A029908 -id:A029908 - cp's OEIS frontend

A082084 a(n)=A029908[n! ]=A029908[A000142[n]] Fixed points of iterated A001414 function if started at factorials as initial values.

0, 2, 5, 5, 5, 19, 5, 7, 7, 11, 13, 13, 23, 13, 19, 101, 61, 5, 19, 5, 11, 5, 5, 7, 23, 7, 5, 7, 283, 293, 5, 5, 7, 367, 379, 389, 23, 7, 463, 5, 13, 11, 29, 5, 23, 7, 7, 19, 5, 5, 5, 5, 61, 7, 73, 47, 857, 7, 947, 5, 29, 7, 13, 5, 5, 19, 137, 7, 5, 7, 11, 23, 353, 53, 11, 1471, 1489

Offset: 1

Labos Elemer, Apr 08 2003

nonn

			Fixed point seems to be always a prime:tested for 1<n<201
n=20!=2432902008176640000: a(20)=5 with fixed-point-list=
{2432902008176640000,154,20,9,6,5,5}

Cf. A001414, A056239, A008475, A082081-A082083, A000961, A000142.

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_] := Apply[Plus, ba[x]*ep[x]] Table[FixedPoint[slog, w! ], {w, 1, 128}]

A084931 Consider trajectory of n under repeated applications of the function f(x) = 'Sum of the prime factors of x (with multiplicity)' (see A029908). Sequence gives composite numbers n that end at a prime m that divides n and m is greater than any m's seen already.

Original entry on oeis.org

15, 21, 182, 494, 1219, 2852, 3182, 9782, 19339, 19982, 22454, 72836, 76814, 102134, 156782, 192182, 423182, 750979, 758894, 1364534, 1465454, 1548782, 2376182, 3379982, 4066934, 4204982

Offset: 1

Sven Simon, Jun 12 2003

With a prime triple (p,p+4,p+6), the number a(n) = 2*p*(p+6) is always in the sequence, f( f( 2*p*(p+6) )) = f( 2*(p+4) ) = p+6. Such prime triples can be found in sequence A022005.
As long as two successive triples (p1,p1 + 4,p1 + 6) and (p2,p2 + 4,p2 + 6) of A022005 have p2 < 1.2*p1, no other numbers occur in the sequence between a(n1) and a(n2), this holds at least for larger p1 > 500. Other types of prime sets occurring in the sequence: (p,p+4,3p-4) with F( F( (p+4)*(3p-4))) = F( 4p ) = p + 4 (p,p+6,p+8) with F( F( 4*p*(p+8) )) = F( 2*(p+6) ) = p + 8.
Large examples of (p,p+4,++6)-triples: (108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7, + 11, + 13 (4135 digits, David Broadhurst) (18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7, + 11, +13 (4134 digits, David Broadhurst) Record examples of prime triples can be found on Tony Forbes's web site. There are triples of type (p,p+4,p+6) too.

			a(10) = 19982: f(f(19982)) = f(f(2*97*103)) = f(2+97+103) = f(202) = f(2*101) = 2+101 = 103.

Cf. A022005, A048133, A084932 (primes reached).

A075860 a(n) is the fixed point reached when the map x -> A008472(x) is iterated, starting from x = n, with the convention a(1)=0.

Original entry on oeis.org

0, 2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 5, 13, 3, 2, 2, 17, 5, 19, 7, 7, 13, 23, 5, 5, 2, 3, 3, 29, 7, 31, 2, 3, 19, 5, 5, 37, 7, 2, 7, 41, 5, 43, 13, 2, 5, 47, 5, 7, 7, 7, 2, 53, 5, 2, 3, 13, 31, 59, 7, 61, 3, 7, 2, 5, 2, 67, 19, 2, 3, 71, 5, 73, 2, 2, 7, 5, 5, 79, 7, 3, 43, 83, 5, 13, 2, 2, 13, 89

Offset: 1

Joseph L. Pe, Oct 15 2002

For n>1, the sequence reaches a fixed point, which is prime.
From Robert Israel, Mar 31 2020: (Start)
a(n) = n if n is prime.
a(n) = n/2 + 2 if n is in A108605.
a(n) = n/4 + 2 if n is in 4*A001359. (End)

			Starting with 60 = 2^2 * 3 * 5 as the first term, add the prime factors of 60 to get the second term = 2 + 3 + 5 = 10. Then add the prime factors of 10 = 2 * 5 to get the third term = 2 + 5 = 7, which is prime. (Successive terms of the sequence will be equal to 7.) Hence a(60) = 7.

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

Cf. A008472 (sum of prime divisors of n), A029908.

f:= proc(n) option remember;
  if isprime(n) then n
  else procname(convert(numtheory:-factorset(n), `+`))
  fi
end proc:
f(1):= 0:
map(f, [$1..100]); # Robert Israel, Mar 31 2020

f[n_] := Module[{a}, a = n; While[ !PrimeQ[a], a = Apply[Plus, Transpose[FactorInteger[a]][[1]]]]; a]; Table[f[i], {i, 2, 100}]
(* Second program: *)
a[n_] := If[n == 1, 0, FixedPoint[Total[FactorInteger[#][[All, 1]]]&, n]];
Array[a, 100] (* Jean-François Alcover, Apr 01 2020 *)

fp(n, pn) = if (n == pn, n, fp(vecsum(factor(n)[, 1]), n));
a(n) = if (n==1, 0, fp(n, 0)); \\ Michel Marcus, Sep 02 2023

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

Better description from Labos Elemer, Apr 09 2003

Name clarified by Michel Marcus, Sep 02 2023

A002217 Starting with n, repeatedly calculate the sum of prime factors (with repetition) of the previous term, until reaching 0 or a fixed point: a(n) is the number of terms in the resulting sequence.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

nonn

			20 -> 2+2+5 = 9 -> 3+3 = 6 -> 2+3 = 5, so a(20) = length of sequence {20,9,6,5} = 4.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Christian N. K. Anderson, Table of n, a(n) for n = 1..10000 (first 1000 terms are from T. D. Noe)
Christian N. K. Anderson, n, the fixed point, a(n), and the trajectories for n = 1..10000.
M. Lal, Iterates of a number-theoretic function, Math. Comp., 23 (1969), 181-183.
Eric Weisstein's World of Mathematics, Sum of Prime Factors

Cf. A001414 (sum of prime factors of n), A029908 (fixed point that is reached).

sopfr[n_] := Times @@@ FactorInteger[n] // Total;
a[1] = 2; a[n_] := Length[ FixedPointList[sopfr, n]] - 1;
Array[a, 105] (* Jean-François Alcover, Feb 09 2018 *)

More terms and better description from Reinhard Zumkeller, Apr 08 2003

Incorrect comment removed by Harvey P. Dale, Aug 16 2011

A082081 a(n) = fixed point when the pseudo-log function A008475[ ] is iterated.

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 8, 9, 7, 11, 7, 13, 9, 8, 16, 17, 11, 19, 9, 7, 13, 23, 11, 25, 8, 27, 11, 29, 7, 31, 32, 9, 19, 7, 13, 37, 7, 16, 13, 41, 7, 43, 8, 9, 25, 47, 19, 49, 27, 9, 17, 53, 29, 16, 8, 13, 31, 59, 7, 61, 9, 16, 64, 11, 16, 67, 7, 8, 9, 71, 17, 73, 16, 11, 23, 11, 11, 79, 7, 81

Offset: 1

Labos Elemer, Apr 08 2003

nonn

Fixed point is always a prime or a power of prime: fixed points are terms of A000961.

			n=10!=3628800:list to fixed point={3628800,369,50,27}.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A008475, A001414, A056239, A029908, A082083.

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]}] sex[x_] := Apply[Plus, ba[x]^ep[x]] Table[FixedPoint[sex, w], {w, 1, 128}]

A036430 Number of iterations needed to reach 1 under the map n -> Omega(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 3, 2, 1, 3, 2, 2, 2, 3, 1, 3, 2, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 2, 1, 3, 2

Offset: 1

Dann Toliver

			16 -> 4 -> 2 -> 1 and thus a(16) = 3.

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

Cf. A029908, A073855.

a(n)=my(s);while(n>1,n=bigomega(n);s++);s \\ Charles R Greathouse IV, Apr 25 2012

a(n) = a(bigomega(n)) + 1 for n > 1. - Vladeta Jovovic, Jul 10 2004

a(n) = O(log* n). - Charles R Greathouse IV, Apr 25 2012

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 21 2003

Formula corrected by Charles R Greathouse IV, Apr 25 2012

A029909 Starting with n (but omitting the primes), repeatedly sum prime factors (counted with multiplicity) until reaching a limit.

Original entry on oeis.org

0, 4, 5, 5, 5, 7, 7, 5, 5, 5, 5, 5, 7, 13, 5, 7, 5, 5, 11, 7, 7, 5, 19, 7, 7, 7, 5, 11, 7, 5, 11, 7, 11, 5, 7, 5, 17, 11, 5, 13, 13, 31, 7, 5, 13, 7, 5, 5, 7, 5, 5, 7, 5, 13, 23, 5, 5, 13, 7, 43, 5, 13, 11, 7, 17, 13, 5, 5, 19, 5, 5, 13, 5, 17, 5, 13, 19, 5, 5, 13, 5, 11, 5, 5, 11, 5

Offset: 1

Dann Toliver

nonn

Is this sequence generating ALL prime numbers (greater than 3) ? Also how many times each prime (greater than 3)is generated in this sequence? - Alexander R. Povolotsky, Nov 05 2008

Records appear to correspond to A006512 (n>2). - Bill McEachen, Jun 10 2025

			20 -> 2+2+5 = 9 -> 3+3 = 6 -> 2+3 = 5.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A029908.

A081758 Sum of prime factors (with repetition) of sum of prime factors (with repetition) of n.

Original entry on oeis.org

2, 3, 4, 5, 5, 7, 5, 5, 7, 11, 7, 13, 6, 6, 6, 17, 6, 19, 6, 7, 13, 23, 6, 7, 8, 6, 11, 29, 7, 31, 7, 9, 19, 7, 7, 37, 10, 8, 11, 41, 7, 43, 8, 11, 10, 47, 11, 9, 7, 9, 17, 53, 11, 8, 13, 13, 31, 59, 7, 61, 14, 13, 7, 8, 8, 67, 10, 15, 9, 71, 7, 73, 16, 13, 23, 8, 8, 79, 13, 7, 43, 83, 9, 13, 11

Offset: 2

Reinhard Zumkeller, Apr 08 2003

nonn

a(n) = A001414(A001414(n)). For further iterations see: A029908, A002217;

a(n)=n iff n is prime or 4.

			18 = 2*3*3 -> 2+3+3 = 2*2*2 -> 2+2+2 = 6: a(18) = 6.

Harvey P. Dale, Table of n, a(n) for n = 2..1000

spf[n_]:=Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[n]]; Table[Nest[Total[spf[#]]&,n,2],{n,2,90}] (* Harvey P. Dale, May 31 2025 *)

A082880 Largest value of A075860(j) when j runs through composite numbers between n-th and (n+1)-th primes. That is, the largest fixed-point[=prime] reached by iteration of A008472(=sum of prime factors) initiated with composite values between two consecutive primes.

Original entry on oeis.org

0, 2, 5, 7, 5, 3, 5, 13, 5, 7, 19, 7, 5, 13, 7, 31, 7, 7, 19, 5, 7, 43, 13, 19, 7, 13, 2, 5, 7, 61, 7, 19, 3, 73, 7, 7, 7, 43, 13, 19, 7, 13, 5, 7, 2, 103, 109, 3, 5, 31, 61, 7, 13, 19, 13, 31, 7, 139, 19, 2, 73, 151, 7, 5, 3, 43, 13, 31, 19, 13, 181, 19, 13, 7, 193, 23, 199, 29, 103, 73

Offset: 1

Labos Elemer, Apr 16 2003

nonn

Cf. A075860, A008472, A082087, A082088, A082881, A029908.

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[Max[0, Table[FixedPoint[sopf, w], {w, 1+Prime[n], Prime[n+1]-1}]], {n, 80}]

a(n) = Max_{x=1+prime(n)..prime(n+1)-1} A075860(x).

A082086 Fixed points when A001414 is iterated and started at factorials of prime numbers.

Original entry on oeis.org

2, 5, 5, 5, 13, 23, 61, 19, 5, 283, 5, 23, 13, 29, 7, 61, 947, 29, 137, 11, 353, 23, 199, 5, 47, 2381, 5, 103, 359, 13, 5, 5, 7, 5, 47, 19, 577, 7, 5, 29, 5, 5, 5, 41, 11, 23, 239, 7, 11, 5, 11, 5, 23, 11821, 31, 5, 5, 13669, 7, 193, 5, 991, 89, 7, 13, 18199, 131, 113, 20849

Offset: 1

Labos Elemer, Apr 08 2003

nonn

			n=100, prime(100) = 541, start at 541!, the list is: {541!,...,46374,195,21,10,7} so a(100)=7.

Cf. A008475, A082081, A000142, A000040, A000961.

sopfr[n_] := Total[Times @@@ FactorInteger[n]]; a[n_] := FixedPoint[sopfr, Prime[n]!]; Array[a, 100] (* Jean-François Alcover, May 06 2017 *)

a(n) = A029908(A000142(A000040(n))) = A029908(prime(n)!).

cp's OEIS Frontend

A082084 a(n)=A029908[n! ]=A029908[A000142[n]] Fixed points of iterated A001414 function if started at factorials as initial values.

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A084931 Consider trajectory of n under repeated applications of the function f(x) = 'Sum of the prime factors of x (with multiplicity)' (see A029908). Sequence gives composite numbers n that end at a prime m that divides n and m is greater than any m's seen already.

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A075860 a(n) is the fixed point reached when the map x -> A008472(x) is iterated, starting from x = n, with the convention a(1)=0.

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A002217 Starting with n, repeatedly calculate the sum of prime factors (with repetition) of the previous term, until reaching 0 or a fixed point: a(n) is the number of terms in the resulting sequence.

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A082081 a(n) = fixed point when the pseudo-log function A008475[ ] is iterated.

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Mathematica

A036430 Number of iterations needed to reach 1 under the map n -> Omega(n).

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A029909 Starting with n (but omitting the primes), repeatedly sum prime factors (counted with multiplicity) until reaching a limit.

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A081758 Sum of prime factors (with repetition) of sum of prime factors (with repetition) of n.

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Mathematica

A082880 Largest value of A075860(j) when j runs through composite numbers between n-th and (n+1)-th primes. That is, the largest fixed-point[=prime] reached by iteration of A008472(=sum of prime factors) initiated with composite values between two consecutive primes.