A064415 -id:A064415 - cp's OEIS frontend

A054725 a(1)=1; a(n) = Sum_{p | n} e * a(p-1), where sum is over all primes p that divide n, and e is the multiplicity of p in n.

1, 1, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 4, 3, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 5, 4, 5, 4, 4, 4, 4, 4, 5, 5, 4, 4, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 4, 5, 5, 5, 5, 5, 4, 6, 5, 5, 5, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 4, 6, 6, 5, 6, 5, 5, 6, 6, 5, 5, 6, 5, 6, 5, 6, 6, 5, 5, 6, 6, 6, 6, 6

Offset: 1

Leroy Quet, Apr 20 2000

nonn

			a(20) = a(2-1) + a(2-1) + a(5-1) = 1 + 1 +2 = 4 because 20 = 2*2*5.

Joerg Arndt, Table of n, a(n) for n = 1..10000

Fold[Append[#1, Total@ Table[#1[[p - 1]], {p, Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger[#2]]}]] &, {1}, Range[2, 105]] (* Michael De Vlieger, Dec 11 2017 *)

a(n)=if (n<=1, 1, my(F=factor(n)); sum(e=1, #F[,1], F[e,2] * a(F[e,1]-1) ) );
vector(200, n, a(n)) \\ Joerg Arndt, Apr 08 2014

a(1) = 1 and a(n) = A064415(n) for n>=2. [Joerg Arndt, Apr 08 2014]

Title clarified by Sean A. Irvine, Feb 18 2022

A334097 a(n) is the exponent of the eventual power of 2 reached when starting from n and using the map k -> k + k/p, where p can be any odd prime factor of k, for example, the largest.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 6, 5, 6, 6, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 7, 7, 6, 7, 7, 7, 6, 7, 7, 7, 7, 6, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 7, 7, 7, 7, 7, 8, 7, 7, 7, 8, 7, 8, 7, 8, 8, 7, 7, 7, 7, 8, 7, 7, 7, 8, 8, 8, 8, 7, 7, 8

Offset: 1

Antti Karttunen, Apr 29 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007814, A052126, A078701, A209229, A331410, A334098, A334862.

Cf. also A064415 (analogous sequence when using the map k -> k - k/p).

Array[Log2@ NestWhile[# + #/FactorInteger[#][[-1, 1]] &, #, !IntegerQ@ Log2@ # &] &, 105] (* Michael De Vlieger, Apr 30 2020 *)

A334097(n) = if(!bitand(n,n-1),valuation(n,2),my(f=factor(n)[, 1]); A334097(n+(n/f[2-(n%2)])));

A334097(n) = if(!bitand(n,n-1),valuation(n,2),A334097(n+(n/vecmax(factor(n)[, 1]))));

A334097(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],f[k,2],f[k,2]*A334097(1+f[k,1]))); };

Totally additive sequence: a(2) = 1, a(p) = a(p+1) for odd primes p, a(m*n) = a(m)+a(n) for m, n > 1.
If A209229(n) == 1, a(n) = A007814(n), otherwise a(n) = a(n+A052126(n)), or equally, a(n) = a(n+(n/A078701(n))).
a(n) = A331410(n) + A334098(n) = A334862(n) +  A064415(n).

A309243 Completely multiplicative with a(p) = p * a(p-1) for any prime number p.

Original entry on oeis.org

1, 2, 6, 4, 20, 12, 84, 8, 36, 40, 440, 24, 312, 168, 120, 16, 272, 72, 1368, 80, 504, 880, 20240, 48, 400, 624, 216, 336, 9744, 240, 7440, 32, 2640, 544, 1680, 144, 5328, 2736, 1872, 160, 6560, 1008, 43344, 1760, 720, 40480, 1902560, 96, 7056, 800, 1632, 1248

Offset: 1

Rémy Sigrist, Jul 17 2019

All terms are distinct and belong to A064522.

			a(2) = 2 * a(1) = 2.
a(5) = 5 * a(4) = 5 * a(2)^2 = 5 * 2^2 = 20.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A000010, A006530, A053585, A064415, A064522.

a(n) = my (f=factor(n), p=f[,1]~, e=f[,2]~); prod (i=1, #p, (p[i] * a(p[i] - 1))^e[i])

a(n) >= n with equality iff n is a power of 2.
a(n) is a multiple of n.
a(n) is a multiple of A000010(n).
A006530(a(n)) = A006530(n).
A053585(a(n)) = A053585(n).
Apparently, A007814(a(n)) = A064415(n).

cp's OEIS Frontend

A054725 a(1)=1; a(n) = Sum_{p | n} e * a(p-1), where sum is over all primes p that divide n, and e is the multiplicity of p in n.

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A334097 a(n) is the exponent of the eventual power of 2 reached when starting from n and using the map k -> k + k/p, where p can be any odd prime factor of k, for example, the largest.

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A309243 Completely multiplicative with a(p) = p * a(p-1) for any prime number p.

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