A078426 -id:A078426 - cp's OEIS frontend

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

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

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.

Original entry on oeis.org

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 *)

A180460 a(n) is the smallest number m such that sigma(m)=22^n, or 0 if m does not exist.

Original entry on oeis.org

1, 0, 0, 10363, 136647, 3018141, 66411009, 1636922343, 31276995183, 688217286267, 15200749439001, 324029599659171, 7264291502741679, 160447401116572437, 3530475812620849113, 75514126111770824037, 1662716417771040164631, 36586320846189859358019, 804905851136700392012493, 17704604426749226872106319

Offset: 0

Walter Kehowski, Sep 06 2010

nonn

Conjecture: Given any even integer E not a power of 2 (see A078426) there exists a positive integer N such that for all n>=N the equation sigma(m)=E^n has at least one solution for m.

			a(4)=136647=3^4*7*241 since sigma(3^4*7*241)=(11^2)(2^3)(2*11^2)=2^4*11^4 and 136647 is the smallest such number.

Max Alekseyev, Table of n, a(n) for n = 0..100

Cf. A048251, A110077, A180265, A180461, A180462, A180463, A180464.

Terms a(37) onward (in b-file) from Max Alekseyev, Mar 04 2014

A063883 Number of ways writing n as a sum of different Mersenne prime exponents (terms of A000043).

Original entry on oeis.org

0, 1, 1, 0, 2, 0, 2, 1, 1, 2, 0, 2, 1, 1, 2, 1, 2, 2, 2, 3, 2, 4, 2, 4, 3, 3, 4, 2, 4, 2, 4, 3, 3, 4, 3, 4, 4, 4, 5, 4, 5, 4, 4, 5, 3, 5, 3, 4, 4, 3, 5, 3, 5, 4, 4, 5, 4, 5, 4, 4, 5, 3, 5, 4, 3, 6, 2, 6, 3, 5, 5, 3, 6, 3, 5, 4, 4, 4, 4, 4, 4, 4, 5, 3, 6, 3, 5, 5, 4, 6, 3, 7, 3, 6, 5, 5, 6, 5, 6, 5, 6, 6, 5, 6, 6

Offset: 1

Labos Elemer, Aug 28 2001

This sequence appears to be growing. However, for 704338 < n < 756839, a(n) is 0. See A078426 for the n such that a(n) = 0. - T. D. Noe, Oct 12 2006

Numbers k such that sigma(k) = 2^n. - Juri-Stepan Gerasimov, Mar 08 2017

			n = 50 = 2 + 5 + 7 + 17 + 19 = 2 + 17 + 31 = 19 + 31, so a(50) = 3. The first numbers for which the number of these Mersenne-exponent partitions is k = 0, 1, 2, 3, 4, 5, 6, 7, 8 are 1, 2, 5, 20, 22, 39, 66, 92, 107, respectively.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000043, A046528, A048947, A054784, A054973, A063889.

Numbers k such that a(k) = m: A078426 (m = 0), A283160 (m = 1).

N:= 500: # to get the first N terms
G:= mul(1+x^i,i=select(t -> numtheory:-mersenne(t)::integer, [$1..N])):
S:= series(G,x,N+1):
seq(coeff(S,x,n),n=1..N); # Robert Israel, Sep 22 2016

exponents[n_] := Reap[For[k = 1, k <= n, k++, If[PrimeQ[2^k-1], Sow[k]]]][[2, 1]]; r[n_] := Module[{ee, x, xx}, ee = exponents[n]; xx = Array[x, Length[ee]]; Reduce[And @@ (0 <= # <= 1 & /@ xx) && xx.ee == n, xx, Integers]]; a[n_] := Which[rn = r[n]; Head[rn] === Or, Length[rn],  Head[rn] === And, 1, Head[rn] === Equal, 1, rn === False, 0, True, Print["error ", rn]]; a[1] = 0; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Feb 05 2014 *)
With[{e = MersennePrimeExponent[Range[10]]}, Rest@ CoefficientList[Product[1 + x^e[[i]], {i, 1, Length[e]}], x]] (* Amiram Eldar, Dec 23 2024 *)

first(lim)=my(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); if(lim>M[#M], error("Need more Mersenne exponents to compute further")); M=select(p->p<=lim, M); Vec(prod(i=1, #M, 1+x^M[i], O(x^(lim\1+1))+1)) \\ Charles R Greathouse IV, Mar 08 2017

a(n) = sum(k=1, 2^n+1, sigma(k)==2^n); \\ Michel Marcus, Mar 07 2017

a(n) = A054973(2^n). - Michel Marcus, Mar 08 2017

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

A094502 a(n) = A000203(A046528(n)): sigma of those numbers whose sigma is a power of 2, in order of appearance.

Original entry on oeis.org

1, 4, 8, 32, 32, 128, 128, 256, 512, 1024, 1024, 4096, 4096, 8192, 16384, 32768, 32768, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 1048576, 2097152, 2097152, 4194304, 4194304, 4194304, 4194304, 8388608

Offset: 1

Labos Elemer, Jun 02 2004

nonn

Observe that certain powers of 2 do not arise as sum of divisors of something: 2,16,64,2048. Are there more? Yes, see A094505 and A078426.

Cf. A000203, A046528, A078426, A094505.

{ta=Table[0, {100}], u=1}; Do[If[IntegerQ[Log[2, DivisorSigma[1, n]]], Print[n];ta[[u]]=n;u=u+1], {n, 1, 100000000}] DivisorSigma[1, ta]

isok(n) = (n==1) || (ispower(sigma(n), , &r) && (r==2));
for(n=1, 1e7, if(isok(n), print1(sigma(n)", "))) \\ Altug Alkan, Nov 01 2015

a(n) = 2^A048947(n). - R. J. Mathar, Sep 22 2016

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

A094505 Powers of 2 which are not the sum of divisors of any other number. Powers of 2 present in A007369.

Original entry on oeis.org

2, 16, 64, 2048

Offset: 1

Labos Elemer, Jun 02 2004

nonn

The next term, 2^470, is too large to include.

Cf. A007369, A000203, A046528, A094502.

a(n) = 2^A078426(n).

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

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

cp's OEIS Frontend

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

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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.

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A180460 a(n) is the smallest number m such that sigma(m)=22^n, or 0 if m does not exist.

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A063883 Number of ways writing n as a sum of different Mersenne prime exponents (terms of A000043).

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

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A094502 a(n) = A000203(A046528(n)): sigma of those numbers whose sigma is a power of 2, in order of appearance.

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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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