A078310 -id:A078310 - cp's OEIS frontend

A078320 Sum of all prime factors of n*rad(n)+1, where rad = A007947 (squarefree kernel).

2, 5, 7, 6, 15, 37, 12, 17, 11, 101, 63, 73, 24, 197, 115, 14, 36, 109, 183, 70, 32, 102, 60, 34, 15, 677, 43, 134, 423, 70, 52, 18, 116, 102, 615, 38, 144, 39, 763, 401, 60, 358, 49, 39, 30, 102, 37, 34, 49, 170, 1303, 55, 288, 23, 108, 162, 30, 678, 1743, 1801

Offset: 1

Reinhard Zumkeller, Nov 23 2002

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

Cf. A001414, A078310, A078314, A078318, A078319.

a078320 = a001414 . a078310  -- Reinhard Zumkeller, Jul 23 2013

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

rad(n)=my(f=factor(n)[,1]); prod(i=1,#f,f[i])
a(n)=my(f=factor(n*rad(n)+1));sum(i=1,#f~,f[i,1]*f[i,2]) \\ Charles R Greathouse IV, Jul 15 2013

a(n) = A001414(A078310(n)).

A078322 a(n) = rad(n*rad(n)+1), where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 10, 3, 26, 37, 10, 17, 14, 101, 122, 73, 170, 197, 226, 33, 290, 109, 362, 201, 442, 485, 530, 145, 42, 677, 82, 393, 842, 901, 962, 65, 1090, 1157, 1226, 217, 1370, 85, 1522, 401, 58, 1765, 370, 969, 26, 2117, 2210, 17, 86, 501, 2602, 1353, 2810, 65

Offset: 1

Reinhard Zumkeller, Nov 23 2002

a(n) = A007947(A078310(n)).

			a(25) = rad(25*rad(25)+1) = rad(25*rad(5^2)+1) = rad(25*5+1) = rad(125+1) = rad(126) = rad(2*3*3*7) = 2*3*7 = 42.

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

Cf. A007947, A078310, A078325.

a078322 = a007947 . a078310
-- Reinhard Zumkeller, Oct 19 2011

rad:= n-> mul(i, i=numtheory[factorset](n)):
a:= n-> rad(n*rad(n)+1):
seq(a(n), n=1..70);  # Alois P. Heinz, May 04 2017

rad[n_] := Times @@ FactorInteger[n][[All, 1]]; Table[ rad[n*rad[n] + 1], {n, 1, 54}] (* Jean-François Alcover, Dec 03 2012 *)

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

A078311 Smallest prime factor of n*rad(n)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 5, 2, 3, 2, 37, 2, 17, 2, 101, 2, 73, 2, 197, 2, 3, 2, 109, 2, 3, 2, 5, 2, 5, 2, 677, 2, 3, 2, 17, 2, 5, 2, 13, 2, 7, 2, 5, 2, 401, 2, 5, 2, 3, 2, 29, 2, 17, 2, 3, 2, 3, 2, 5, 2, 5, 2, 5, 2, 1801, 2, 5, 2, 3, 2, 4357, 2, 3, 2, 13, 2, 433, 2, 5477, 2, 3, 2, 5, 2, 3, 2, 5, 2, 3529, 2, 13

Offset: 1

Reinhard Zumkeller, Nov 23 2002

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

Cf. A007947, A020639, A078310, A078312.

a078311 = a020639 . 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) = factor(1 + n * vecprod(factor(n)[, 1]))[1,1]; \\ Amiram Eldar, Apr 10 2025

a(n) = A020639(A078310(n)).

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)).

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)).

A078317 Number of divisors of n*rad(n)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

2, 2, 4, 3, 4, 2, 6, 2, 6, 2, 4, 2, 8, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 4, 12, 2, 4, 4, 4, 4, 8, 4, 8, 4, 4, 4, 8, 6, 4, 2, 6, 4, 12, 8, 9, 4, 16, 3, 8, 4, 4, 8, 8, 6, 8, 4, 16, 4, 4, 2, 4, 4, 6, 4, 4, 2, 8, 6, 4, 6, 4, 2, 16, 2, 4, 8, 8, 4, 4, 6, 6, 6, 16, 2, 4, 4, 8, 4, 8, 4, 8, 8, 12, 2, 4, 2, 8, 2, 12, 8

Offset: 1

Reinhard Zumkeller, Nov 23 2002

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

Cf. A000005, A007947, A078310, A078313, A078314, A078318.

a078317 = a000005 . a078310  -- Reinhard Zumkeller, Jul 23 2013

a[n_] := DivisorSigma[0, 1 + n * Times @@ FactorInteger[n][[;;, 1]]]; Array[a, 100] (* Amiram Eldar, Apr 10 2025 *)

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

a(n) = A000005(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

A078321 Euler's totient of n*rad(n)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

1, 4, 4, 6, 12, 36, 20, 16, 12, 100, 60, 72, 64, 196, 112, 20, 112, 108, 180, 132, 192, 384, 208, 112, 36, 676, 40, 260, 420, 832, 432, 48, 432, 1056, 612, 180, 544, 1088, 760, 400, 812, 1408, 720, 576, 312, 2016, 768, 272, 168, 332, 1300, 800, 1120, 240

Offset: 1

Reinhard Zumkeller, Nov 23 2002

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

Cf. A000010, A007947, A078310.

a078321 = a000010 . a078310  -- Reinhard Zumkeller, Jul 23 2013

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

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

a(n) = A000010(A078310(n)).

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)).

A258565 Arithmetic derivative of powerful numbers, cf. A001694.

Original entry on oeis.org

0, 4, 12, 6, 32, 10, 27, 80, 60, 14, 192, 156, 108, 140, 216, 22, 75, 448, 384, 26, 252, 380, 540, 240, 405, 1024, 912, 34, 756, 147, 38, 700, 960, 1296, 420, 572, 800, 2304, 46, 2112, 500, 1836, 945, 780, 1458, 1792, 2320, 58, 3024, 1860, 62, 1628, 2592

Offset: 1

Reinhard Zumkeller, Jun 03 2015

nonn

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

Cf. A003415, A001694, A068720, A068328, A078310.

Haskell
```
a258565 = a003415 . a001694
```

ader[n_] := Switch[n, 0|1, 0, _, If[PrimeQ[n], 1, Sum[Module[{p, e}, {p, e} = pe; n e/p], {pe, FactorInteger[n]}]]];
ader /@ Join[{1}, Select[Range[1000], AllTrue[FactorInteger[#][[All, 2]], # > 1&]&]] (* Jean-François Alcover, Oct 12 2021 *)

a(n) = A003415(A001694(n)).

cp's OEIS Frontend

A078320 Sum of all prime factors of n*rad(n)+1, where rad = A007947 (squarefree kernel).

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A078322 a(n) = rad(n*rad(n)+1), where rad = A007947 (squarefree kernel).

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A078311 Smallest prime factor of 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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A078316 Maximum exponent in the prime factorization of n*rad(n)+1, where rad = A007947 (squarefree kernel).

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A078317 Number of divisors of n*rad(n)+1, where rad = A007947 (squarefree kernel).

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Mathematica

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

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