A180162 -id:A180162 - cp's OEIS frontend

A180221 Numbers that can be written as sum of one or more distinct elements of A000043. Numbers k for which sigma(A180162(k))=2^k, k>=2.

2, 3, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Walter Kehowski, Aug 16 2010

nonn

The distinct values of log_2(sigma(m)), where m > 1 is a term of A046528. - Amiram Eldar, Jun 02 2020

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1855 from Walter A. Kehowski)

Cf. A000043, A180162, A046528, A180169. Complement of A078426.

n = 10; p = MersennePrimeExponent[Range[n]]; Rest[-1 + Position[CoefficientList[Series[Product[(1 + x^p[[k]]), {k, 1, n}], {x, 0, p[[-1]]}], x], ?(# > 0 &)] // Flatten] (* _Amiram Eldar, Jun 02 2020 *)

A180169 Sigma(A180162(n)), where A180162(n) is the smallest N such that sigma(N) is an n-th power, or 0 if no such N can be found.

Original entry on oeis.org

1, 3, 4, 8, 1296, 32, 46656, 128, 256, 512, 1024, 362797056, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824, 2147483648

Offset: 0

Walter Kehowski, Aug 14 2010

			A180162(4)=510 and so A180169(4)=sigma(510)=2^4*3^4.

Ray Chandler, Table of n, a(n) for n = 0..469

Cf. A065496, A000203, A180162.

Empirical o.g.f.: (-1-1*x+2*x^2-1280*x^4+2560*x^5-46592*x^6+93184*x^7-362795008*x^11+725590016*x^12)/(2*x-1). - Simon Plouffe, Feb 26 2011.

a(23) onwards from Ray Chandler, Aug 20 2010

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.

Original entry on oeis.org

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

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

cp's OEIS Frontend

A180221 Numbers that can be written as sum of one or more distinct elements of A000043. Numbers k for which sigma(A180162(k))=2^k, k>=2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A180169 Sigma(A180162(n)), where A180162(n) is the smallest N such that sigma(N) is an n-th power, or 0 if no such N can be found.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions