A069527 Smallest multiple of n with digit sum = 8, or 0 if no such number exists, e.g. a(3k)= 0.

8, 8, 0, 8, 35, 0, 35, 8, 0, 80, 44, 0, 26, 224, 0, 80, 17, 0, 152, 80, 0, 44, 161, 0, 125, 26, 0, 224, 116, 0, 62, 224, 0, 170, 35, 0, 0, 152, 0, 80, 1025, 0, 215, 44, 0, 1610, 611, 0, 1421, 350, 0, 260, 53, 0, 440, 224, 0, 116, 413, 0, 305, 62, 0, 512, 260, 0, 134, 1700, 0

Offset: 1

Author

Amarnath Murthy, Apr 01 2002

Keywords

• base
• nonn

Comments

The number ABCDEF (A through F are digits) is divisible by 37 if the number XYZ (where X=A+D, Y=B+E, Z=C+F) is divisible by 37. If the digit sum of XYZ is S, then the digit sum of ABCDEF is S+9k for some k. A quick check of all multiples of 37 with three or fewer digits shows that none have a digit sum of 8. Thus no multiple of 37 has a digit sum of 8 and a(37) is undefined as is a(37p) for all p. - Christopher Lund (clund(AT)san.rr.com), Apr 16 2002

a(n) = 0 if n is a multiple of 3, 37, 271 or 4649. - Robert Israel, Feb 14 2024

Links

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

Crossrefs

Cf. A069521, A069522, A069523, A069524, A069525, A069526.

Programs

Maple

unfinished:= true: V:= Vector(1000): V0:= select(t -> igcd(t, 3*37*271*4649) = 1, {$1..1000}):
for i1 from 0 while unfinished do
  for i2 from 0 to i1 while unfinished do
    for i3 from 0 to i2 while unfinished do
      for i4 from 0 to i3 while unfinished do
        for i5 from 0 to i4 while unfinished do
          for i6 from 0 to i5 while unfinished do
            for i7 from 0 to i6 while unfinished do
              for i8 from 0 to i7 while unfinished do
              v:= 10^i1 + 10^i2 + 10^i3 + 10^i4 + 10^i5 + 10^i6 + 10^i7 + 10^i8;
              dv:= numtheory:-divisors(v);
              for s in V0 intersect dv do
                V[s]:= v;
              od;
              V0:= V0 minus dv;
              unfinished:= evalb(V0 <> {});
od od od od od od od od:
convert(V,list); # Robert Israel, Feb 14 2024

Formula

a(n) = n*A088397(n).

Extensions

More terms from Christopher Lund (clund(AT)san.rr.com), Apr 16 2002