[,2]);vecmin(factor(factorback(f)+1)[,2]) \\ Charles R Greathouse IV, May 20 2013

a(n) = A051904(A078310(n)).

A078318 Sum of divisors of n*rad(n)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

3, 6, 18, 13, 42, 38, 93, 18, 56, 102, 186, 74, 324, 198, 342, 48, 540, 110, 546, 272, 756, 588, 972, 180, 312, 678, 126, 528, 1266, 972, 1596, 84, 1980, 1260, 1842, 256, 2484, 1842, 2286, 402, 2613, 2124, 3534, 1440, 1281, 2220, 4536, 307, 660, 672

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

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

Cf. A000203, A007947, A078310, A078317, A078319, A078320.

a078318 = a000203 . a078310  -- Reinhard Zumkeller, Jul 23 2013

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

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

a(n) = A000203(A078310(n)).

A078324 Primes of the form m*rad(m)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 17, 37, 73, 101, 109, 197, 257, 401, 433, 577, 677, 1153, 1297, 1373, 1601, 1801, 2593, 2917, 3137, 3457, 3529, 3889, 4001, 4357, 5477, 7057, 8101, 8713, 8837, 9001, 10369, 12101, 13457, 14401, 15377, 15877, 16001, 16901, 17497, 17957, 18253, 18433, 20809

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

			12*rad(12)+1 = 12*rad(3*2^2)+1 = 12*3*2+1 = 72+1 = 73, therefore 73 is a term.
a(33) = 10369 = 10368 + 1: A078310(1728) = (2*3)*(2^6*3^3) = 10368.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Intersection of A000040 and A224866.

Cf. A010051, A078310, A078325.

a078324 n = a078324_list !! (n-1)
a078324_list = filter ((== 1) . a010051') a224866_list
-- Reinhard Zumkeller, Jul 23 2013

powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Prime[Range[2400]], powQ[# - 1] &] (* Amiram Eldar, Jul 31 2022 *)

is(n) = isprime(n) && ispowerful(n-1); \\ Amiram Eldar, Jul 31 2022

Missing terms 10369, 16001, 17497 and 18433 inserted by Reinhard Zumkeller, Jul 23 2013

A224866 Numbers of the form m*rad(m)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 9, 10, 17, 26, 28, 33, 37, 50, 65, 73, 82, 101, 109, 122, 126, 129, 145, 170, 197, 201, 217, 226, 244, 257, 289, 290, 325, 344, 362, 393, 401, 433, 442, 485, 501, 513, 530, 577, 626, 649, 676, 677, 730, 785, 801, 842, 865, 901, 962, 969, 973, 1001

Offset: 1

Reinhard Zumkeller, Jul 23 2013

nonn

A078310 in natural order.

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

Subsequences: A078325, A078324.

Cf. A001694, A078310, A007947.

a224866 n = a224866_list !! (n-1)
a224866_list = [x | x <- [2..] , let x' = x - 1, let k = a007947 x',
                    let (y,m) = divMod x' k, m == 0, a007947 y == k]

powQ[n_] := n == 1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Range[1001], powQ[# - 1] &] (* Amiram Eldar, Jul 31 2022 *)

is(n) = n>1 && ispowerful(n-1) \\ Charles R Greathouse IV, Aug 08 2013, corrected by Amiram Eldar, Jul 31 2022

a(n) = A001694(n) + 1.

A078319 Sum of distinct prime factors of n*rad(n)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 7, 3, 15, 37, 7, 17, 9, 101, 63, 73, 24, 197, 115, 14, 36, 109, 183, 70, 32, 102, 60, 34, 12, 677, 43, 134, 423, 70, 52, 18, 116, 102, 615, 38, 144, 22, 763, 401, 31, 358, 44, 39, 15, 102, 37, 17, 45, 170, 1303, 55, 288, 18, 108, 162, 20, 678, 1743, 1801, 1863

Offset: 1

Reinhard Zumkeller, Nov 23 2002

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

Cf. A007947, A008472, A078310, A078313, A078318, A078320.

a078319 = a008472 . a078310  -- Reinhard Zumkeller, Jul 23 2013

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

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

a(n) = A008472(A078310(n)).

cp's OEIS Frontend

A064549 a(n) = n * Product_{primes p|n} p.

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A073353 Sum of n and its squarefree kernel.

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A078325 Squarefree numbers of the form m*rad(m)+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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A078315 Minimum exponent in prime factorization of n*rad(n)+1, where rad = A007947 (the radical or squarefree kernel).

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A078318 Sum of divisors of n*rad(n)+1, where rad = A007947 (squarefree kernel).

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