A221644 Let abcd... be the decimal expansion of k. Sequence lists numbers k such that 1/a + 2/b + 3/c + 4/d + ... is an integer.

1, 11, 12, 24, 33, 111, 113, 121, 123, 139, 142, 146, 155, 184, 212, 216, 222, 226, 241, 243, 331, 333, 369, 414, 424, 482, 486, 649, 662, 666, 848, 1111, 1112, 1114, 1128, 1131, 1132, 1134, 1168, 1177, 1196, 1211, 1212, 1214, 1228, 1231, 1232, 1234, 1268

Offset: 1

Author

Michel Lagneau, Aug 08 2013

Keywords

• nonn
• base
• less

Comments

The repunits numbers 1, 11, 111, 1111, ... (A002275) are in the sequence.

The first nine terms 1, 12, 123, 1234, ... of A007908 are in the sequence.

If a number of the form ab1 is in the sequence, the corresponding number of the form ab3 is also in the sequence.

If a number of the form abc1 is in the sequence, the corresponding number of the form abc2 is also in the sequence.

If a number of the form abc11 is in the sequence, the corresponding number of the form abc15 is also in the sequence.

The first nine terms 1, 12, 113, 1114, 11115, ... of A198375 are in the sequence.

In the general case, if n = abcd...q is in the sequence where q is the k-th decimal digit of n, the number abcd...qr is also in the sequence if k+1 is divisible by r; for example, 82812 is in the sequence => 828121, 828122, 828123 and 828126 are also in the sequence because 6 is divisible by 1, 2, 3 and 6.

Examples

			184 is in the sequence because 1/1 + 2/8 + 3/4 = 2.
		

Links

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

Crossrefs

Cf. A002275, A007908, A198375.

Programs

Maple

with(numtheory):for n from 1 to 2000 do: d:=convert(n, base, 10):n1:=nops(d):p:=product('d[i]', 'i'=1..n1):if p<>0 then s:=sum('i/d[n1-i+1] ', 'i'=1..n1):if s=floor(s) then printf(`%d, `,n):else fi:fi:od:

Mathematica

Select[Range[1300],FreeQ[IntegerDigits[#],0]&&IntegerQ[Total[ Range[ IntegerLength[ #]]/ IntegerDigits[ #]]]&] (* Harvey P. Dale, May 16 2018 *)

PARI

isok(n) = my(d=digits(n)); vecmin(d) && (denominator(sum(k=1, #d, k/d[k])) == 1); \\ Michel Marcus, Sep 14 2017