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%I #23 May 16 2019 05:55:39
%S 15,56,165,195,207,224,255,270,280,285,286,345,368,435,465,555,615,
%T 616,645,672,705,708,728,795,836,850,858,885,915,920,952,958,1005,
%U 1035,1064,1065,1095,1185,1242,1245,1288,1335,1365,1400,1430,1449,1455,1506,1515,1545
%N Numbers k having at least one divisor d such that sigma(k) = sigma(k-d).
%C 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.
%C 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.
%C 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
%H Robert Israel, <a href="/A304295/b304295.txt">Table of n, a(n) for n = 1..10000</a>
%e One divisor of 15 is 1 and sigma(15) = sigma(15 - 1) = 24.
%e One divisor of 56 is 2 and sigma(56) = sigma(56 - 2) = 120.
%p with(numtheory): P:=proc(n) local a,k; a:=divisors(n);
%p for k from 1 to nops(a) do if sigma(n)=sigma(n-a[k]) then RETURN(n);
%p fi; od; end: seq(P(i),i=1..1545);
%p # Alternative:
%p filter:= proc(n) local s,d;
%p s:= numtheory:-sigma(n);
%p for d in numtheory:-divisors(n) do
%p if numtheory:-sigma(n-d)=s then return true fi
%p od;
%p false
%p end proc:
%p select(filter, [$1..10000]); # _Robert Israel_, Jun 01 2018
%t Select[Range[1600], Function[k, AnyTrue[Divisors@ k, DivisorSigma[1, k] == DivisorSigma[1, k - #] &]]] (* _Michael De Vlieger_, May 14 2018 *)
%o (PARI) isok(n) = sumdiv(n, d, if (n>d, sigma(n-d) == sigma(n))) > 0; \\ _Michel Marcus_, May 14 2018
%Y Cf. A000203, A304294.
%K nonn,easy
%O 1,1
%A _Paolo P. Lava_, May 14 2018
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