A180221 -id:A180221 - cp's OEIS frontend

A078426 Numbers k such that there is no solution to the equation sigma(x) = 2^k, where sigma(x) denotes the sum of the divisors of x.

1, 4, 6, 11, 470, 475, 477, 480, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 522, 525, 527, 532, 1077, 1082

Offset: 1

Shyam Sunder Gupta, Dec 29 2002

nonn

Numbers that are not a sum of distinct Mersenne exponents (A000043). - Vladeta Jovovic, Jan 01 2003
Because there is a large gap between the 31st and 32nd Mersenne exponents, all k between 704338 and 756839 are in this sequence. - T. D. Noe, Oct 12 2006
A000203(A180162(a(n))) = 6^a(n), for n > 1. - Walter Kehowski, Aug 16 2010
Using all known Mersenne exponents, there are exactly 52935 terms in this sequence. When a new Mersenne prime (with exponent q) is found, there will be no new terms if the sum of the previous Mersenne exponents (A109472) is greater than q - 22.

			a(2)=4 because no positive integer value of x can satisfy sigma(x) = 2^4 = 16.

S. Kravitz, "Beware of the Fifth", Solution to Problem 2309, Journal of Recreational Mathematics, 29(1):76 Baywood NY 1998.

T. D. Noe, Table of n, a(n) for n = 1..300

Cf. A000203, A007369, A046528, A063883, A180221 (complement).

e={2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217,4253, 4423,9689,9941,11213,19937,21701,23209,44497,86243,110503,132049,216091, 756839,859433,1257787,1398269}; u={0}; Do[u=Union[u, u+e[[k]]], {k,Length[e]}]; Complement[Range[e[[-1]]], u]

More terms from Vladeta Jovovic, Jan 01 2003
Edited by N. J. A. Sloane, Aug 23 2010
Edited by Max Alekseyev, Jan 24 2014

A175432 a(n) = the greatest number k such that sigma(n) = m^k for any m >= 1 (sigma = A000203).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Jaroslav Krizek, May 10 2010

nonn

a(A175431(n)) = 1 for n >= 1.
a(A065496(n)) > 1 for n >= 1.
It appears that the record values in this sequence, 1, 2, 3, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, ..., is A180221 with a 1 prepended, at least through term #469. Is this a theorem? - Ray Chandler, Aug 20 2010

			For n = 7, a(7) = 3 because sigma(7) = 8 = 2^3.

Antti Karttunen, Table of n, a(n) for n = 1..16384

For locations of records see A169981.

Cf. A000203, A052409, A175433.

Array[Apply[GCD, FactorInteger[DivisorSigma[1, #]][[All, -1]]] &, 105] (* Michael De Vlieger, Nov 05 2017 *)

a(n)=max(ispower(sigma(n)),1) \\ Charles R Greathouse IV, Feb 14 2013

a(n) = A052409(A000203(n)). - N. J. A. Sloane, Aug 19 2010

a(n) = log_A175433(n) [A000203(n)].

A247956 a(n) is the smallest number k such that sigma(k) = 2^n or 0 if no such k exists.

Original entry on oeis.org

1, 0, 3, 7, 0, 21, 0, 93, 217, 381, 651, 0, 2667, 8191, 11811, 24573, 57337, 82677, 172011, 393213, 761763, 1572861, 2752491, 5332341, 11010027, 21845397, 48758691, 85327221, 199753347, 341310837, 677207307, 1398273429, 3220807683, 6192353757, 10836557067

Offset: 0

Jaroslav Krizek, Sep 28 2014

nonn

See A078426 for numbers n such that there is no solution to the equation sigma(x) = 2^n.

If a(n) > 0, then it is a term of A046528 (numbers that are a product of distinct Mersenne primes).

			a(0) = 1 because 1 is the smallest number k with sigma(1) = 1 = 2^0.
a(5) = 21 because 21 is the smallest number k with sigma(k) = 32 = 2^5.
a(6) = 0 because there is no number k with sigma(k) = 64 = 2^6.

Amiram Eldar, Table of n, a(n) for n = 0..300
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000203, A046528, A078426, A180162, A180221, A182221, A295043.

a(n) = for (k=1, 2^n, if (sigma(k)== 2^n, return (k))); return (0); \\ Michel Marcus, Oct 03 2014, Oct 31 2015

a(n) = max(0, invsigmaMin(1<Amiram Eldar, Dec 31 2024, using Max Alekseyev's invphi.gp (see links).

a(A078426(n)) = 0.

a(A182221(n)) > 0.

a(0) = 1 prepended by Michel Marcus, Oct 31 2015

A295043 a(n) is the largest number k such that sigma(k) = 2^n or 0 if no such k exists.

Original entry on oeis.org

1, 0, 3, 7, 0, 31, 0, 127, 217, 381, 889, 0, 3937, 8191, 11811, 27559, 57337, 131071, 253921, 524287, 1040257, 1777447, 4063201, 7281799, 16646017, 32247967, 66584449, 116522119, 225735769, 516026527, 1073602561, 2147483647, 4294434817, 7515217927, 15032385529

Offset: 0

Jaroslav Krizek, Nov 13 2017

nonn

If a(n) > 0, then it is a term of A046528 (numbers that are a product of distinct Mersenne primes).

			a(0) = 1 because 1 is the largest number k with sigma(k) = 1 = 2^0.
a(5) = 31 because 31 is the largest number k with sigma(k) = 32 = 2^5.
a(6) = 0 because there is no number k with sigma(k) = 64 = 2^6.

Amiram Eldar, Table of n, a(n) for n = 0..250
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Cf. A000043, A000203, A046528, A048947, A057637, A180221.
Cf. A247956 (the smallest number k instead of the largest).
Cf. A078426 (no solution to the equation sigma(x)=2^n).
A000668 (Mersenne primes) is a subsequence.

a(n) = {local(r, k); r=0; for(k=1, 2^n, if(sigma(k) == 2^n, r=k)); return(r)}; \\ Michael B. Porter, Nov 14 2017

a(n) = forstep(k=2^n, 1, -1, if (sigma(k)==2^n, return (k))); return (0) \\ Rémy Sigrist, Jan 08 2018

a(n) = invsigmaMax(1<Amiram Eldar, Dec 20 2024, using Max Alekseyev's invphi.gp

a(A078426(n)) = 0.
a(A180221(n)) > 0.
a(n) <= 2^n - 1 with equality when n is a Mersenne exponent (A000043). - Michael B. Porter, Nov 14 2017

A247473 Numbers of the form 2^k (k>=0) that are a sum of divisors of n for some n.

Original entry on oeis.org

1, 4, 8, 32, 128, 256, 512, 1024, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 34359738368

Offset: 1

Jaroslav Krizek, Feb 27 2015

nonn

Set of values A094502(n) = sigma(A046528(n)) in increasing order.
Complement of A094505 with respect to A000079 (powers of 2).
Corresponding values of numbers k>0 are in A180221.

			32 = 2^5 is in sequence because there are numbers n = 21 and 31 with sigma(n) = 32.

Cf. A000079, A000203, A046528, A078426, A094502, A094505, A180221.

Set(Sort([SumOfDivisors(n): n in [A046528(n)]]))

Set(Sort([SumOfDivisors(n): n in[1..10000], k in [0..100] | SumOfDivisors(n) eq 2^k]))

Magma
```
[1] cat [2^n: n in[A180221(n)]]
```

a(1) = 1, for n>=2, a(n) = 2^A180221(n-1).

cp's OEIS Frontend

A078426 Numbers k such that there is no solution to the equation sigma(x) = 2^k, where sigma(x) denotes the sum of the divisors of x.

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A175432 a(n) = the greatest number k such that sigma(n) = m^k for any m >= 1 (sigma = A000203).

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A247956 a(n) is the smallest number k such that sigma(k) = 2^n or 0 if no such k exists.

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A295043 a(n) is the largest number k such that sigma(k) = 2^n or 0 if no such k exists.

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A247473 Numbers of the form 2^k (k>=0) that are a sum of divisors of n for some n.

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