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A342555 2*a(n) is the start of 3 consecutive numbers (even-odd-even) that are sums of divisors, i.e., terms of A000203.

%I A342555 #9 May 16 2021 01:55:29
%S A342555 3,6,15,19,63,153,199,255,423,480,511,546,861,1111,1189,1400,1770,
%T A342555 1875,1935,1995,2047,2556,3475,3619,4005,4095,4920,5151,5215,6649,
%U A342555 8046,8191,8646,8749,9765,11175,11199,14028,14197,15391,15427,15470,16383,19494,25878,26557,26799
%N A342555 2*a(n) is the start of 3 consecutive numbers (even-odd-even) that are sums of divisors, i.e., terms of A000203.
%C A342555 There exists one exceptional case of 4 consecutive numbers 12, 13, 14, 15, where 13 would start an odd-even-odd progression.
%H A342555 Hugo Pfoertner, <a href="/A342555/b342555.txt">Table of n, a(n) for n = 1..1000</a>
%H A342555 Max Alekseyev, <a href="https://home.gwu.edu/~maxal/gpscripts/invphi.gp">PARI/GP Scripts for Miscellaneous Math Problems: invphi.gp</a>, Oct. 2005.
%H A342555 Jeppe Stig Nielsen, <a href="/A085790/a085790.txt">List of numbers with divisor sum k</a>, k<=10000.
%e A342555 a(1) = 3, because 2*3 = 6 is the start of the first occurrence of a row of 3 consecutive numbers, all of which are in A000203. 6 = sigma(5), 7 = sigma(4), 8 = sigma(7).
%e A342555 a(2) = 6: 2*6 = 12 = sigma(6) = sigma(11), 13 = sigma(9), 14 = sigma(13). 15 = sigma(8), which would be at the end of the row 13, 14, 15, is excluded by the even-odd-even condition.
%e A342555 a(3) = 15: 2*15 = 30 = sigma(29), 31 = sigma(16) = sigma(25), 32 = sigma(21) = sigma(31).
%e A342555 See Jeppe Stig Nielsen's list for more examples.
%o A342555 (PARI) a342555(nterms) = {my(N=vector(3, i, invsigmaNum(i+1)), n=0, k=4); while(n<=nterms, if(vecmin(N)>0 && !(k%2), print1((k-2)/2, ", "); n++); k++; N[1+k%3] = invsigmaNum(k))}; \\ see Alekseyev link for invsigmaNum()
%o A342555 a342555(46)
%Y A342555 Cf. A000203, A085790, A342560.
%K A342555 nonn
%O A342555 1,1
%A A342555 _Hugo Pfoertner_, May 14 2021