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%I #28 Nov 22 2019 02:58:02
%S 1,0,2,3,0,5,4,7,0,0,0,6,9,13,8,0,0,10,0,19,0,0,0,14,0,0,0,12,0,29,16,
%T 21,0,0,0,22,0,37,18,27,0,20,0,43,0,0,0,33,0,0,0,0,0,34,0,28,49,0,0,
%U 24,0,61,32,0,0,0,0,67,0,0,0,30,0,73,0,0,0,45,0,57,0,0,0,44,0,0,0,0,0
%N Smallest k such that sigma(k) = n, or 0 if there is no such k, where sigma = A000203 = sum of divisors.
%C Column 1 of A299762. - _Omar E. Pol_, Mar 14 2018
%C This is a right inverse of sigma = A000203 on A002191 = range(sigma): if n is in A002191, then there is some x with sigma(x) = n, and by definition a(n) is the smallest such x, so sigma(a(n)) = n. - _M. F. Hasler_, Nov 22 2019
%D R. K. Guy, Unsolved Problems Theory of Numbers, B1.
%H T. D. Noe, <a href="/A051444/b051444.txt">Table of n, a(n) for n = 1..10000</a>
%H Max Alekseyev, <a href="https://home.gwu.edu/~maxal/gpscripts/invphi.gp">PARI/GP Scripts for Miscellaneous Math Problems: invphi.gp</a>, Oct. 2005
%e sigma(1) = 1, so a(1) = 1.
%e There is no k with sigma(k) = 2, since sigma(k) >= k + 1 for all k > 1 and sigma(1) = 1, so a(2) = 0.
%e sigma(4) = 7, and 4 is the smallest (since only) such number, so a(7) = 4.
%e 6 and 12 are the only k with sigma(k) = 12, so 6 is the smallest and a(12) = 6.
%t Do[ k = 1; While[ DivisorSigma[ 1, k ] != n && k < 10^4, k++ ]; If[ k != 10^4, Print[ k ], Print[ 0 ] ], {n, 1, 100} ]
%o (PARI) a(n)=for(k=1,n,if(sigma(k)==n,return(k))); 0 \\ _Charles R Greathouse IV_, Mar 09 2014
%o (PARI) A051444(n)=if(n=invsigma(n),vecmin(n)) \\ See Alekseyev link for invsigma(). An update including invsigmaMin = A051444 is planned. - _M. F. Hasler_, Nov 21 2019
%Y Cf. A000203, A002192, A007626, A007369 (positions of zeros), A299762.
%K nonn,nice
%O 1,3
%A _Jud McCranie_
%E Edited by _M. F. Hasler_, Nov 22 2019
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