Displaying 1-10 of 12 results found.
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.
+20
6
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
EXAMPLE
For n = 7:
- sigma(889) = 1024,
- tau(889) = 4,
- 1024 = 4^5,
- so a(7) = 5.
PROG
(PARI) See Links section.
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
COMMENTS
This sequence is a permutation of the natural numbers.
EXAMPLE
For n = 8:
- so a(8) = 11.
PROG
(PARI) See Links section.
Numbers that are a product of distinct Mersenne primes (elements of A000668).
+10
38
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
COMMENTS
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
REFERENCES
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.
LINKS
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.
EXAMPLE
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.
MAPLE
mersennes:= [seq(numtheory:-mersenne([i]), i=1..10)]:
sort(select(`<`, map(convert, combinat:-powerset(mersennes), `*`), numtheory:-mersenne([11]))); # Robert Israel, May 01 2016
MATHEMATICA
{1}~Join~TakeWhile[Times @@@ Rest@ Subsets@ # // Sort, Function[k, k <= Last@ #]] &@ Select[2^Range[0, 31] - 1, PrimeQ] (* Michael De Vlieger, May 01 2016 *)
PROG
(PARI) isok(n) = (n==1) || (ispower(sigma(n), , &r) && (r==2)); \\ Michel Marcus, Dec 10 2013
EXTENSIONS
Further terms from Jon Hart, Sep 22 2006
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.
+10
5
1, 3, 7, 2667, 27559, 677207307, 225735769, 698915267211, 29587412978599, 811637999283747, 16907189874529, 12200855315219510767697163, 254155396405925065290841, 878412242330556407427, 1074593611687774330088252281, 16138807601873739769, 37471768236581557067194399
COMMENTS
See A051281 for numbers m such that sigma(m) = tau(m)^k where k = integer. a(n) = 0 for n = 76, 81, ...
EXAMPLE
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.
PROG
(PARI) See Links section.
Numbers k such that phi(k) (the totient function A000010) is a power of the number of divisors of k ( A000005).
+10
4
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
MATHEMATICA
Join[{1}, Select[Range[2, 5000], IntegerQ[Log[DivisorSigma[0, #], EulerPhi[#]]]&]] (* Harvey P. Dale, Aug 06 2017 *)
PROG
(PARI) 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.
(PARI) 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(PARI) 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 (PARI) 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
CROSSREFS
Cf. A000005, A000010, A019434, A020488, A032447, A036913, A051281, A068559, A068560, A114063, A286627.
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.
+10
4
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
COMMENTS
See A051281 for numbers m such that sigma(m) = tau(m)^k where k = integer. a(n) = 0 for n = 76, 81, ...
EXAMPLE
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.
MATHEMATICA
Table[Block[{m = n}, While[#2 != #1^n & @@ DivisorSigma[{0, 1}, m], m++]; m], {n, 10}] (* Michael De Vlieger, Nov 05 2021 *)
PROG
(Magma) [1] cat [Min([m: m in[2..10^6] | &+Divisors(m) eq #Divisors(m)^n]): n in [2..10]]
Numbers n such that there is an integer k with the property that k^tau(n) = sigma(n).
+10
3
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
COMMENTS
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
EXAMPLE
a(4) = 862 because sigma(862) = 1296 = 6^tau(862) = 6^4; k = 6.
PROG
(PARI) 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 */ (PARI) 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
CROSSREFS
Cf. A000005 (tau(n): number of divisors of n). Cf. A000203 (sigma(n): sum of divisors of n).
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.
+10
3
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
EXAMPLE
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.
PROG
(PARI) A286628(n) = if(1==n, n, valuation(sigma(n), numdiv(n)));
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.
+10
3
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
COMMENTS
As a flat sequence, this is a permutation of A051281.
EXAMPLE
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;
...
PROG
(PARI) See Links section.
Numbers m for which sigma(m) - m = tau(m)^k for some integer k > 0.
+10
0
4, 26, 56, 90, 122, 568, 2042, 8186, 32762, 37432, 68652, 299576, 2097146, 8388602, 19173944, 33554426, 67751984, 78536544824, 306296525088, 15640174780344, 39998905951528, 120948840863188
COMMENTS
39614081257132168796771975162 is also a term. - Donovan Johnson, Nov 28 2012 19495118728903626376363904 = 2^7*152305615069559581065343 is a term. - Martin Ehrenstein, Jul 31 2023
MATHEMATICA
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 *)
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