A334420 -id:A334420 - cp's OEIS frontend

A334421 a(n) is the smallest number m with n divisors d such that sigma(d)/tau(d) is an integer for all divisors d.

1, 3, 49, 15, 923521, 147, 88245939632761, 105, 8281, 2770563, 4345096786921664259621718196367601, 735, 9024585590445680759701490904755712009585829774768244676951841, 264737818898283, 45252529, 1155

Offset: 1

Jaroslav Krizek, May 01 2020

nonn

Sequences of numbers m with n such divisors for 1 < n < 6:
n = 2: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... (A065091 - odd primes).
n = 3: 49, 169, 361, 961, 1369, 1849, 3721, 4489, 5329, 6241, 9409, ...
n = 4: 15, 21, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, ...
n = 5: 923521, 13845841, 519885601, 1073283121, 1982119441, ...
Other terms: a(18) = 24843, a(24) = 8085, a(32) = 15015.
a(p) = (k*p# + 1)^(p-1) for some k > 0 and p# is the product of primes <= k. - David A. Corneth, May 01 2020 [Proof: a(p) must be of the form q^(p-1), where q is a prime. Thus r | (q^r - 1)/(q - 1) if r <= p is prime. Suppose that q - 1 is not divisible by r, then 0 == q^r - 1 == q - 1 (mod r), a contradiction! Therefore, q - 1 is divisible by any primes r <= p. In conclusion, q = k*p# + 1 for some k > 0. - Jinyuan Wang, May 02 2020]
a(17) = 4084081^16 and a(19) = 106696591^18 are too large to be included. - Amiram Eldar, May 02 2020

			a(4) = 15 because number 15 with divisors 1, 3, 5 and 15 is the smallest number with 4 such divisors: sigma(1)/tau(1) = 1/1 = 1, sigma(3)/tau(3) = 4/2 = 2, sigma(5)/tau(5) = 6/2 = 3, sigma(15)/tau(15) = 24/4 = 6.

Cf. A000005 (tau), A000203 (sigma), A003601, A306639, A324499, A324500, A334420.

[Min([m: m in[1..1000] | &+[&+Divisors(d) mod #Divisors(d): d in Divisors(m)] eq 0 and #Divisors(m) eq k]): k in [1..4]]

a(13)-a(14) from Amiram Eldar, May 02 2020

A338170 a(n) is the number of divisors d of n such that tau(d) divides sigma(d).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 3, 4, 1, 2, 3, 2, 3, 4, 3, 2, 3, 2, 2, 3, 3, 2, 6, 2, 1, 4, 2, 4, 3, 2, 3, 4, 3, 2, 7, 2, 4, 5, 3, 2, 3, 3, 2, 4, 2, 2, 5, 4, 4, 4, 2, 2, 8, 2, 3, 4, 1, 4, 7, 2, 3, 4, 6, 2, 3, 2, 2, 4, 3, 4, 6, 2, 3, 3, 2, 2, 7, 4, 3, 4, 4, 2, 7, 4, 4, 4, 3, 4, 4, 2, 4, 5, 3, 2, 6, 2, 2, 8

Offset: 1

Jaroslav Krizek, Oct 14 2020

nonn

a(n) is the number of arithmetic divisors d of n.
a(n) = tau(n) = A000005(n) for numbers n from A334420.
See A338171 and A338172 for sum and product such divisors.
a(n) = 1 iff n = 2^k (A000079). - Bernard Schott, Dec 06 2020

			a(6) = 3 because there are 3 arithmetic divisors of 6 (1, 3 and 6):
sigma(1)/tau(1) =  1/1 = 1; sigma(3)/tau(3) = 4/2 = 2; sigma(6)/tau(6) = 12/4 = 3.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Inverse Möbius transform of A245656.
Cf. A000005 (tau), A000203 (sigma), A003601 (arithmetic numbers).
Cf. A334420, A334421.
Cf. A337326 (smallest numbers m with n such divisors).

[#[d: d in Divisors(n) | IsIntegral(&+Divisors(d) / #Divisors(d))]: n in [1..100]];

a[n_] := DivisorSum[n, 1 &, Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, Oct 15 2020 *)

a(n) = sumdiv(n, d, !(sigma(d) % numdiv(d))); \\ Michel Marcus, Oct 15 2020

a(n) = Sum_{d|n} c(d), where c(n) is the arithmetic characteristic of n (A245656).

a(p) = 2 for odd primes p (A065091).

Data section extended up to 105 terms by Antti Karttunen, Dec 12 2021

A338171 a(n) is the sum of those divisors d of n such that tau(d) divides sigma(d).

Original entry on oeis.org

1, 1, 4, 1, 6, 10, 8, 1, 4, 6, 12, 10, 14, 22, 24, 1, 18, 10, 20, 26, 32, 34, 24, 10, 6, 14, 31, 22, 30, 60, 32, 1, 48, 18, 48, 10, 38, 58, 56, 26, 42, 94, 44, 78, 69, 70, 48, 10, 57, 6, 72, 14, 54, 91, 72, 78, 80, 30, 60, 140, 62, 94, 32, 1, 84, 142, 68, 86

Offset: 1

Jaroslav Krizek, Oct 14 2020

a(n) is the sum of arithmetic divisors d of n.
a(n) = sigma(n) = A000203(n) for numbers n from A334420.
See A338170 and A338172 for number and product such divisors.

			a(6) = 10 because there are 3 arithmetic divisors of 6 (1, 3 and 6): sigma(1)/tau(1) =  1/1 = 1; sigma(3)/tau(3) = 4/2 = 2; sigma(6)/tau(6) = 12/4 = 3. Sum of this divisors is 10.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005 (tau), A000203 (sigma), A003601 (arithmetic numbers).

Cf. A334420, A334421.

[&+[d: d in Divisors(n) | IsIntegral(&+Divisors(d) / #Divisors(d))]: n in [1..100]];

f:= proc(n) uses numtheory;
convert(select(t -> sigma(t) mod tau(t) = 0, divisors(n)),`+`) end proc:
map(f, [$1..100]); # Robert Israel, Oct 27 2020

a[n_] := DivisorSum[n, # &, Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, Oct 15 2020 *)

a(n) = sumdiv(n, d, d*!(sigma(d) % numdiv(d))); \\ Michel Marcus, Oct 15 2020

a(p) = p + 1 for odd primes p (A065091).

A338172 a(n) is the product of those divisors d of n such that tau(d) divides sigma(d).

Original entry on oeis.org

1, 1, 3, 1, 5, 18, 7, 1, 3, 5, 11, 18, 13, 98, 225, 1, 17, 18, 19, 100, 441, 242, 23, 18, 5, 13, 81, 98, 29, 40500, 31, 1, 1089, 17, 1225, 18, 37, 722, 1521, 100, 41, 1555848, 43, 10648, 10125, 1058, 47, 18, 343, 5, 2601, 13, 53, 26244, 3025, 5488, 3249, 29

Offset: 1

Jaroslav Krizek, Oct 14 2020

nonn

a(n) is the product of arithmetic divisors d of n.

a(n) = pod(n) = A007955(n) for numbers n from A334420.

			a(6) = 18 because there are 3 arithmetic divisors of 6 (1, 3 and 6): sigma(1)/tau(1) =  1/1 = 1; sigma(3)/tau(3) = 4/2 = 2; sigma(6)/tau(6) = 12/4 = 3. Product of this divisors is 18.

Cf. A000005 (tau), A000203 (sigma), A003601 (arithmetic numbers).
Cf. A334420, A334421.
See A338170 and A338171 for number and sum of such divisors.

[&*[d: d in Divisors(n) | IsIntegral(&+Divisors(d) / #Divisors(d))]: n in [1..100]];

a[n_] := Times @@ Select[Divisors[n],  Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, Oct 15 2020 *)

a(n) = my(d=divisors(n)); prod(k=1, #d, if (sigma(d[k]) % numdiv(d[k]), 1, d[k])); \\ Michel Marcus, Oct 15 2020

a(p) = p for odd primes p (A065091).

A342144 Numbers m with integer solution to x^x == (x+1)^(x+1) (mod m) with x > 0.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 123, 127, 129, 131, 133, 137, 139, 141, 143, 145, 147, 149, 151, 155, 157, 159, 161, 163, 167

Offset: 1

Owen C. Keith, Mar 01 2021

nonn

Some values of m have multiple solutions.
For example, for m = 49, 25^25 == 26^26 (mod 49) and 37^37 == 38^38 (mod 49).
All terms are odd.
First differs from A334420 at a(70) which is 167 for this sequence and 165 for A334420.
First differs from A056911 at a(21) which is 49 for this sequence and 51 for A056911.

			3 is a term since 1^1 == 2^2 (mod 3).
5 is a term since 11^11 == 12^12 (mod 5).

Cf. A056911, A084820, A174824, A334420, A338445.

seqQ[n_] := AnyTrue[Range[LCM[n, CarmichaelLambda[n]]+1], PowerMod[#, #, n] == PowerMod[# + 1, # + 1, n] &]; Select[Range[145], seqQ]

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A334421 a(n) is the smallest number m with n divisors d such that sigma(d)/tau(d) is an integer for all divisors d.

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A338170 a(n) is the number of divisors d of n such that tau(d) divides sigma(d).

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A338171 a(n) is the sum of those divisors d of n such that tau(d) divides sigma(d).

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A338172 a(n) is the product of those divisors d of n such that tau(d) divides sigma(d).

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A342144 Numbers m with integer solution to x^x == (x+1)^(x+1) (mod m) with x > 0.

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