A248004 -id:A248004 - 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

A248006 Least positive integer m such that m + n divides phi(m*n), where phi(.) is Euler's totient function.

Original entry on oeis.org

3, 4, 3, 6, 5, 8, 9, 6, 9, 4, 11, 7, 5, 16, 7, 9, 5, 12, 7, 18, 21, 8, 15, 13, 27, 14, 11, 10, 14, 32, 7, 14, 5, 12, 35, 10, 13, 24, 7, 14, 13, 11, 9, 42, 45, 16, 11, 30, 13, 12, 19, 27, 33, 8, 15, 22, 28, 4, 35, 28, 18, 64, 7, 14, 21, 28, 19, 10

Offset: 3

Zhi-Wei Sun, Sep 29 2014

nonn

Conjecture: For any n > 2, a(n) exists and a(n) <= n.
See also A248007 and A248008 for similar conjectures. - Zhi-Wei Sun, Sep 29 2014
The conjecture is true: One can show that 2*n divides phi(n^2) for all n > 2. So, a(n) is at most n. - Derek Orr, Sep 29 2014
a(n) >= 3 for all n. - Robert Israel, Sep 29 2014

			a(5) = 3 since 3 + 5 divides phi(3*5) = 8.

Zhi-Wei Sun, Table of n, a(n) for n = 3..10000
Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Cf. A000010, A248004, A248007, A248008.

f:= proc(n)
local m;
for m from 3 do
  if numtheory:-phi(m*n) mod (m+n) = 0 then return m fi
od
end proc;
seq(f(n),n=3..100); # Robert Israel, Sep 29 2014

Do[m=1;Label[aa];If[Mod[EulerPhi[m*n],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,3,70}]

a(n)=m=1;while(eulerphi(m*n)%(m+n),m++);m
vector(100,n,a(n+2)) \\ Derek Orr, Sep 29 2014

A248029 Least positive integer m such that m + n divides phi(m)*sigma(n), where phi(.) and sigma(.) are given by A000010 and A000203.

Original entry on oeis.org

1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 2, 1, 10, 9, 15, 1, 8, 1, 1, 11, 14, 1, 6, 6, 16, 5, 14, 1, 6, 1, 10, 15, 11, 13, 16, 1, 7, 9, 5, 1, 6, 1, 12, 7, 26, 1, 14, 8, 12, 21, 46, 1, 6, 17, 4, 23, 32, 1, 24, 1, 34, 41, 63, 7, 6, 1, 16, 11, 2

Offset: 2

Zhi-Wei Sun, Sep 29 2014

nonn

Conjecture: For any n > 1, we have a(n) <= n.

The existence of a(n) is easy; in fact, for m = sigma(n) - n, obviously m + n divides phi(m)*sigma(n). - Zhi-Wei Sun, Oct 02 2014

			a(8) = 7 since 7 + 8 = 15 divides phi(7)*sigma(8) = 6*15 = 90.

Zhi-Wei Sun, Table of n, a(n) for n = 2..10000
Zhi-Wei Sun, A new theorem on the prime-counting function, arXiv:1409.5685, 2014.

Cf. A000010, A000203, A248004, A248007, A248008, A248030.

Do[m=1;Label[aa];If[Mod[EulerPhi[m]*DivisorSigma[1,n],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,2,70}]

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

A248030 Least positive integer m such that m + n divides sigma(m)*phi(n), where sigma(.) and phi(.) are given by A000203 and A000010.

Original entry on oeis.org

2, 12, 4, 2, 3, 6, 2, 10, 3, 2, 21, 8, 3, 22, 13, 8, 9, 6, 8, 12, 3, 8, 10, 4, 5, 10, 21, 8, 20, 26, 4, 8, 7, 14, 13, 12, 8, 4, 33, 8, 23, 6, 20, 12, 3, 16, 22, 72, 7, 10, 13, 4, 27, 42, 5, 24, 15, 26, 57, 18, 11, 38, 27, 20, 31, 4, 21, 36, 19, 2

Offset: 1

Zhi-Wei Sun, Sep 29 2014

nonn

Conjecture: a(n) < 2*n for any n > 2.

Numbers n such that a(n) > n: 1, 2, 3, 8, 11, 14, 48, 227, 908, 4478, ... The next number, if it exists, is greater than 10^5. - Derek Orr, Sep 29 2014

			a(2) = 12 since 12 + 2 = 14 divides sigma(12)*phi(2) = 28.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000203, A248004, A248007, A248008, A248029.

Do[m=1;Label[aa];If[Mod[DivisorSigma[1,m]*EulerPhi[n],m+n]==0,Print[n," ",m];Goto[bb]];m= m+1; Goto[aa];Label[bb];Continue,{n,1,70}]
lpim[n_]:=Module[{m=1,ephn=EulerPhi[n]},While[Mod[ephn*DivisorSigma[1,m],m+n]!=0, m++]; m]; Array[lpim,70] (* Harvey P. Dale, Feb 14 2024 *)

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

A248144 Least positive integer m such that m + n divides p(m*n), where p(.) is the partition function given by A000041.

Original entry on oeis.org

4, 5, 3, 1, 2, 14, 2, 10, 1, 4, 17, 5, 9, 1, 1, 6, 4, 10, 16, 357, 1, 197, 14, 1, 3, 9, 6, 1123, 15, 93, 4, 1, 8, 46, 77, 99, 18, 53, 10, 76, 4, 2, 15, 152, 4, 3, 10, 29, 6, 12, 4, 1, 25, 1, 252, 64, 106, 11, 11, 136

Offset: 1

Zhi-Wei Sun, Oct 02 2014

nonn

Conjecture: a(n) exists for any n > 0.

			 a(6) = 14 since 6 + 14 = 20 divides p(6*14) = 26543660.

Zhi-Wei Sun, Table of n, a(n) for n = 1..496

Cf. A000041, A248004, A248008, A248143, A248175.

Do[m=1;Label[aa];If[Mod[PartitionsP[m*n],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]

A248590 Least positive integer m such that prime(m) == 1 (mod m + n).

Original entry on oeis.org

3, 4, 19, 10, 5, 6, 13, 15, 7, 8, 31, 17, 9, 19, 20, 38, 22, 10, 11, 24, 78, 80, 25, 12, 28, 30, 13, 14, 599, 97, 15, 31, 32, 178, 33, 16, 102, 104, 35, 108, 17, 18, 38, 39, 361, 40, 19, 41, 73, 20, 21, 43, 45, 78, 134, 22, 391, 47, 23, 84

Offset: 1

Zhi-Wei Sun, Oct 09 2014

nonn

Conjecture: (i) a(n) exists for any n > 0. Moreover, a(n) < n*(n-1) if n > 3.

(ii) For any n > 0, there is a positive integer m such that prime(m) == -1 (mod m + n). Moreover, we may require m < n*(n-1) if n > 1.

			a(3) = 19 since prime(19) = 67 == 1 (mod 19 + 3).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A247824, A248004, A248588.

Do[m=1;Label[aa];If[Mod[Prime[m]-1,m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]

A248175 Least positive integer m such that m + n divides q(m*n), where q(.) is the strict partition function given by A000009.

Original entry on oeis.org

11, 4, 9, 2, 12, 10, 9, 16, 3, 6, 1, 5, 2, 18, 7, 8, 5, 14, 11, 36, 2, 34, 4, 8, 31, 6, 15, 36, 23, 2, 9, 14, 17, 22, 11, 18, 1, 22, 11, 7, 1, 22, 12, 7, 55, 7, 19, 40, 15, 6, 31, 12, 43, 10, 25, 40, 7, 91, 61, 20

Offset: 1

Zhi-Wei Sun, Oct 03 2014

nonn

Conjecture: (i) a(n) exists for any n > 0.

(ii) For each n > 0, there is a positive integer m such that m + n divides q(m) + q(n).

			a(3) = 9 since 9 + 3 = 12 divides q(9*3) = 192 = 12*16.

Zhi-Wei Sun, Table of n, a(n) for n = 1..405

Cf. A000009, A247824, A248004, A248143, A248144.

Do[m=1;Label[aa];If[Mod[PartitionsQ[m*n],m+n]==0,Print[n," ",m];Goto[bb]];m=m+1;Goto[aa];Label[bb];Continue,{n,1,60}]

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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.

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

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A248006 Least positive integer m such that m + n divides phi(m*n), where phi(.) is Euler's totient function.

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A248029 Least positive integer m such that m + n divides phi(m)*sigma(n), where phi(.) and sigma(.) are given by A000010 and A000203.

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A248030 Least positive integer m such that m + n divides sigma(m)*phi(n), where sigma(.) and phi(.) are given by A000203 and A000010.

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A248144 Least positive integer m such that m + n divides p(m*n), where p(.) is the partition function given by A000041.

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A248590 Least positive integer m such that prime(m) == 1 (mod m + n).

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A248175 Least positive integer m such that m + n divides q(m*n), where q(.) is the strict partition function given by A000009.

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