OFFSET
1,1
COMMENTS
The first number that admits two different divisors is 1335: two of its divisors are 1 and 89, and sigma(1335) = sigma(1335 - 1) = sigma(1335 - 89) = 2160.
The first number that admits three different divisors is 145515: three of its divisors are 89, 109, and 9701, and sigma(145515) = sigma(145515 - 89) = sigma(145515 - 109) = sigma(145515 - 9701) = 237600.
If k is in the sequence, and d a divisor such that sigma(k)=sigma(k-d), then k*m is in the sequence for any m coprime to k and k-d. - Robert Israel, May 16 2018
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
One divisor of 15 is 1 and sigma(15) = sigma(15 - 1) = 24.
One divisor of 56 is 2 and sigma(56) = sigma(56 - 2) = 120.
MAPLE
with(numtheory): P:=proc(n) local a, k; a:=divisors(n);
for k from 1 to nops(a) do if sigma(n)=sigma(n-a[k]) then RETURN(n);
fi; od; end: seq(P(i), i=1..1545);
# Alternative:
filter:= proc(n) local s, d;
s:= numtheory:-sigma(n);
for d in numtheory:-divisors(n) do
if numtheory:-sigma(n-d)=s then return true fi
od;
false
end proc:
select(filter, [$1..10000]); # Robert Israel, Jun 01 2018
MATHEMATICA
Select[Range[1600], Function[k, AnyTrue[Divisors@ k, DivisorSigma[1, k] == DivisorSigma[1, k - #] &]]] (* Michael De Vlieger, May 14 2018 *)
PROG
(PARI) isok(n) = sumdiv(n, d, if (n>d, sigma(n-d) == sigma(n))) > 0; \\ Michel Marcus, May 14 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paolo P. Lava, May 14 2018
STATUS
approved