A051444 Smallest k such that sigma(k) = n, or 0 if there is no such k, where sigma = A000203 = sum of divisors.

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, 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, 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

Offset

1,3

Comments

Column 1 of A299762. - Omar E. Pol, Mar 14 2018

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

References

R. K. Guy, Unsolved Problems Theory of Numbers, B1.

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems: invphi.gp, Oct. 2005

Example

sigma(1) = 1, so a(1) = 1.
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.
sigma(4) = 7, and 4 is the smallest (since only) such number, so a(7) = 4.
6 and 12 are the only k with sigma(k) = 12, so 6 is the smallest and a(12) = 6.

Mathematica

Do[ k = 1; While[ DivisorSigma[ 1, k ] != n && k < 10^4, k++ ]; If[ k != 10^4, Print[ k ], Print[ 0 ] ], {n, 1, 100} ]

Prog

(PARI) a(n)=for(k=1, n, if(sigma(k)==n, return(k))); 0 \\ Charles R Greathouse IV, Mar 09 2014
(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

Crossrefs

Cf. A000203, A002192, A007626, A007369 (positions of zeros), A299762.

Sequence in context: A322333 A346615 A240667 * A299762 A057637 A258913

Adjacent sequences: A051441 A051442 A051443 * A051445 A051446 A051447

Keyword

nonn,nice

Author

Jud McCranie

Extensions

Edited by M. F. Hasler, Nov 22 2019

Status

approved