A330901
Numbers k such that k and k+2 have the same deficiency (A033879).
Original entry on oeis.org
2, 6497, 12317, 91610, 133787, 181427, 404471, 439097, 485237, 1410119, 2696807, 6220607, 6827369, 6954767, 9770027, 10302419, 10449347, 10887977, 11014007, 16745387, 18959111, 25883519, 27334469, 39508037, 40311149, 40551617, 42561437, 44592209, 47717471, 48912107
Offset: 1
2 is a term since 2 and 4 have the same deficiency: A033879(2) = 2*2 - sigma(2) = 4 - 3 = 1, and A033879(4) = 2*4 - sigma(4) = 8 - 7 = 1.
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def[n_] := 2*n - DivisorSigma[1, n]; Select[Range[10^5], def[#] == def[# + 2] &]
SequencePosition[Table[2n-DivisorSigma[1,n],{n,48920000}],{x_,,x}][[;;,1]] (* Harvey P. Dale, Apr 26 2025 *)
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j1=1;j2=1;for(k=3,50000000,j=k+k-sigma(k);if(j==j1,print1(k-2,", "));j1=j2;j2=j) \\ Hugo Pfoertner, May 01 2020
A112646
Special solutions to abs(sigma(x+1) - sigma(x)) = 2 where x + 1 = 3^y.
Original entry on oeis.org
8, 26, 2186, 1594322, 7509466514979724803946715958257546, 13915193059764305937984450503671774362956903094026
Offset: 1
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ta={{0}};Do[s=Abs[DivisorSigma[1, 3^n]-DivisorSigma[1, -1+3^n]]; If[Equal[s, 2], ta=Append[ta, n];Print[ -1+3^n]], {n, 1, 100000000}]; ta=Delete[ta, 1]
A112647
a(n)=x is the smallest solution to abs(sigma(x+1)-sigma(x))=n, or 0 if no solution exists.
Original entry on oeis.org
14, 2, 1, 3, 6, 9, 5, 7, 62
Offset: 0
n=5: least solution is 9 because sigma for 9 and 9+1=10 are 13 and 13+5=18.
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N = 2*10^8; % to search sigma(n) for n <= N
M = 100; % to get a(1) to a(M)
Sigma = ones(1,N);
for n=2:N
inds = [n:n:N];
Sigma(inds) = Sigma(inds) + n;
end
DSigma = abs(Sigma(2:end) - Sigma(1:end-1));
A = zeros(1,M);
for v = 1:M
r = find(DSigma == v,1,'first');
if numel(r) > 0
A(v) = r;
end
end
A % Robert Israel, May 24 2016
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f[x_] :=Abs[DivisorSigma[1, n+1] - DivisorSigma[1, n]]; t=Table[0, {258}]; Do[s=f[n]; If[s<258 && t[[s+1]]==0, t[[s+1]]=n], {n, 10^7}]; t (* edited by Giovanni Resta, Oct 29 2019 *)
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