A306824 Integer k such that digsum(k) = digsum (k/p(1)) = digsum (k/p(2)) = ... for all prime factors p(i) of k, where digsum(k) = A007953(k) is the digital sum of k.

1, 27, 54, 81, 108, 135, 162, 216, 243, 270, 324, 351, 361, 405, 432, 513, 540, 621, 702, 703, 810, 1026, 1053, 1080, 1215, 1242, 1458, 1620, 1728, 1944, 2071, 2079, 2106, 2133, 2160, 2187, 2403, 2413, 2592, 2700, 2701, 2916, 3024, 3051, 3105, 3267, 3321, 4023, 4033, 4050, 4158

Offset

1,2

Comments

a(n) == 0 or 1 (mod 9). If a(n) == 0 (mod 9), a(n) == 0 (mod 27).

Links

Table of n, a(n) for n=1..51.

Example

4158 = 2*3^3*7*11 is in the sequence because 4 + 1 + 5 + 8 = 18, and:
4158/2 = 2079 and digsum(2079) = 18;
4158/3 = 1386 and digsum(1386) = 18;
4158/7 = 594 and digsum(594) = 18;
4158/11 = 378 and digsum(378) = 18.

Maple

with(numtheory):nn:=4200:
  for k from 1 to nn do:
     d:=factorset(k):n1:=nops(d):it:=0:
     b:=convert(k, base, 10):n2:=nops(b):s:=sum(‘b[i]’, ‘i’=1..n2):
      for i from 1 to n1 do:
        x:=k/d[i]:b1:=convert(x, base, 10):n3:=nops(b1):
        s1:=sum(‘b1[i]’, ‘i’=1..n3):
        if s1=s
         then
         it:=it+1:
         else
        fi:
      od:
       if it=n1
        then
        printf(`%d, `, k):
       else
      fi:
od:

Mathematica

sod[n_] := Total@IntegerDigits@n; Select[Range[1, 5000], {sod[#]} == Union[sod /@ (#/First /@ FactorInteger[#])] &] (* Giovanni Resta, Mar 12 2019 *)

Prog

(PARI) isok(k) = {my(pf = factor(k)[, 1]~, sd = sumdigits(k)); for (i=1, #pf, if (sumdigits(k/pf[i]) != sd, return (0)); ); return (1); } \\ Michel Marcus, Mar 12 2019

Crossrefs

Cf. A007953, A305548, A306761 (a subsequence).

Sequence in context: A183032 A040702 A305548 * A043323 A033903 A175806

Adjacent sequences: A306821 A306822 A306823 * A306825 A306826 A306827

Keyword

nonn,base

Author

Michel Lagneau, Mar 12 2019

Status

approved