A063883 -id:A063883 - 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

A054784 Integers n such that sigma(2n) - sigma(n) is a power of 2, where sigma is the sum of the divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 31, 32, 42, 48, 56, 62, 64, 84, 93, 96, 112, 124, 127, 128, 168, 186, 192, 217, 224, 248, 254, 256, 336, 372, 381, 384, 434, 448, 496, 508, 512, 651, 672, 744, 762, 768, 868, 889, 896, 992, 1016, 1024, 1302, 1344, 1488

Offset: 1

Labos Elemer, May 22 2000

nonn

If n is a squarefree product of Mersenne primes multiplied by a power of 2, then sigma(2n) - sigma(n) is a power of 2.
The reverse is also true. All numbers in this sequence have this form. - Ivan Neretin, Aug 12 2016
From Antti Karttunen, Sep 01 2021: (Start)
Numbers k such that the sum of their odd divisors [A000593(k)] is a power of 2.
Numbers k whose odd part [A000265(k)] is in A046528.
(End)

			For n=12, sigma(2n) = sigma(24) = 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 and sigma(n) = sigma(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28. So sigma(2n) - sigma(n) = 60 - 28 = 32 = 2^5 is a power of 2, and therefore 12 is in the sequence. - _Michael B. Porter_, Aug 15 2016

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000203, A000265, A000396 (even terms form a subsequence), A000593, A000668, A046528, A063883, A209229, A306204, A331410, A336923 (characteristic function).
Positions of zeros in A336922. Positions of 0's and 1's in A336361.
Cf. also A003401.

N:= 10^6: # to get all terms <= N
M:= select(isprime, [seq(2^i-1, i=select(isprime, [$2..ilog2(N+1)]))]):
R:= map(t -> seq(2^i*t, i=0..floor(log[2](N/t))), map(convert,combinat:-powerset(M),`*`)):
sort(convert(R,list)); # Robert Israel, Aug 12 2016

Sort@Select[Flatten@Outer[Times, p2 = 2^Range[0, 11], Times @@ # & /@ Subsets@Select[p2 - 1, PrimeQ]], # <= Max@p2 &] (* Ivan Neretin, Aug 12 2016 *)
Select[Range[1500],IntegerQ[Log2[DivisorSigma[1,2#]-DivisorSigma[1,#]]]&] (* Harvey P. Dale, Apr 23 2019 *)

A209229(n) = (n && !bitand(n,n-1));
isA054784(n) = A209229(sigma(n>>valuation(n,2))); \\ Antti Karttunen, Aug 28 2021

Numbers n such that A000203(2*n) - A000203(n) = 2^w for some w.

Sum_{n>=1} 1/a(n) = 2 * Product_{p in A000668} (1 + 1/p) = 2 * A306204 = 3.1711177758... . - Amiram Eldar, Jan 11 2023

A063869 Least k such that sigma(k)=m^n for some m>1.

Original entry on oeis.org

2, 3, 7, 217, 21, 2667, 93, 217, 381, 651, 2752491, 2667, 8191, 11811, 24573, 57337, 82677, 172011, 393213, 761763, 1572861, 2752491, 5332341, 11010027, 21845397, 48758691, 85327221, 199753347, 341310837, 677207307, 1398273429, 3220807683

Offset: 1

Labos Elemer, Aug 27 2001

nonn

For n=2 to 20 sigma(a(n)) = m^n with m=2 or m=4. Computed terms are products of Mersenne primes (A000668). Is this true for larger n? Validity of a(11) was tested individually.
The Nagell-Ljunggren conjecture implies that sigma(x) is never 3^n for n>1. If this is true, then m=2 and m=4 are the smallest possible solutions. When A063883(n)>0, we can take m=2 and, as explained by Brown, find k to be a product of Mersenne primes (i.e. one of the numbers in A046528). When A063883(n)=0, which is true for the n in A078426, then m=4 and we have a(n)=a(2n) because 4=2^2. - T. D. Noe, Oct 18 2006
Sierpiński says that he proved sigma(x) is never 3^r for r>1. Hence m=2 and m=4 are the smallest possible solutions. When A063883(n)>0, we can take m=2 and, as explained by Brown, find k to be a product of Mersenne primes (i.e. one of the numbers in A046528). When A063883(n)=0, which is true for the n in A078426, then m=4 and we have a(n)=a(2n) because 4=2^2. - T. D. Noe, Oct 18 2006

			For n = 11, sigma(a(n)) = sigma(2752491) = sigma(3 * 7 * 131071) = 4^11.

T. D. Noe, Table of n, a(n) for n=1..500
K. S. Brown, Sum of Divisors Equals a Power of 2
W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964, page 165.

Cf. A006532, A020477, A019422, A019423, A019424, A048257, A048258, A000668, A046528.

d={2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253}; nn=3700; t=Table[Infinity, {nn}]; t[[1]]=2; u={0}; k=1; While[2+d[[k]]<=nn, mer=2^d[[k]]-1; Do[a=u[[i]]+d[[k]]; If[a<=nn, If[u[[i]]==0, t[[a]]=Min[t[[a]], mer], t[[a]]=Min[t[[a]], t[[u[[i]]]]*mer]]], {i, Length[u]}]; u=Union[u, u+d[[k]]]; k++ ]; Do[If[t[[i]]==Infinity, t[[i]]=t[[2i]]], {i, nn}]; t (* T. D. Noe, Oct 13 2006 *)
c[] = 0; c[1] = 2; r = 1; Do[S = If[# > 1, Rest@ Divisors@ #, 0] &[GCD @@ FactorInteger[DivisorSigma[1, i]][[All, -1]]]; If[Length[S] > 0, Map[If[c[#] == 0, Set[c[#], i]] &, S]; If[# > r, r = #] &@ Max@ S], {i, 2^22}]; TakeWhile[Array[c, r], # > 0 &] (* _Michael De Vlieger, May 23 2022 *)

a(n) = my(k=2); while (!ispower(sigma(k), n), k++); k; \\ Michel Marcus, May 23 2022

a(n) = Min{x : A000203(x)=m^n} for some m.

a(24) corrected by T. D. Noe, Oct 15 2006

A283160 Numbers k such that there is exactly one solution to the equation sigma(x) = 2^k, where sigma(x) denotes the sum of the divisors of x.

Original entry on oeis.org

0, 2, 3, 8, 9, 13, 14, 16, 465, 467, 468, 472, 473, 478, 479, 481, 521, 523, 524, 529, 530, 534, 535, 537, 1072, 1074, 1075, 1079, 1080, 1085, 1086, 1088, 1128, 1130, 1131, 1136, 1137, 1141, 1142, 1144, 1744, 1746, 1747, 1751, 1752, 1757, 1758, 1760, 1800, 1802, 1803, 1808, 1809, 1813, 1814

Offset: 1

Juri-Stepan Gerasimov, Mar 01 2017

nonn

Numbers n such that there is a unique subset S of the Mersenne exponents A000043 summing to n. - Charles R Greathouse IV, Mar 07 2017

			8 is in this sequence that k = 217 is the only number having sigma(k) = 2^8.

Charles R Greathouse IV, Table of n, a(n) for n = 1..168

Cf. A000203, A046528, A063883, A078426, A247956.

With[{e = MersennePrimeExponent[Range[16]]}, Select[Range[0, e[[-1]]], SeriesCoefficient[Series[Product[1 + x^e[[i]], {i, 1, Length[e]}], {x, 0, #}], #] == 1 &]] (* Amiram Eldar, Dec 20 2024 *)

is(n)=my(N=2^n,s); for(k=1,N, if(sigma(k)==N && s++>1, return(0))); s \\ Charles R Greathouse IV, Mar 07 2017

list(lim)=my(v=List(),M=[2,3,5,7,13,17,19,31,61,89,107,127,521,607,1279,2203,2281,3217,4253,4423,9689,9941,11213,19937,21701,23209,44497,86243,110503,132049,216091,756839,859433,1257787,1398269,2976221,3021377,6972593,13466917,20996011,24036583,25964951,30402457,32582657,37156667], x='x,P); if(lim>M[#M], error("Need more Mersenne exponents to compute further")); M=select(p->p<=lim,M); P=prod(i=1,#M,1+x^M[i],O(x^(lim\1+1))+1); for(i=0,lim, if(polcoeff(P,i)==1, listput(v,i))); P=0; Vec(v) \\ Charles R Greathouse IV, Mar 07 2017

A063883(a(n)) = 1.

a(9)-a(55) from Charles R Greathouse IV, Mar 07 2017

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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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A054784 Integers n such that sigma(2n) - sigma(n) is a power of 2, where sigma is the sum of the divisors of n.

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A063869 Least k such that sigma(k)=m^n for some m>1.

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A283160 Numbers k such that there is exactly one solution to the equation sigma(x) = 2^k, where sigma(x) denotes the sum of the divisors of x.

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