A008472 -id:A008472 - cp's OEIS frontend

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.

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

A323171 Numerator of the average of distinct prime factors of n (A008472(n)/A001221(n)).

Original entry on oeis.org

2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 5, 13, 9, 4, 2, 17, 5, 19, 7, 5, 13, 23, 5, 5, 15, 3, 9, 29, 10, 31, 2, 7, 19, 6, 5, 37, 21, 8, 7, 41, 4, 43, 13, 4, 25, 47, 5, 7, 7, 10, 15, 53, 5, 8, 9, 11, 31, 59, 10, 61, 33, 5, 2, 9, 16, 67, 19, 13, 14, 71, 5, 73, 39, 4, 21, 9, 6, 79, 7, 3, 43, 83, 4, 11, 45, 16, 13, 89, 10, 10, 25, 17, 49, 12, 5

Offset: 2

Antti Karttunen, Jan 05 2019

			Fractions begins with 2, 3, 2, 5, 5/2, 7, 2, 3, 7/2, 11, 5/2, 13, ...

Antti Karttunen, Table of n, a(n) for n = 2..20000

Cf. A323172 (denominators).

Cf. A001221, A008472, A123528.

a[n_] := Numerator[Mean[FactorInteger[n][[;; , 1]]]]; Array[a, 100, 2] (* Amiram Eldar, Sep 17 2024 *)

A008472(n) = vecsum(factor(n)[, 1]); \\ From A008472
A323171(n) = (numerator(A008472(n)/omega(n)));

A323172 Denominator of the average of distinct prime factors of n (A008472(n)/A001221(n)).

Original entry on oeis.org

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

Offset: 2

Antti Karttunen, Jan 05 2019

Antti Karttunen, Table of n, a(n) for n = 2..65539

Cf. A323171 (numerators).

Cf. A001221, A008472, A123529, A322591.

a[n_] := Denominator[Mean[FactorInteger[n][[;; , 1]]]]; Array[a, 100, 2] (* Amiram Eldar, Sep 17 2024 *)

A008472(n) = vecsum(factor(n)[, 1]); \\ From A008472
A323172(n) = (denominator(A008472(n)/omega(n)));

A089352 Numbers that are divisible by the sum of their distinct prime factors (A008472).

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 70, 71, 73, 79, 81, 83, 84, 89, 90, 97, 101, 103, 105, 107, 109, 113, 120, 121, 125, 127, 128, 131, 137, 139, 140, 149, 150, 151, 157, 163, 167, 168, 169, 173

Offset: 1

Ramin Naimi (rnaimi(AT)oxy.edu), Dec 26 2003

The Koninck & Luca bound of x / exp(c(1 + o(1))sqrt(log x log log x)) on A158804 applies equally to this sequence. - Charles R Greathouse IV, Sep 08 2012

			84=2*2*3*7 is divisible by 2+3+7.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jean-Marie de Koninck, Florian Luca, Integers divisible by the sum of their prime factors, Mathematika 52:1-2 (2005), pp. 69-77.

Cf. A008472 (sopf).

Different from A071139.

primeDivisors[n_] := Select[Divisors[n], PrimeQ]; primeSumDivQ[n_] := 0 == Mod[n, Apply[Plus, primeDivisors[n]]]; Select[Range[2, 300], primeSumDivQ]
Select[Range[2, 175], Divisible[#, Plus @@ First /@ FactorInteger[#]] &] (* Jayanta Basu, Aug 13 2013 *)

is(n)=my(f=factor(n)[,1]);n%sum(i=1,#f,f[i])==0 \\ Charles R Greathouse IV, Feb 01 2013

Name edited by Michel Marcus, Jul 15 2020

A075657 Numbers n such that sum of digits (A007953) is a divisor of sum of prime divisors (A008472).

Original entry on oeis.org

2, 3, 5, 7, 10, 42, 70, 84, 91, 100, 104, 110, 114, 115, 130, 143, 148, 154, 160, 170, 182, 185, 212, 215, 221, 222, 228, 230, 234, 238, 250, 266, 295, 304, 312, 326, 336, 372, 402, 412, 425, 437, 460, 468, 485, 494, 516, 555, 558, 583, 700, 702, 721, 730

Offset: 1

Floor van Lamoen, Sep 23 2002 and Sep 30 2002

			digsum(10) = 1 + 0 = 1, PrimeDivisors(10) = PrimeDivisors(2 *5) = {2,5} and sopf(10) = 2 + 5 = 7 = 7*1.
digsum(154) = 1 + 5 + 4 = 10, PrimeDivisors(154) = PrimeDivisors(2 * 7 * 11) = {2,7,11} and sopf(154) = 2+7+11 = 20 = 2*10.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

[m:m in [2..730]| &+PrimeDivisors(m) mod &+Intseq(m) eq 0]; // Marius A. Burtea, Jul 11 2019

Select[Range[2,800],Divisible[Total[Select[Divisors[#],PrimeQ]], Total[ IntegerDigits[#]]]&] (* Harvey P. Dale, Sep 23 2012 *)

A134382 a(n) is the smallest number k larger than a(n-1) such that nd(k)sopf(k)=sigma(k), where d is the number of divisors (A000005) and sopf the sum of prime factors without repetition (A008472).

Original entry on oeis.org

20, 140, 464, 660, 1276, 1365, 2204, 2508, 2805, 2907, 5590, 5698, 5742, 6006, 7395, 8680, 14645, 15052, 18875, 19170, 19740, 23871, 34579, 34804, 35164, 35244, 35934, 38121, 106805, 114953, 261536, 503082

Offset: 1

Enoch Haga, Oct 23 2007

Sequence suggested by Puzzle 419 in Carlos Rivera's The Prime Puzzles & Problems Connection.
For n=33, the search for terms k that satisfy 33*d(k)*sopf(k)=sigma(k), without being greater than a(32), gives 21070, 25585, 30702, 36120, 41710, 49256, 52269, 68906, 74692, 92785, 95702, 111342, 117626, 383086 with no other terms up to 10^9. So this sequence might well be complete. - Michel Marcus, Oct 02 2019
I confirm that the solutions for n=33 listed above are complete, thus the sequence stops at n=32. - Max Alekseyev, Sep 18 2024

Carlos Rivera, Puzzle 419. Four SOPF questions, Prime Puzzles.

Subsequence of A070222. - R. J. Mathar, Feb 05 2010

Cf. A134383, A134384, A134385, A134386.

A008472 := proc(n) local divs,i ; if n = 1 then 0; else divs := ifactors(n)[2] ; add( op(1,i),i=divs) ; fi ; end: A134382 := proc(n) option remember ; local k,kmin ; if n = 1 then kmin := 1 ; else kmin := procname(n-1)+1 ; fi ; for k from kmin do if numtheory[sigma](k) = n* numtheory[tau](k)*A008472(k) then RETURN(k) ; fi ; od: end: for n from 1 to 30 do print( A134382(n)) ; od: # R. J. Mathar, Nov 16 2007, Jun 24 2009

sopf[1] = 0; sopf[n_] := Total[FactorInteger[n][[All, 1]]]; a[n_] := a[n] = For[k = If[n == 1, 1, a[n-1] + 1], True, k++, If[DivisorSigma[1, k] == n*DivisorSigma[0, k]*sopf[k], Return[k]]]; Table[Print[a[n]]; a[n], {n, 1, 32}] (* Jean-François Alcover, Sep 12 2013 *)

lista(nn) = {lasta = 2; for (n=1, nn, k = lasta; while ((f = factor(k)) && (n*numdiv(k)*sum(j=1,#f~,f[j,1]) != sigma(k)), k++); print1(k, ", "); lasta = k;);} \\ Michel Marcus, Feb 25 2016

a(n) > a(n-1): n*A000005(a(n))*A008472(a(n)) = A000203(a(n)). - R. J. Mathar, Nov 16 2007, Jun 24 2009

Edited by R. J. Mathar, Nov 16 2007
A-number in formula and Maple program corrected by R. J. Mathar, Jun 24 2009
a(32) from R. J. Mathar, Feb 05 2010
full,fini keywords added by Max Alekseyev, Sep 18 2024

A058977 For a rational number p/q let f(p/q) = sum of distinct prime factors (A008472) of p+q divided by number of distinct prime factors (A001221) of p+q; a(n) is obtained by iterating f, starting at n/1, until an integer is reached, or if no integer is ever reached then a(n) = 0.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jan 14 2001

A247462 gives number of iterations needed to reach a(n). - Reinhard Zumkeller, Sep 17 2014

			f(5/1) = 5/2 and f(5/2) = 7, so a(5)=7.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
P. Schogt, The Wild Number Problem: math or fiction?, arXiv preprint arXiv:1211.6583 [math.HO], 2012. - From _N. J. A. Sloane_, Jan 03 2013

Cf. A058971, A058972.
Cf. A008472, A001221.
Cf. A247462, A247468.

import Data.Ratio ((%), numerator, denominator)
a058977 = numerator . until ((== 1) . denominator) f . f . fromIntegral
   where f x = a008472 z % a001221 z
               where z = numerator x + denominator x
-- Reinhard Zumkeller, Aug 29 2014

nxt[n_]:=Module[{s=Numerator[n]+Denominator[n]},Total[Transpose[ FactorInteger[ s]][[1]]]/PrimeNu[s]]; Table[NestWhile[nxt,nxt[n],!IntegerQ[#]&],{n,90}] (* Harvey P. Dale, Mar 15 2013 *)

f2(p,q) = my(f=factor(p+q)[,1]~); vecsum(f)/#f;
f1(r) = f2(numerator(r), denominator(r));
loop(list) = {my(v=Vecrev(list)); for (i=2, #v, if (v[i] == v[1], return(1)););}
a(n) = {my(ok=0, m=f2(n,1), list=List()); while(denominator(m) != 1, m = f1(m); listput(list, m); if (loop(list), return (0));); return(m);} \\ Michel Marcus, Feb 09 2022

More terms from Matthew Conroy, Apr 18 2001

A068936 Numbers having the sum of distinct prime factors not greater than the sum of exponents in prime factorization, A008472(k) <= A001222(k).

Original entry on oeis.org

1, 4, 8, 16, 27, 32, 48, 64, 72, 81, 96, 108, 128, 144, 162, 192, 216, 243, 256, 288, 320, 324, 384, 432, 486, 512, 576, 640, 648, 729, 768, 800, 864, 972, 1024, 1152, 1280, 1296, 1458, 1536, 1600, 1728, 1792, 1944, 2000, 2048, 2187, 2304, 2560, 2592, 2916

Offset: 1

Reinhard Zumkeller, Mar 08 2002

nonn

The product of any two terms is also a term. - Amiram Eldar, May 14 2025

			a(5) = 27 = 3^3, 3 = 3.
a(10) = 81 = 3^4, 3 < 4.
a(100) = 16000 = 2^7 * 5^3,  2+5 < 7+3.
a(1000) = 10321920 = 2^15 * 3^2 * 5 * 7, 2+3+5+7 < 15+2+1+1.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)

Cf. A001222, A008472, A068935, A068937, A068938.

Subsequences: A054411, A100716, A257278.

a068936 n = a068936_list !! (n-1)
a068936_list = [x | x <- [1..], a008472 x <= a001222 x]
-- Reinhard Zumkeller, Nov 10 2013

fQ[n_] := Block[{f = FactorInteger@n}, Plus @@ Last /@ f >= Plus @@ First /@ f]; Select[ Range@3000, fQ@ # &] (* Robert G. Wilson v, Jan 16 2006 *)
Select[Range@ 3000, First@ Differences@ Map[Total, Transpose@ FactorInteger@ #] >= 0 &] (* Michael De Vlieger, Dec 08 2016 *)

isok(k) = {my(f = factor(k)); vecsum(f[,1]) <= bigomega(f);} \\ Amiram Eldar, May 14 2025

More terms from Robert G. Wilson v, Jan 16 2006

A076387 Numbers n such that sum of digits in base 9 is a divisor of sum of prime divisors (A008472).

Original entry on oeis.org

2, 3, 5, 7, 9, 21, 27, 65, 69, 70, 81, 84, 90, 110, 123, 126, 130, 133, 154, 189, 222, 228, 243, 252, 259, 264, 327, 329, 333, 340, 342, 343, 350, 365, 372, 381, 402, 434, 450, 516, 528, 580, 588, 618, 621, 650, 684, 729, 730, 731, 738, 740, 741, 756, 765, 774

Offset: 1

Floor van Lamoen, Oct 08 2002

The sequence is infinite because, for m = 9^k, k >= 0, digsum(m_9) = 1. - Marius A. Burtea, Jul 10 2019

			21 = 23_9, digsum(23_9) = 5, PrimeDivisors(21) = {3, 7}, sopf(21) = 3+7 = 10 = 5*2.

Marius A. Burtea, Table of n, a(n) for n = 1..5189

Cf. A075657, A076380 - A076387.

[n: n in [1..800]| &+PrimeDivisors(n) mod &+Intseq(n,9) eq 0] ; // Marius A. Burtea, Jul 10 2019

A076387 := proc(n) local i,j,t,t1, sod, sopd; t := NULL; for i from 2 to n do t1 := i; sod := 0; while t1 <> 0 do sod := sod + (t1 mod 9); t1 := floor(t1/9); od; sopd := 0; j := 1; while ithprime(j) <= i do if i mod ithprime(j) = 0 then sopd := sopd+ithprime(j); fi; j := j+1; od; if sopd mod sod = 0 then t := t,i; fi; od; t; end;

{for(ixp=2,783,
casi=ixp;cvst=0;dsu=0;M=factor(ixp);smt=0;
for(i=1,matsize(M)[1],smt=smt+M[i, 1]);
while(casi!=0,
cvd=casi%9;dsu=dsu+cvd;casi=(casi-cvd)/9);
if(smt%dsu==0,print1(ixp,", ")))} \\ Douglas Latimer, May 08 2012

A068938 Numbers having the sum of distinct prime factors greater than the sum of exponents in prime factorization, A008472(n)>A001222(n).

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Reinhard Zumkeller, Mar 08 2002

nonn

			12 is included because 12 = 2^2 * 3^1 and 2+3 > 2+1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A068935, A068936, A054411, A068937.

sdfQ[n_]:=Module[{fi=Transpose[FactorInteger[n]]},Total[fi[[1]]] > Total[ fi[[2]]]]; Select[Range[80],sdfQ] (* Harvey P. Dale, Jul 23 2013 *)

More terms from David Wasserman, Jun 17 2002

cp's OEIS Frontend

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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A323171 Numerator of the average of distinct prime factors of n (A008472(n)/A001221(n)).

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A323172 Denominator of the average of distinct prime factors of n (A008472(n)/A001221(n)).

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A089352 Numbers that are divisible by the sum of their distinct prime factors (A008472).

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A075657 Numbers n such that sum of digits (A007953) is a divisor of sum of prime divisors (A008472).

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A134382 a(n) is the smallest number k larger than a(n-1) such that n*d(k)*sopf(k)=sigma(k), where d is the number of divisors (A000005) and sopf the sum of prime factors without repetition (A008472).

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A058977 For a rational number p/q let f(p/q) = sum of distinct prime factors (A008472) of p+q divided by number of distinct prime factors (A001221) of p+q; a(n) is obtained by iterating f, starting at n/1, until an integer is reached, or if no integer is ever reached then a(n) = 0.

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A134382 a(n) is the smallest number k larger than a(n-1) such that nd(k)sopf(k)=sigma(k), where d is the number of divisors (A000005) and sopf the sum of prime factors without repetition (A008472).