A248006 -id:A248006 - cp's OEIS frontend

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.

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

Zhi-Wei Sun, Sep 29 2014

nonn

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

			a(6) = 4 since 4 + 6 = 10 divides sigma(4*6) = 60.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Cf. A000203, A248004, A248006, A248007.

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}]

a(n)=m=1;while(sigma(m*n)%(m+n),m++);m
vector(100,n,a(n)) \\ Derek Orr, Sep 29 2014

A248007 Least positive integer m such that m + n divides phi(m)*phi(n), where phi(.) is Euler's totient function.

Original entry on oeis.org

5, 8, 3, 14, 9, 20, 11, 10, 9, 16, 7, 18, 5, 12, 3, 38, 21, 8, 15, 58, 9, 20, 11, 18, 14, 32, 7, 14, 13, 12, 35, 22, 9, 24, 7, 46, 13, 31, 3, 42, 45, 16, 11, 30, 13, 44, 19, 27, 25, 40, 15, 26, 28, 36, 35, 28, 9, 64, 7, 54, 21, 28, 19, 26

Offset: 7

Zhi-Wei Sun, Sep 29 2014

nonn

Conjecture: a(n) exists for any n > 6. - Zhi-Wei Sun, Sep 29 2014
Numbers n for which a(n) > n: 10, 12, 22, 26, 42, 78, 166, 266, 290. The next term in this mini-sequence, if it exists, is greater than 3*10^4. I conjecture this list is finite. - Derek Orr, Sep 29 2014
a(2^n) <= 2^n for all n > 2. Also, if a(i) = j, then a(j) <= i. - Derek Orr, Sep 29 2014

			a(10) = 14 since 10 + 14 divides phi(10)*phi(14) = 4*6 = 24.

Zhi-Wei Sun, Table of n, a(n) for n = 7..10000

Cf. A000010, A248004, A248006, A248008.

Do[m=1; Label[aa]; If[Mod[EulerPhi[m]*EulerPhi[n], m+n]==0, Print[n, " ", m]; Goto[bb]]; m=m+1; Goto[aa]; Label[bb]; Continue, {n, 7, 70}]

a(n)=m=1;while((eulerphi(m)*eulerphi(n))%(m+n),m++);m
vector(100,n,a(n+6)) \\ Derek Orr, Sep 29 2014

cp's OEIS Frontend

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.

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