A051281 -id:A051281 - cp's OEIS frontend

A334455 a(n) is unique integer k such that sigma(A051281(n)) = tau(A051281(n))^k (where sigma is the sum of divisors (A000203) and tau the number of divisors (A000005)), with a(1) = 1.

1, 2, 3, 5, 7, 4, 5, 4, 6, 13, 5, 8, 17, 6, 9, 19, 10, 10, 7, 11, 11, 8, 8, 12, 12, 13, 9, 9, 7, 6, 15, 31, 8, 16, 11, 17, 12, 18, 18, 19, 13, 13, 13, 8, 10, 10, 11, 11, 22, 9, 12, 24, 10, 25, 17, 17, 13, 13, 14, 14, 14, 19, 12, 12, 15, 15, 61, 21, 16, 32, 13

Offset: 1

Rémy Sigrist, Nov 10 2020

nonn

			For n = 7:
- A051281(7) = 889,
- sigma(889) = 1024,
- tau(889) = 4,
- 1024 = 4^5,
- so a(7) = 5.

Rémy Sigrist, Table of n, a(n) for n = 1..12885
Rémy Sigrist, Colored scatterplot of the first 12885 terms (where the color is function of tau(A051281(n)))
Rémy Sigrist, PARI program for A334455

Cf. A000005, A000203, A051281, A225369.

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a(n) = log(A000203(A051281(n))) / log(A000005(A051281(n))) for n > 1.

A349917 a(n) is the unique k such that A349838(n) = A051281(k).

Original entry on oeis.org

1, 2, 3, 6, 8, 4, 7, 11, 9, 14, 30, 5, 19, 29, 12, 22, 23, 33, 44, 15, 27, 28, 50, 17, 18, 45, 46, 53, 20, 21, 35, 47, 48, 24, 25, 37, 51, 63, 64, 91, 10, 26, 41, 42, 43, 57, 58, 71, 85, 59, 60, 61, 78, 79, 31, 65, 66, 82, 83, 100, 34, 69, 13, 36, 55, 56, 74

Offset: 1

Rémy Sigrist, Dec 05 2021

This sequence is a permutation of the natural numbers.

			For n = 8:
- A349838(8) = 27559 = A051281(11),
- so a(8) = 11.

Rémy Sigrist, Table of n, a(n) for n = 1..3309
Rémy Sigrist, PARI program for A349917
Index entries for sequences that are permutations of the natural numbers

Cf. A051281, A349838.

PARI
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A046528 Numbers that are a product of distinct Mersenne primes (elements of A000668).

Original entry on oeis.org

1, 3, 7, 21, 31, 93, 127, 217, 381, 651, 889, 2667, 3937, 8191, 11811, 24573, 27559, 57337, 82677, 131071, 172011, 253921, 393213, 524287, 761763, 917497, 1040257, 1572861, 1777447, 2752491, 3120771, 3670009, 4063201, 5332341, 7281799, 11010027, 12189603

Offset: 1

Labos Elemer

nonn

Or, numbers n such that the sum of the divisors of n is a power of 2, see A094502.
Or, numbers n such that the number of divisors of n and the sum of the divisors of n are both powers of 2.
n is a product of distinct Mersenne primes iff sigma(n) is a power of 2: see exercise in Sivaramakrishnan, or Shallit.
Sequence gives n > 1 such that sigma(n) = 2*phi(sigma(n)). - Benoit Cloitre, Feb 22 2002
The graph of this sequence shows a discontinuity at the 4097th number because there is a large relative gap between the 12th and 13th Mersenne primes, A000043. Other discontinuities can be predicted using A078426. - T. D. Noe, Oct 12 2006
Supersequence of A051281 (numbers n such that sigma(n) is a power of tau(n)). Conjecture: numbers n such that sigma(n) = tau(n)^(a/b), where a, b are integers >= 1. Example: sigma(93) = 128 = tau(93)^(7/2) = 4^(7/2). - Jaroslav Krizek, May 04 2013

			a(20) = 82677 = 3*7*31*127, whose sum of divisors is 131072 = 2^17;
a(27) = 1040257 = 127*8191, whose sum of divisors is 1048576 = 2^20.

J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problem 264 pp. 188, Ellipses Paris 2004.
R. Sivaramakrishnan, Classical Theory of Arithmetic Functions. Dekker, 1989.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from T. D. Noe)
Kevin S. Brown, Sum of Divisors Equals a Power of 2.
C. D. H. Cooper, Problem E 2493, The American Mathematical Monthly, Vol. 81, No. 8 (1974), p. 902; W. J. Dodge, solution, ibid., Vol. 82, No. 8 (1975), pp. 855-856.
Jeffrey Shallit, Problem 1319, Diophantine Equation, sigma(n) = 2^m, Math. Magazine, 63 (1990), 129.
Eric Weisstein's World of Mathematics, Divisor Function.

Cf. A000668, A000043, A056652 (product of Mersenne primes), A306204.

mersennes:= [seq(numtheory:-mersenne([i]),i=1..10)]:
sort(select(`<`,map(convert,combinat:-powerset(mersennes),`*`),numtheory:-mersenne([11]))); # Robert Israel, May 01 2016

{1}~Join~TakeWhile[Times @@@ Rest@ Subsets@ # // Sort, Function[k, k <= Last@ #]] &@ Select[2^Range[0, 31] - 1, PrimeQ] (* Michael De Vlieger, May 01 2016 *)

isok(n) = (n==1) || (ispower(sigma(n), , &r) && (r==2)); \\ Michel Marcus, Dec 10 2013

Sum_{n>=1} 1/a(n) = Product_{p in A000668} (1 + 1/p) = 1.5855588879... (A306204) - Amiram Eldar, Jan 06 2021

More terms from Benoit Cloitre, Feb 22 2002
Further terms from Jon Hart, Sep 22 2006
Entry revised by N. J. A. Sloane, Mar 20 2007
Three more terms from Michel Marcus, Dec 10 2013

A349007 a(1) = 1; for n > 1, a(n) is the largest number m such that sigma(m) = tau(m)^n or 0 if no such m exists.

Original entry on oeis.org

1, 3, 7, 2667, 27559, 677207307, 225735769, 698915267211, 29587412978599, 811637999283747, 16907189874529, 12200855315219510767697163, 254155396405925065290841, 878412242330556407427, 1074593611687774330088252281, 16138807601873739769, 37471768236581557067194399

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) = 2667 because 2667 is the largest number m such that sigma(m) = tau(m)^4; sigma(2667) = 4096 = tau(2667)^4  = 8^4.

Rémy Sigrist, PARI program for A349007

Subsequence of A051281.

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

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A289276 Numbers k such that phi(k) (the totient function A000010) is a power of the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 3, 5, 8, 10, 17, 18, 24, 30, 34, 63, 76, 85, 128, 136, 170, 257, 315, 333, 364, 380, 436, 444, 514, 640, 680, 972, 1285, 1542, 1820, 1824, 1836, 1875, 2142, 2220, 2907, 3285, 3488, 3796, 4369, 4788, 4860

Offset: 1

Franklin T. Adams-Watters, Jun 30 2017

A019434 is a subsequence. - David A. Corneth, Jun 30 2017

Is the frequency of e such that A000005(a(n))^e = A000010(a(n)) finite? - David A. Corneth, Jul 01 2017

Charles R Greathouse IV, Table of n, a(n) for n = 1..5319 (first 526 terms from Antti Karttunen)

Cf. A000005, A000010, A019434, A020488, A032447, A036913, A051281, A068559, A068560, A114063, A286627.

Join[{1},Select[Range[2,5000],IntegerQ[Log[DivisorSigma[0,#],EulerPhi[#]]]&]] (* Harvey P. Dale, Aug 06 2017 *)

ispowerof(n, k)= if(k==1, return(n==1)); while(n>=k, if(n%k!=0, return(0)); n\=k); n==1
isa(n) = ispowerof(eulerphi(n),numdiv(n)) \\ Quick program, fast enough for early values.

is(n) = if(n==1, return(1)); my(f = factor(n); phi = eulerphi(f), ndiv = numdiv(f), e = logint(phi, ndiv)); ndiv^e == phi \\ David A. Corneth, Jun 30 2017, changed per suggestion of Charles R Greathouse IV

isA289276(n)= if(n==1, return(1)); my(phi = eulerphi(n), ndiv = numdiv(n), v = valuation(phi, ndiv)); ndiv^v == phi; \\ (A variant of above program). - Antti Karttunen, Jun 30 2017

list(lim)=my(v=List([1])); forfactored(n=2,lim\1, my(phi = eulerphi(n), ndiv = numdiv(n)); if(ndiv^valuation(phi,ndiv) == phi, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Jul 01 2017

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.

Original entry on oeis.org

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 *)

A225239 Numbers n such that there is an integer k with the property that k^tau(n) = sigma(n).

Original entry on oeis.org

1, 3, 217, 862, 1177, 1207, 1219, 3937, 8743, 9481, 13822, 18137, 19567, 19849, 20057, 20257, 20299, 20437, 33607, 57337, 91847, 96217, 100579, 103897, 154969, 157921, 158623, 228889, 233047, 304117, 324817, 325579, 329057, 330529, 537817, 595417, 608287

Offset: 1

Jaroslav Krizek, May 04 2013

nonn

Corresponding values of k: 1, 2, 4, 6, 6, 6, 6, 8, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 14, 16, 18, 18, 18, 18, 20, 20, 20, 22, 22, 24, 24, 24, 24, 24, 28, 28, 28, ... (see A225369).
Conjecture: all terms are squarefree numbers.
Conjecture is false: p = (312^6 / 13) - 1 = 70955197267967 is prime, so sigma(9*p) = sigma(9)*sigma(p) = 13*(p+1) = 312^6 = 312^tau(9*p). - Charlie Neder, Oct 05 2018

			a(4) = 862 because sigma(862) = 1296 = 6^tau(862) = 6^4; k = 6.

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A000005 (tau(n): number of divisors of n).
Cf. A000203 (sigma(n): sum of divisors of n).
Cf. A051281 (sigma(n) is a power of tau(n)), A225369.

c=1; write("b225239.txt", c " " 1); for(n=2, 1943881801, s=sigma(n); if(ispower(s), nd=numdiv(n); r=round(sqrtn(s, nd)); if(r^nd==s, c++; write("b225239.txt", c " " n)))) /* Donovan Johnson, May 05 2013 */

isok(n) = if (n==1, return(1)); my(s=sigma(n)); if(ispower(s), my(nd=numdiv(n)); r=sqrtnint(s, nd); (r^nd==s);); \\ Michel Marcus, Feb 19 2020

A286628 a(n) = exponent of the highest power of A000005(n) (number of divisors of n) dividing A000203(n) (sum of divisors of n), a(1) = 1.

Original entry on oeis.org

1, 0, 2, 0, 1, 1, 3, 0, 0, 0, 2, 0, 1, 1, 1, 0, 1, 0, 2, 1, 2, 1, 3, 0, 0, 0, 1, 0, 1, 1, 5, 0, 2, 0, 2, 0, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 4, 0, 1, 0, 1, 0, 1, 1, 1, 1, 2, 0, 2, 1, 1, 2, 0, 0, 1, 1, 2, 1, 2, 1, 3, 0, 1, 0, 0, 0, 2, 1, 4, 0, 0, 0, 2, 0, 1, 1, 1, 0, 1, 0, 2, 1, 3, 2, 1, 1, 1, 0, 1, 0, 1, 1, 3, 0, 2, 0, 2, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 2, 0

Offset: 1

Antti Karttunen, Jun 30 2017

nonn

a(1) = 1 by convention.

			A000005(6) = 4, A000203(6) = 12, 4^1 is the highest power of 4 which divides 12, thus a(6) = 1.
A000005(7) = 2, A000203(7) = 8, 2^3 is the highest power of 2 which divides 8, thus a(7) = 3.
A000005(8) = 4, A000203(8) = 15, 4^0 = 1 is the highest power of 4 which divides 15, thus a(8) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A051281, A054025, A286561, A286627.

Cf. A049642 (positions of zeros), A003601 (of nonzeros).

A286628(n) = if(1==n,n,valuation(sigma(n), numdiv(n)));

a(n) = A286561(A000203(n), A000005(n)).

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

A219668 Numbers m for which sigma(m) - m = tau(m)^k for some integer k > 0.

Original entry on oeis.org

4, 26, 56, 90, 122, 568, 2042, 8186, 32762, 37432, 68652, 299576, 2097146, 8388602, 19173944, 33554426, 67751984, 78536544824, 306296525088, 15640174780344, 39998905951528, 120948840863188

Offset: 1

Zdenek Cervenka, Nov 27 2012

39614081257132168796771975162 is also a term. - Donovan Johnson, Nov 28 2012

19495118728903626376363904 = 2^7*152305615069559581065343 is a term. - Martin Ehrenstein, Jul 31 2023

Cf. A051281.

f[n_] := FullSimplify[Log[DivisorSigma[1, n] - n]/Log[DivisorSigma[0, n]]]; Select[Range[2, 1000], IntegerQ[f[#]] && f[#] > 0 &] (* T. D. Noe, Nov 27 2012 *)

a(16)-a(17) from Donovan Johnson, Nov 28 2012

a(18)-a(22) from Martin Ehrenstein, Jul 31 2023

cp's OEIS Frontend

A334455 a(n) is unique integer k such that sigma(A051281(n)) = tau(A051281(n))^k (where sigma is the sum of divisors (A000203) and tau the number of divisors (A000005)), with a(1) = 1.

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A349917 a(n) is the unique k such that A349838(n) = A051281(k).

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A046528 Numbers that are a product of distinct Mersenne primes (elements of A000668).

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A349007 a(1) = 1; for n > 1, a(n) is the largest number m such that sigma(m) = tau(m)^n or 0 if no such m exists.

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A289276 Numbers k such that phi(k) (the totient function A000010) is a power of the number of divisors of k (A000005).

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Mathematica

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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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Magma

Mathematica

A225239 Numbers n such that there is an integer k with the property that k^tau(n) = sigma(n).

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