A008472 -id:A008472 - cp's OEIS frontend

A076380 Sum of digits in base 2 is a divisor of sum of prime divisors (A008472).

2, 4, 8, 14, 15, 16, 26, 28, 32, 33, 35, 38, 39, 42, 45, 51, 52, 56, 64, 65, 66, 74, 75, 76, 81, 84, 91, 95, 98, 104, 112, 114, 119, 126, 128, 129, 130, 132, 134, 135, 146, 148, 152, 153, 154, 161, 168, 170, 175, 194, 196, 198, 206, 208, 215, 221, 222, 224, 225

Offset: 0

Floor van Lamoen, Oct 08 2002

Prime divisors counted without multiplicity. - Harvey P. Dale, Jan 08 2019

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A075657, A076381 - A076387.

A076380 := 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 2); t1 := floor(t1/2); 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;

Select[Range[2,250],Divisible[Total[FactorInteger[#][[All,1]]],Total[ IntegerDigits[ #,2]]]&] (* Harvey P. Dale, Jan 08 2019 *)

A068935 Numbers having the sum of distinct prime factors less than the sum of exponents in prime factorization, A008472(n) < A001222(n).

Original entry on oeis.org

8, 16, 32, 64, 81, 96, 128, 144, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576, 640, 648, 729, 768, 864, 972, 1024, 1152, 1280, 1296, 1458, 1536, 1600, 1728, 1944, 2048, 2187, 2304, 2560, 2592, 2916, 3072, 3200, 3456, 3584, 3888, 4000, 4096, 4374, 4608

Offset: 1

Reinhard Zumkeller, Mar 08 2002

nonn

			144 is included because 144 = 2^4 * 3^2 and 2+3 < 4+2.

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

Cf. A068936, A054411, A068937, A068938.

okQ[n_]:=Module[{tfi=Transpose[FactorInteger[n]]},Total[First[tfi]]Harvey P. Dale, Jan 17 2011 *)

isok(n) = vecsum(factor(n)[,1]) < bigomega(n); \\ Michel Marcus, Apr 25 2016

More terms from Michel Marcus, Apr 25 2016

A068937 Numbers having the sum of distinct prime factors not less than the sum of exponents in prime factorization, A008472(n)>=A001222(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 1

Reinhard Zumkeller, Mar 08 2002

nonn

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

Cf. A068935, A068936, A054411, A068938.

Select[Range[76], Plus @@ (f = FactorInteger[#])[[;;,1]] >= Plus @@ f[[;;,2]] &] (* Amiram Eldar, Nov 14 2019 *)

More terms from David Wasserman, Jun 17 2002

A065925 Smallest k such that sopf(n+k) = sopf(k), where sopf = A008472.

Original entry on oeis.org

5, 2, 7, 4, 114, 2, 5, 8, 13, 10, 25, 4, 5, 2, 19, 16, 85, 6, 5, 5, 209, 22, 25, 3, 493, 26, 31, 4, 20, 2, 5, 32, 7, 34, 516, 12, 33, 38, 10, 10, 99, 6, 5, 44, 57, 46, 25, 6, 5, 50, 49, 52, 52, 18, 855, 8, 61, 58, 295, 4, 261, 2, 91, 64, 602, 6, 5, 68, 21, 10, 25, 9, 7, 74, 13, 76

Offset: 1

Jason Earls, Nov 28 2001

nonn

			a(6) = 2 because A008472(2) = A008472(6+2) = 2, but A008472(1) = 0 doesn't equal A008472(6+1) = 7.

J. Earls, Mathematical Bliss, Pleroma Publications, 2009, pages 99-100. ASIN: B002ACVZ6O [From Jason Earls, Nov 26 2009]

Harry J. Smith, Table of n, a(n) for n=1..1000
Carlos Rivera, Conjecture 25. sopf(n) = sopf(n+k), The Prime Puzzles and Problems Connection.

Cf. A008472, A065926, A065927.

Table[k = 1; While[Total[FactorInteger[n + k][[All, 1]]] != Total[FactorInteger[k][[All, 1]]], k++]; k, {n, 76}] (* Michael De Vlieger, Jan 11 2017 *)

sopf(n) = local(fac, i); fac=factor(n); sum(i=1,matsize(fac)[1],fac[i,1])
A065925(m)={local(k,n); for(k=1,m,n=1; while(sopf(n)!=sopf(n+k), n++); print1(n,","))} \\ Klaus Brockhaus

from sympy import primefactors
from itertools import count, dropwhile
def sopf(n): return sum(p for p in primefactors(n))
def a(n):
  k = 1
  while sopf(n+k) != sopf(k): k += 1
  return k
print([a(n) for n in range(1, 77)]) # Michael S. Branicky, May 02 2021

A075653 a(n) = n + sopf(n), where sopf is the sum of the distinct prime factors of n (A008472).

Original entry on oeis.org

1, 4, 6, 6, 10, 11, 14, 10, 12, 17, 22, 17, 26, 23, 23, 18, 34, 23, 38, 27, 31, 35, 46, 29, 30, 41, 30, 37, 58, 40, 62, 34, 47, 53, 47, 41, 74, 59, 55, 47, 82, 54, 86, 57, 53, 71, 94, 53, 56, 57, 71, 67, 106, 59, 71, 65, 79, 89, 118, 70, 122, 95, 73, 66, 83, 82, 134, 87, 95

Offset: 1

Joseph L. Pe, Oct 11 2002

For 1 <= k <= n, add sigma(k) if k is a prime factor of n, otherwise add 1. For example, a(6) = 11 since for k = 1,2,.. we have 1 + sigma(2) + sigma(3) + 1 + 1 + 1 = 1 + (1+2) + (1+3) + 1 + 1 + 1 = 11. - Wesley Ivan Hurt, Oct 18 2021

			6 + sum of prime factors of 6 = 6 + 2 + 3 = 11, so a(6) = 11.

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

Cf. A008472, A075254, A075255, A076694.

Flatten[Append[{1}, Table[n + Apply[Plus, Transpose[FactorInteger[n]][[1]]], {n, 2, 100}]]]
Join[{1},Table[n+Total[FactorInteger[n][[All,1]]],{n,2,70}]] (* Harvey P. Dale, Sep 29 2016 *)

a(n) = n + vecsum(factor(n)[,1]); \\ Michel Marcus, Feb 22 2017

a(n) = n + A008472(n).

a(p) = 2p for primes p. - Wesley Ivan Hurt, Oct 18 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])].

A226479 Numbers n such that (sopf(n)*d(n))^2 = sigma(n) where sopf(n) = sum of distinct prime factors of n (A008472) and d(n) = number of divisors of n.

Original entry on oeis.org

22446139, 26116291, 28097023, 30236557, 31090489, 31124341, 39618558, 41628195, 49941589, 51777957, 61137673, 62224039, 66960589, 71096795, 71334867, 71585139, 72304400, 82266591, 83045869, 92346023, 92837591, 105183961, 114762567, 117908994, 123563821

Offset: 1

Donovan Johnson, Jun 09 2013

nonn

Suggested by N. J. A. Sloane.

			n = 22446139 = 31*67*101*107. sopf(n) = 31+67+101+107 = 306. d(n) = 16. (sopf(n)*d(n))^2 = (306*16)^2 = 23970816 = sigma(n).

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000005, A000203, A006532, A008472, A126028, A226480.

A260310 Pairs with balanced sums of prime divisors (A008472) and inverse prime divisors (A069359), ordered by larger members.

Original entry on oeis.org

3, 8, 7, 16, 11, 18, 7, 27, 17, 45, 29, 50, 41, 54, 53, 60, 31, 64, 71, 84, 29, 99, 107, 132, 61, 147, 41, 153, 131, 162, 53, 207, 157, 220, 113, 225, 179, 228, 239, 240, 131, 242, 79, 243, 73, 245, 127, 255, 127, 256, 229, 264, 223, 280, 113, 297, 199, 315, 73, 325, 317, 336, 181, 338, 43, 343, 269, 348

Offset: 1

Juri-Stepan Gerasimov, Jul 22 2015

nonn

Consider pairs (x,y) of numbers where sum(p|x) p + sum(q|y) q = x*sum(p|x) 1/p + y*sum(q|y) 1/q where p, q are primes and sum(p|x) p > sum(q|y) q.
Or, pairs of numbers x and y where A008472(x) + A008472(y) = A069359(x) + A069359(y) where A008472(x) > A008472(y).
A001222(a(2n -1)) = 1 and A001222(a(2n)) >= 3.
For the vast majority of the time, a(2n-1) is prime. There seems to be about 1 pair per decade.
Conjecture: a(2n) < a(2n+2) for all n>0, but there are many times (1/10.84) that a(2n) + 1 = a(2n+2).
Conjecture: if a(2n-1) is prime then a(2n) is composite and vice versa. And when a(2n-1) is composite, it is congruent to 0 (mod 6). - Robert G. Wilson v, Jul 22 2015
The first conjecture appears to be satisfied because if both x and y were prime then the sum of the A008472 were the sum of the two primes and the sum of the A069359 were two. - R. J. Mathar, Aug 03 2015

			3 and 8 is first pair of this sequence because A008472(3) + A008472(8) = 3 + 2 = 5 is equal to A069359(3) + A069359(8) = 1 + 4 = 5.

Robert G. Wilson v, Table of n, a(n) for n = 1..19812

Cf. A001222, A008472, A069359.

f[n_] := f[n] = Block[{fi = FactorInteger[n][[All, 1]]}, {Plus @@ fi, n*Plus @@ (1/fi)}] /; n > 0; k =3; lst = {}; While[ k < 400, j = 2; While[ j < k, If[ f[k][[1]] + f[j][[1]] == f[k][[2]] + f[j][[2]] && f[k][[1]] != f[k][[2]], AppendTo[lst, {j,k}]]; j++]; k++]; lst // Flatten (* Robert G. Wilson v, Jul 22 2015 *)

Corrected and edited by Robert G. Wilson v, Jul 22 2015

A328051 Numbers m such that sigma(m)/(d(m)*sopf(m)) is an integer, where d is the number of divisors (A000005) and sopf the sum of prime factors without repetition (A008472).

Original entry on oeis.org

20, 35, 42, 54, 140, 189, 195, 207, 209, 276, 378, 464, 470, 500, 506, 510, 527, 540, 608, 660, 672, 741, 846, 864, 875, 899, 923, 945, 989, 1029, 1120, 1276, 1316, 1323, 1334, 1349, 1365, 1519, 1539, 1564, 1595, 1715, 1725, 1736, 1755, 1815, 1880, 1887, 1914, 2058

Offset: 1

Michel Marcus, Oct 03 2019

nonn

This sequence is motivated by the short fate of A134382.

			For n=20, sigma(20)/(d(20)*sopf(20)) = 42/(6*7) = 1, an integer, so 20 is a term.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A008472, A134382, A328052.

[k: k in [2..2100]|IsIntegral(DivisorSigma(1,k)/(#Divisors(k)*(&+PrimeDivisors(k))))]; // Marius A. Burtea, Oct 03 2019

f[p_, e_] := (p^(e + 1) - 1)/((e + 1)*(p - 1)); Select[Range[2, 2100], IntegerQ[ Times @@ (f @@@ (fct = FactorInteger[#])) / Plus @@ (fct[[;; , 1]])] &] (* Amiram Eldar, Oct 03 2019 *)

sopf(f) = sum(j=1, #f~, f[j, 1]); \\ A008472
isok(m) = if (m>1, my(f=factor(m)); (sigma(f) % (numdiv(f)*sopf(f))) == 0);

cp's OEIS Frontend

A076380 Sum of digits in base 2 is a divisor of sum of prime divisors (A008472).

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A068935 Numbers having the sum of distinct prime factors less than the sum of exponents in prime factorization, A008472(n) < A001222(n).

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A068937 Numbers having the sum of distinct prime factors not less than the sum of exponents in prime factorization, A008472(n)>=A001222(n).

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A065925 Smallest k such that sopf(n+k) = sopf(k), where sopf = A008472.

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A075653 a(n) = n + sopf(n), where sopf is the sum of the distinct prime factors of n (A008472).

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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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A226479 Numbers n such that (sopf(n)*d(n))^2 = sigma(n) where sopf(n) = sum of distinct prime factors of n (A008472) and d(n) = number of divisors of n.

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