A075860 -id:A075860 - cp's OEIS frontend

A029908 Starting with n, repeatedly sum prime factors (with multiplicity) until reaching 0 or a fixed point. Then a(n) is the fixed point (or 0).

0, 2, 3, 4, 5, 5, 7, 5, 5, 7, 11, 7, 13, 5, 5, 5, 17, 5, 19, 5, 7, 13, 23, 5, 7, 5, 5, 11, 29, 7, 31, 7, 5, 19, 7, 7, 37, 7, 5, 11, 41, 7, 43, 5, 11, 7, 47, 11, 5, 7, 5, 17, 53, 11, 5, 13, 13, 31, 59, 7, 61, 5, 13, 7, 5, 5, 67, 7, 5, 5, 71, 7, 73, 5, 13, 23, 5, 5, 79, 13, 7, 43, 83, 5, 13

Offset: 1

Dann Toliver

nonn

That is, the sopfr function (see A001414) applied repeatedly until reaching 0 or a fixed point.
For n > 1, the sequence reaches a fixed point which is either 4 or a prime.
A002217(n) is number of terms in sequence from n to a(n). - Reinhard Zumkeller, Apr 08 2003
Because sopfr(n) <= n (with equality at 4 and the primes), the first appearance of all primes is in the natural order: 2,3,5,7,11,... . - Zak Seidov, Mar 14 2011
The terms 0, 2, 3 and 4 occur exactly once, because no number > 5 can have factors that sum to be < 5, and therefore can never enter a trajectory that will drop below 5. - Christian N. K. Anderson, May 19 2013
For all primes p, where p is contained in A001359, then a(p^2) = p + 2. (A006512). Proof: p^2 has prime factors (p, p). This sums to 2p. 2p has factors (2, p). This sums to p + 2. Since p was the lesser of a twin prime, then p + 2 is the greater of a twin prime. - Ryan Bresler, Nov 04 2021

			20 -> 2+2+5 = 9 -> 3+3 = 6 -> 2+3 = 5, so a(20)=5.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Eric Weisstein's World of Mathematics, Sum of Prime Factors.

Cf. A001414 (sum of prime factors of n).
Cf. A081758, A002217, A075860.
Cf. A001414, A056239, A008475, A082081, A082083.

f:= proc(n) option remember;
if isprime(n) then n
else `procname`(add(x[1]*x[2], x = ifactors(n)[2]))
fi
end proc:
f(1):= 0: f(4):= 4:
map(f, [$1..100]); # Robert Israel, Apr 27 2015

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] ep[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] slog[x_] := slog[x_] := Apply[Plus, ba[x]*ep[x]] Table[FixedPoint[slog, w], {w, 1, 128}]
f[n_] := Plus @@ Flatten[ Table[ #[[1]], {#[[2]]}] & /@ FactorInteger@n]; Array[ FixedPoint[f, # ] &, 87] (* Robert G. Wilson v, Jan 18 2006 *)
fz[n_]:=Plus@@(#[[1]]*#[[2]]&/@FactorInteger@n); Array[FixedPoint[fz,#]&,1000] (* Zak Seidov, Mar 14 2011 *)

from sympy import factorint
def a(n, pn):
    if n == pn:
        return n
    else:
        return a(sum(p*e for p, e in factorint(n).items()), n)
print([a(i, None) for i in range(1, 100)]) # Gleb Ivanov, Nov 05 2021

A082087 a(n) is the fixed point if function A008472 (= sum of prime factors with no repetition) is iterated when started at initial value n!.

Original entry on oeis.org

2, 5, 5, 7, 7, 17, 17, 17, 17, 3, 3, 41, 41, 41, 41, 31, 31, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 5, 5, 7, 7, 7, 7, 7, 7, 197, 197, 197, 197, 2, 2, 281, 281, 281, 281, 43, 43, 43, 43, 43, 43, 7, 7, 7, 7, 7, 7, 5, 5, 5, 5, 5, 5, 5, 5, 73, 73, 73, 73, 2, 2, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 13, 13, 13, 13

Offset: 1

Labos Elemer, Apr 09 2003

nonn

			n=73: iteration list=
{73!=61234458376886086861524070385274672740778091784697328983823014963978384987221689274204160000000000000000,639,74,39,16,2,2}

Cf. A007504, A008472, A034387, A075860.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sopf[x_] := Apply[Plus, ba[x]] Table[FixedPoint[sopf, w! ], {w, 2, 100}]

a(n) = A075860(A000142(n)).

A082088 a(n) is the fixed point if function A008472[=sum of prime factors with no repetition] is iterated when started at initial value prime[n]!.

Original entry on oeis.org

2, 5, 7, 17, 3, 41, 31, 5, 7, 5, 7, 197, 2, 281, 43, 7, 5, 5, 73, 2, 7, 7, 13, 5, 7, 5, 3, 7, 13, 3, 7, 7, 7, 7, 571, 7, 7, 5, 7, 7, 5, 7, 5, 7, 2, 7, 19, 3, 3, 67, 5, 2, 71, 43, 7, 71, 239, 7, 7, 7699, 2, 5, 313, 8893, 2, 3, 199, 5, 5, 2, 5, 2, 3, 7, 6361, 3, 97, 5, 19, 97, 7, 2593, 5, 5

Offset: 1

Labos Elemer, Apr 09 2003

nonn

			n=100,p(100)=541,start at 541! and get iteration list=
{541!,24133} ended immediately in a(100)=24133;
n=99,p(99)-523,start at 523! and get a list of
{523!,23592,988,34,19}, a(99)=19.

Cf. A008472, A034387, A007504, A075860, A082087.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sopf[x_] := Apply[Plus, ba[x]] Table[FixedPoint[sopf, Prime[w]! ], {w, 2, 100}]

a(n)=A082087[A000142(p[n])].

A321944 Starting from n, repeatedly compute the sum of the prime divisors until a fixed point or 0 is reached; a(n) is the number of terms, including n.

Original entry on oeis.org

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

Offset: 1

Wilmer Emiro Castrillon Calderon, Dec 12 2018

nonn

a(n) is 1 + the number of iterations of n -> A008472(n) until n = A008472(n) or n=0.
The fixed points are in A075860.
For n>1 the fixed point is a prime number.

			For n=21: 21->{3,7} 3+7=10, 10->{2,5} 2+5=7, 7->{7} 7; 3 terms found {21,10,7}, therefore a(21) = 3.
For n=2: 2->{2} 2, 1 term found {2}, therefore a(2) = 1.
For n=1: 1->{} 0, 2 term found {1,0}, therefore a(1) = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A008472, A075860, A002217.

f:= proc(n) option remember;
  if isprime(n) then 1
  else 1+procname(convert(numtheory:-factorset(n),`+`))
  fi
end proc:
f(1):= 2:
map(f, [$1..100]); # Robert Israel, Mar 30 2020

s[n_] := DivisorSum[n, # &, PrimeQ[#] &]; a[1] = 2; a[n_] := Length[ FixedPointList[s, n]] - 1; Array[a, 60, 0] (* Amiram Eldar, Dec 12 2018 *)

a(n)={my(k=1); while(n&&!isprime(n), k++; n=vecsum(factor(n)[, 1])); k} \\ Andrew Howroyd, Dec 12 2018

A082089 a(n)-th prime is the fixed point if function A008472[=sum of prime factors with no repetition] is iterated when started at factorial of n-th prime.

Original entry on oeis.org

1, 3, 4, 7, 2, 13, 11, 3, 4, 3, 4, 45, 1, 60, 14, 4, 3, 3, 21, 1, 4, 4, 6, 3, 4, 3, 2, 4, 6, 2, 4, 4, 4, 4, 105, 4, 4, 3, 4, 4, 3, 4, 3, 4, 1, 4, 8, 2, 2, 19, 3, 1, 20, 14, 4, 20, 52, 4, 4, 977, 1, 3, 65, 1108, 1, 2, 46, 3, 3, 1, 3, 1, 2, 4, 829, 2, 25, 3, 8, 25, 4, 378, 3, 3, 29, 3, 6, 8, 1, 1, 28

Offset: 2

Labos Elemer, Apr 09 2003

nonn

a(n) < n holds usually, except few large values arising unexpectedly.

			n=100, p(100)=541, starts at factorial of 100th prime and ends in 24133, the 2687th prime, so a(100)=2687;
n=99, initial value=523!, fixed point is 19, the 8th prime, a(99)=8.

Cf. A008472, A034387, A007504, A075860, A082087, A082088.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sopf[x_] := Apply[Plus, ba[x]] Table[PrimePi[FixedPoint[sopf, Prime[w]! ]], {w, 2, 100}]

a(n) = A000720(A082087(A000142(A000040(n)))) = pi(A082087(p(n)!)).

cp's OEIS Frontend

A029908 Starting with n, repeatedly sum prime factors (with multiplicity) until reaching 0 or a fixed point. Then a(n) is the fixed point (or 0).

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A082087 a(n) is the fixed point if function A008472 (= sum of prime factors with no repetition) is iterated when started at initial value n!.

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A082088 a(n) is the fixed point if function A008472[=sum of prime factors with no repetition] is iterated when started at initial value prime[n]!.

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A321944 Starting from n, repeatedly compute the sum of the prime divisors until a fixed point or 0 is reached; a(n) is the number of terms, including n.

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A082089 a(n)-th prime is the fixed point if function A008472[=sum of prime factors with no repetition] is iterated when started at factorial of n-th prime.

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