A304295 Numbers k having at least one divisor d such that sigma(k) = sigma(k-d).

15, 56, 165, 195, 207, 224, 255, 270, 280, 285, 286, 345, 368, 435, 465, 555, 615, 616, 645, 672, 705, 708, 728, 795, 836, 850, 858, 885, 915, 920, 952, 958, 1005, 1035, 1064, 1065, 1095, 1185, 1242, 1245, 1288, 1335, 1365, 1400, 1430, 1449, 1455, 1506, 1515, 1545

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

Cf. A000203, A304294.

Sequence in context: A227219 A250956 A243318 * A228322 A219630 A219849

Adjacent sequences: A304292 A304293 A304294 * A304296 A304297 A304298

Keyword

nonn,easy

Author

Paolo P. Lava, May 14 2018

Status

approved