A198097 n such that d(1)*1! + d(2)*2! + ... + d(k)*k! is a perfect square, where d(i) are the decimal digits of n.

1, 4, 9, 10, 14, 21, 27, 33, 40, 46, 52, 65, 71, 79, 84, 90, 98, 100, 104, 108, 111, 133, 137, 140, 162, 166, 191, 195, 210, 212, 225, 241, 249, 254, 270, 278, 283, 301, 323, 327, 330, 352, 356, 381, 385, 400, 402, 415, 431, 439, 444, 460, 468, 473, 497, 513

Offset

1,2

Comments

A198095 is included in this sequence.

If n is in the sequence, then so is 10*n. - Robert Israel, Aug 16 2020

Links

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

Example

137 is in the sequence because 1*1! + 3*2! + 7*3! = 1 + 6 + 42 = 7^2.

Maple

for n from 1 to 520 do :l:=length(n):n0:=n:s:=0:for m from l by -1  to 1 do:q:=n0:u:=irem(q, 10):v:=iquo(q, 10):n0:=v :s:=s+u*m!:od: z:=sqrt(s):if  z=floor(z) then  printf(`%d, `, n):else fi:od:
# Simpler:
filter:= proc(n)
  local L, k;
  L:= convert(n, base, 10);
  issqr(add(L[-k]*k!, k=1..nops(L)))
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 16 2020

Crossrefs

Cf. A198095, A198044.

Sequence in context: A243170 A356711 A054294 * A358762 A157908 A064973

Adjacent sequences: A198094 A198095 A198096 * A198098 A198099 A198100

Keyword

nonn,base

Author

Michel Lagneau, Oct 21 2011

Status

approved