A283718 Numbers m such that sum of digits of 27*m is 27.

37, 74, 107, 111, 137, 144, 147, 148, 174, 177, 181, 184, 185, 207, 211, 214, 217, 218, 221, 222, 237, 244, 247, 248, 251, 254, 255, 257, 258, 259, 274, 277, 281, 284, 285, 287, 288, 291, 292, 294, 295, 296, 307, 311, 314, 317

Offset

1,1

Comments

There are sets with 10 consecutive numbers, e.g, starting with a(252): {1064, 1065, 1066, 1067, 1068, 1069, 1070, 1071, 1072, 1073}.

Other such sets start at n = {411, 524, 657, 722, 874, 944, 1041, 1109, 1218, 1283, 1351, 1466, 1536};

corresponding a(n) are {1464, 1764, 2064, 2164, 2464, 2564, 2764, 2864, 3064, 3164, 3264, 3464, 3564}, all congruent to 64 mod 100.

Any explanation?

Explanation: if m is in the sequence and the last 3 digits of 27 m are k28 with k <= 7, then 27 (m+i) has last 3 digits k28, k55, k82, (k+1)09, (k+1)36, (k+1)63, (k+1)90, (k+2)17, (k+2)44, (k+2)71, all summing to k+10, and its other digits are the same as those of 27 m. 28 is the only number from 0 to 99 with this property. In order for 27 m == 28 (mod 100) we need m == 28/27 == 64 (mod 100).- Robert Israel, Mar 15 2017

Links

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

Formula

A007953(27*a(n)) = 27.

Example

137 is in the sequence because 27*137 = 3699 and 3 + 6 + 9 + 9 = 27. - Indranil Ghosh, Mar 15 2017

Maple

filter:= n -> convert(convert(27*n, base, 10), `+`)=27:
select(filter, [$1..1000]); # Robert Israel, Mar 15 2017

Mathematica

Select[Range[317], Plus@@ IntegerDigits@ (27#) == 27 &] (* Indranil Ghosh, Mar 15 2017 *)

Prog

(PARI) {for(n=37, 317, if(sumdigits(27*n)==27, print1(n, ", ")))}
\\ Indranil Ghosh, Mar 15 2017
(Python)
def D(n): return sum([int(i) for i in str(n)])
for n in range(37, 317):
....if D(27*n)==27:print str(n)+", ", # Indranil Ghosh, Mar 15 2017
(Magma) [m: m in [0..500] | &+Intseq(27*m) eq 27]; // Bruno Berselli, Mar 15 2017

Crossrefs

Cf. A007953.

Sequence in context: A123815 A289732 A087531 * A085959 A145936 A071884

Adjacent sequences: A283715 A283716 A283717 * A283719 A283720 A283721

Keyword

nonn,base

Author

Zak Seidov, Mar 15 2017

Status

approved