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A248008
Least positive integer m such that m + n divides sigma(m*n), where sigma(k) denotes the sum of all positive divisors of k.
9
2, 1, 1, 3, 1, 4, 1, 7, 4, 14, 1, 18, 1, 10, 9, 15, 1, 12, 1, 1, 11, 5, 1, 4, 6, 4, 6, 2, 1, 18, 1, 28, 6, 14, 13, 13, 1, 12, 17, 22, 1, 22, 1, 10, 3, 10, 1, 30, 8, 12, 9, 18, 1, 2, 17, 6, 7, 26, 1, 52, 1, 22, 28, 38, 19, 12, 1, 22, 36, 26
OFFSET
1,1
COMMENTS
Conjecture: a(n) exists for any n > 0.
a(n) = 1 if and only if n is in A230606. Also, if a(i) = j, a(j) <= i. - Derek Orr, Sep 29 2014
Numbers n such that a(n) > n: 1, 10, 12, 108, 1139, ... The next number, if it exists, is greater than 2*10^4. - Derek Orr, Sep 29 2014
LINKS
Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.
EXAMPLE
a(6) = 4 since 4 + 6 = 10 divides sigma(4*6) = 60.
MATHEMATICA
Do[m=1; Label[aa]; If[Mod[DivisorSigma[1, m*n], m+n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 1, 70}]
PROG
(PARI)
a(n)=m=1; while(sigma(m*n)%(m+n), m++); m
vector(100, n, a(n)) \\ Derek Orr, Sep 29 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Sep 29 2014
STATUS
approved