A349007 -id:A349007 - cp's OEIS frontend

A349006 a(1) = 1; for n > 1, a(n) is the smallest number m such that sigma(m) = tau(m)^n or 0 if no such m exists.

1, 3, 7, 217, 31, 3937, 127, 57337, 253921, 917497, 3670009, 16252897, 8191, 61079603913818329, 1073602561, 4294434817, 131071, 66571993057, 524287, 1208766717309082486038529, 9222228542614937599, 17590038552577, 500367932999371587367, 281472829095937, 1125897758834689

Offset: 1

Jaroslav Krizek, Nov 05 2021

nonn

See A051281 for numbers m such that sigma(m) = tau(m)^k where k = integer.

a(n) = 0 for n = 76, 81, ...

			a(4) = 217 because 217 is the smallest number m such that sigma(m) = tau(m)^4; sigma(217) = 256 = tau(217)^4  = 4^4.

Cf. A000005 (tau), A000203 (sigma), A051281, A334455, A349007.

[1] cat [Min([m: m in[2..10^6] | &+Divisors(m) eq #Divisors(m)^n]): n in [2..10]]

Table[Block[{m = n}, While[#2 != #1^n & @@ DivisorSigma[{0, 1}, m], m++]; m], {n, 10}] (* Michael De Vlieger, Nov 05 2021 *)

A349838 Irregular table read by rows; the n-th row contains in ascending order the integers m > 1 such that sigma(m) = tau(m)^n; the first row contains 1.

Original entry on oeis.org

1, 3, 7, 217, 2667, 31, 889, 27559, 3937, 172011, 677207307, 127, 1777447, 225735769, 57337, 11010027, 12189603, 3612185689, 698915267211, 253921, 113770279, 116522119, 29587412978599, 917497, 1040257, 931892355289, 954432676729, 811637999283747

Offset: 1

Rémy Sigrist, Dec 01 2021

The n-th row has A349837(n) terms.

As a flat sequence, this is a permutation of A051281.

			Table begins:
    1;
    3;
    7;
    217, 2667;
    31, 889, 27559;
    3937, 172011, 677207307;
    127, 1777447, 225735769;
    57337, 11010027, 12189603, 3612185689, 698915267211;
    253921, 113770279, 116522119, 29587412978599;
    917497, 1040257, 931892355289, 954432676729, 811637999283747;
    ...

Rémy Sigrist, PARI program for A349838

Cf. A000005 (tau), A000203 (sigma), A051281, A334455, A349006, A349007, A349837.

PARI
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See Links section.
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T(n, 1) = A349006(n) when A349837(n) > 0.
T(n, A349837(n)) = A349007(n) when A349837(n) > 0.
A334455(T(n, k)) = n.

A349837 a(n) is the number of integers m > 1 such that sigma(m) = tau(m)^n, with a(1) = 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 3, 5, 4, 5, 5, 7, 9, 5, 6, 2, 7, 7, 5, 6, 5, 9, 6, 11, 6, 12, 10, 12, 14, 11, 13, 21, 16, 20, 19, 20, 19, 21, 21, 19, 17, 21, 18, 22, 19, 22, 20, 17, 17, 15, 23, 14, 17, 12, 16, 9, 11, 12, 10, 9, 9, 7, 10, 3, 8, 7, 6, 3, 7, 4, 4, 4, 3, 3, 2

Offset: 1

Rémy Sigrist, Dec 01 2021

nonn

a(n) is the number of integers m such that A334455(m) = n.

Conjecture: a(n) > 0 except for n = 76 and n = 81.

Rémy Sigrist, Table of n, a(n) for n = 1..5000
Rémy Sigrist, PARI program for A349837

Cf. A000005 (tau), A000203 (sigma), A334455, A349006, A349007, A349838.

PARI
```
See Links section.
```

A343945 a(1) = 1; for n >= 2, a(n) = floor(log(sigma(n)) / log(tau(n))).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 3, 2, 2, 2, 4, 2, 4, 2, 2, 2, 4, 1, 3, 2, 2, 2, 4, 2, 5, 2, 2, 2, 2, 2, 5, 2, 2, 2, 5, 2, 5, 2, 2, 3, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 3, 5, 2, 5, 3, 2, 2, 3, 2, 6, 2, 3, 2, 6, 2, 6, 3, 2, 2, 3, 2, 6, 2, 2, 3, 6, 2, 3, 3, 3

Offset: 1

Jaroslav Krizek, Dec 16 2021

nonn

See A051281 for numbers m such that sigma(m) = tau(m)^k where k = integer, i.e., numbers m such that floor(log(sigma(m)) / log(tau(m))) = log(sigma(m)) / log(tau(m)).

			a(6) = floor(log(sigma(6)) / log(tau(6))) = floor(log(12) / log(4)) = floor(1.792481...) = 1.
a(7) = floor(log(sigma(7)) / log(tau(7))) = floor(log(8) / log(2)) = floor(3) = 3.

Cf. A000005(tau), A000203(sigma), A051281, A334455, A349006, A349007.

[1] cat [Floor(Log(&+Divisors(n)) / Log(#Divisors(n))): n in [2..100]]

a[1] = 1; a[n_] := Floor[Log @@ DivisorSigma[{0, 1}, n]]; Array[a, 100] (* Amiram Eldar, Dec 16 2021 *)

cp's OEIS Frontend

A349006 a(1) = 1; for n > 1, a(n) is the smallest number m such that sigma(m) = tau(m)^n or 0 if no such m exists.

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A349838 Irregular table read by rows; the n-th row contains in ascending order the integers m > 1 such that sigma(m) = tau(m)^n; the first row contains 1.

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A349837 a(n) is the number of integers m > 1 such that sigma(m) = tau(m)^n, with a(1) = 1.

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PARI

A343945 a(1) = 1; for n >= 2, a(n) = floor(log(sigma(n)) / log(tau(n))).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica