A229335 -id:A229335 - cp's OEIS frontend

A316225 Numbers k that divide the sum of sums of elements of all subsets of divisors of k (A229335).

1, 2, 4, 6, 8, 16, 24, 28, 32, 40, 64, 96, 120, 128, 224, 256, 288, 360, 384, 496, 512, 640, 672, 1024, 1536, 1792, 1920, 2016, 2048, 2176, 3744, 4096, 4320, 4680, 5632, 5760, 6144, 6528, 8128, 8192, 10240, 10880, 14336, 15872, 16384, 16896, 18432, 18688

Offset: 1

Amiram Eldar, Dec 21 2018

nonn

Harborth proved that this sequence is infinite. He showed that the terms are numbers n such that n|sigma(n)*2^(d(n) - 1), where d(n) is the number of divisors of n (A000005) and sigma(n) is their sum (A000203), and that the even terms, numbers of the form r*2^m where r is odd and m > 0, are those with m = ord_2(r/gcd(r, sigma(r)))*i with i = 1, 2, ... (ord_2(k) is the multiplicative order of 2 mod k, A002326). Thus this sequence includes all the powers of 2, all the numbers of the form n = 2^m*(2^(m + 1) - 1) which include the even perfect numbers.

Pollack and Pomerance call these numbers "H-perfect numbers". They prove that k is H-perfect if and only if denominator(sigma(k)/k) is a power of 2. - Amiram Eldar, Jun 02 2020

Amiram Eldar, Table of n, a(n) for n = 1..500
Heiko Harborth, Eine Bemerkung zu den vollkommenen Zahlen, Elemente der Mathematik, Vol. 31 (1976), pp. 119-121 (in German).
Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.

Cf. A000005, A000203, A000396, A002326, A229335.

divSumSubQ[n_] := Divisible[DivisorSigma[1, n] * 2^(DivisorSigma[0, n] - 1), n]; Select[Range[100000], divSumSubQ]

isok(n) = (sigma(n)*2^(numdiv(n)-1) % n) == 0; \\ Michel Marcus, Dec 21 2018

A061652 Even superperfect numbers: 2^(p-1) where 2^p-1 is a Mersenne prime (A000668).

Original entry on oeis.org

2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, 81129638414606681695789005144064, 85070591730234615865843651857942052864

Offset: 1

Jason Earls, Jun 16 2001

It is conjectured that there are no odd superperfect numbers, in which case this coincides with A019279.
The number of divisors of a(n) is equal to A000043(n). - Omar E. Pol, Feb 29 2008
The sum of divisors of a(n) is equal to A000668(n), the n-th Mersenne prime. - Omar E. Pol, Mar 11 2008
Largest proper divisor of A072868(n). - Omar E. Pol, Apr 25 2008
Indices of hexagonal numbers (A000384) that are also even perfect numbers. [Omar E. Pol, Aug 26 2008]
Except for the first perfect number 6, this sequence is the greatest common divisor of a perfect number (A000396) and its arithmetic derivative (A003415). - Giorgio Balzarotti, Apr 21 2011
If n is in the sequence then n is a solution to the equation phi(sigma(x)) = 2x-2. It seems that there is no other solution to this equation. - Jahangeer Kholdi, Sep 09 2014
The sum of sums of elements of subsets of divisors of a(n), i.e. A229335(a(n)), is a perfect number (A000396). - Jaroslav Krizek, Nov 02 2017

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 147.

Vincenzo Librandi, Table of n, a(n) for n = 1..18
G. L. Cohen and H. J. J. te Riele, Iterating the sum-of-divisors function, Experimental Mathematics, 5 (1996), pp. 93-100.
Eric Weisstein's World of Mathematics, Superperfect Number
Index to divisibility sequences

Cf. A000043, A000384, A000396, A000668, A019279, A032742, A072868.

2^(Select[Range[512], PrimeQ[2^# - 1] &] - 1) (* Alonso del Arte, Apr 22 2011 *)
2^(MersennePrimeExponent[Range[15]]-1) (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 20 2021 *)

forprime(p=2,1e3,if(ispseudoprime(2^p-1),print1(2^(p-1)", "))) \\ Charles R Greathouse IV, Mar 14 2012

a(n) = 2^(A090748(n)). - Lekraj Beedassy, Dec 07 2007
a(n) = (1 + A000668(n))/2. - Omar E. Pol, Mar 11 2008
a(n) = 2^A000043(n)/2 = A072868(n)/2 = A032742(A072868(n)). - Omar E. Pol, Apr 25 2008

A237290 Sum of positive numbers k <= sigma(n) that are a sum of any subset of distinct divisors of n.

Original entry on oeis.org

1, 6, 8, 28, 12, 78, 16, 120, 52, 144, 24, 406, 28, 192, 192, 496, 36, 780, 40, 903, 256, 288, 48, 1830, 124, 336, 320, 1596, 60, 2628, 64, 2016, 384, 432, 384, 4186, 76, 480, 448, 4095, 84, 4656, 88, 2688, 2184, 576, 96, 7750, 228, 2976, 576, 3136, 108, 7260

Offset: 1

Jaroslav Krizek, Mar 02 2014

nonn

			For n = 5, a(5) = 1 + 5 + 6 = 12 (each of the numbers 1, 5 and 6 is the sum of a subset of distinct divisors of 5).
The numbers n = 14 and 15 is an interesting pair of consecutive numbers with identical value of sigma(n) such that simultaneously a(14) = a(15) and A237289(14) = A237289(15).
a(14) = 1+2+3+7+8+9+10+14+15+16+17+21+22+23+24 = a(15) = 1+3+4+5+6+8+9+15+16+18+19+20+21+23+24 = 192.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 200 terms from Vincenzo Librandi)
Jon Maiga, Computer-generated formulas for A237290, Sequence Machine.

Cf. A000203, A119348, A005153, A119347 (count of the same numbers), A184387, A229335, A237287, A237289.

isSumDist := proc(n,k)
    local dvs,s ;
    dvs := numtheory[divisors](n) ;
    for s in combinat[powerset](dvs) do
        add(m,m=op(s)) ;
        if % = k then
            return true;
        end if;
    end do:
    false ;
end proc:
A237290 := proc(n)
    local a;
    a := 0 ;
    for k from 1 to numtheory[sigma](n) do
        if isSumDist(n,k) then
            a := a+k;
        end if;
    end do:
end proc:
seq(A237290(n),n=1..20) ; # R. J. Mathar, Mar 13 2014

a[n_] := Plus @@ Union[Plus @@@ Subsets@ Divisors@ n]; Array[a, 54] (* Giovanni Resta, Mar 13 2014 *)

padbin(n, len) = {b = binary(n); while(length(b) < len, b = concat(0, b);); b;}
a(n) = {vks = []; d = divisors(n); nbd = #d; for (i=1, 2^nbd-1, b = padbin(i, nbd); onek = sum(j=1, nbd, d[j]*b[j]); vks = Set(concat(vks, onek));); sum(i=1, #vks, vks[i]);} \\ Michel Marcus, Mar 09 2014

A237290(n) = { my(c=[0]); fordiv(n,d, c = Set(concat(c,vector(#c,i,c[i]+d)))); vecsum(c); }; \\ after Chai Wah Wu's Python-code, Antti Karttunen, Nov 29 2024

from sympy import divisors
def A237290(n):
    ds = divisors(n)
    c, s = {0}, sum(ds)
    for d in ds:
        c |=  {a+d for a in c}
    return sum(a for a in c if 1<=a<=s) # Chai Wah Wu, Jul 05 2023

a(n) = A184387(n) - A237289(n).
a(p) = 2(p+2) for odd primes p.
a(n) = A184387(n) for practical numbers n (A005153), a(n) < A184387(n) for numbers n that are not practical (A237287).
a(n) = A000203(n) * (A119347(n)+1) / 2. [Found by Sequence Machine and easily seen to be true. Compare for example to the formulas of A229335.] - Antti Karttunen, Nov 29 2024

A229254 Numbers k such that k and k+2 have the same number (A000005) and sum of divisors (A000203).

Original entry on oeis.org

33, 54, 918, 1240, 3304, 4148, 4187, 7169, 12565, 15085, 19688, 24881, 25019, 26609, 38982, 51835, 53963, 59987, 76360, 77057, 96728, 143369, 150419, 167560, 170561, 205727, 215069, 220817, 278920, 418307, 564857, 731320, 785270, 907254, 910315, 986153

Offset: 1

Jaroslav Krizek, Sep 20 2013

nonn

Also numbers k such that A229335(k) = A229335(k+2).

Intersection of A007373 and A062832.

			Divisors of 54 = {1, 2, 3, 6, 9, 18, 27, 54}, divisors of 56 = {1, 2, 4, 7, 8, 14, 28, 56}, both have 8 divisors and sum = 120.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A007373, A062832, A000005, A000203, A229335.

Select[Range[10000], DivisorSigma[0, #] == DivisorSigma[0, # + 2] && DivisorSigma[1, #] == DivisorSigma[1, # + 2] &]

isok(n) = (numdiv(n) == numdiv(n+2)) && (sigma(n) == sigma(n+2)); \\ Michel Marcus, Sep 20 2013

More terms from Michel Marcus, Sep 20 2013

A229338 Product of products of elements of subsets of divisors of n.

Original entry on oeis.org

1, 4, 9, 4096, 25, 2821109907456, 49, 281474976710656, 531441, 10000000000000000, 121, 39939223824273992215667642551956428337968885602521915290518994217942463316460321327052965050967304175616, 169, 2177953337809371136, 6568408355712890625, 1461501637330902918203684832716283019655932542976

Offset: 1

Jaroslav Krizek, Sep 20 2013

nonn

			For n = 4; divisors of 4: {1, 2, 4}; subsets of divisors of n: {}, {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; product of products of elements of subsets = 1*1*2*4*2*4*8*8 = 4096.

Cf. A229335 (sum of sums of the elements), A229336 (product of sums of the elements), A229337 (sum of products of the elements).

Table[Times@@Times@@@Subsets[Divisors[n]],{n,20}] (* Harvey P. Dale, Mar 05 2015 *)

Conjecture: a(n) = n^s(n); where s(n) = A057711(tau(n)) = A057711(A000005(n)) = tau(n)*2^(tau(n)-2).

A237289 Sum of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

Original entry on oeis.org

0, 0, 2, 0, 9, 0, 20, 0, 39, 27, 54, 0, 77, 108, 108, 0, 135, 0, 170, 0, 272, 378, 252, 0, 372, 567, 500, 0, 405, 0, 464, 0, 792, 1053, 792, 0, 665, 1350, 1148, 0, 819, 0, 902, 882, 897, 2052, 1080, 0, 1425, 1395, 2052, 1715, 1377, 0, 2052, 0, 2600, 3375, 1710

Offset: 1

Jaroslav Krizek, Mar 02 2014

nonn

			For n = 5, a(5) = 2 + 3 + 4 = 9 (numbers 2, 3 and 4 are not a sum of any subset of distinct divisors of 5).
Numbers n = 14 and 15 are an interesting pair of consecutive numbers with identical value of sigma(n) such that simultaneously a(14) = a(15) and A237290(14) = A237290(15).
a(14) = 4+5+6+11+12+13+18+19+20 = a(15) = 2+7+10+11+12+13+14+17+22 = 108.
a(6) = 0 as 6 is practical; the sums into distinct divisors from 1 through 12 are 1 = 1, 2 = 2, 3 = 3, 4 = 1 + 3, 5 = 2 + 3, 6 = 1 + 2 + 3, 7 through 12 are (1 through 6) + 6. So none are not a sum distinct divisors of 6. - _David A. Corneth_, Jul 22 2025

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A119348, A005153, A184387, A237287, A237290.

isSumDist := proc(n,k)
    local dvs ;
    dvs := numtheory[divisors](n) ;
    for s in combinat[powerset](dvs) do
        add(m,m=op(s)) ;
        if % = k then
            return true;
        end if;
    end do:
    false ;
end proc:
A237289 := proc(n)
    local a;
    a := 0 ;
    for k from 1 to numtheory[sigma](n) do
        if not isSumDist(n,k) then
            a := a+k;
        end if;
    end do:
    a ;
end proc:
seq(A237289(n),n=1..20) ; # R. J. Mathar, Mar 13 2014

a[n_] := Block[{d = Divisors@n, s}, s = Plus @@ d; s*(s + 1)/2 - Plus @@ Union[Plus @@@ Subsets@d]]; m = Array[a, 59] (* Giovanni Resta, Mar 13 2014 *)

from sympy import divisors
def A237289(n):
    ds = divisors(n)
    c, s = {0}, sum(ds)
    for d in ds:
        c |=  {a+d for a in c}
    return (s*(s+1)>>1)-sum(a for a in c if 1<=a<=s) # Chai Wah Wu, Jul 05 2023

a(n) = A184387(n) - A237290(n).
a(p) = p(p - 1) / 2 - 1 for p = prime > 2.
a(n) = 0 for practical numbers (A005153), a(n) > 0 for numbers that are not practical (A237287).
a(n) = A184387(n) - A229335(n) for numbers n such that A119347(n) = A100587(n).

a(55) and a(57)-a(59) corrected by Giovanni Resta, Mar 13 2014

A229336 Product of sums of elements of nonempty subsets of divisors of n.

Original entry on oeis.org

1, 6, 12, 5040, 30, 77598259200, 56, 1307674368000, 168480, 12703122432000, 132, 52875224823823084892891318660312910903645116196873830400000000000000, 182, 440505199411200, 493242753024000, 8222838654177922817725562880000000, 306

Offset: 1

Jaroslav Krizek, Sep 20 2013

nonn

Number of nonempty subsets of divisors of n = A100587(n).

			For n = 2^2 = 4; divisors of 4: {1, 2, 4}; nonempty subsets of divisors of n: {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; product of sums of elements of subsets = 1*2*4*3*5*6*7 = 5040 = (2^3 - 1)! = 7!.

Cf. A229335 (sum of sums of elements of nonempty subsets of divisors of n),
A229337 (sum of products of elements of nonempty subsets of divisors of n),
A229338 (product of products of elements of nonempty subsets of divisors of n).

Table[Times@@(Total/@Rest[Subsets[Divisors[n]]]),{n,20}] (* Harvey P. Dale, Jan 22 2023 *)

a(2^n) = (2^(n+1) - 1)!.

A229337 Sum of products of elements of nonempty subsets of divisors of n.

Original entry on oeis.org

1, 5, 7, 29, 11, 167, 15, 269, 79, 395, 23, 10919, 27, 719, 767, 4589, 35, 31919, 39, 41579, 1407, 1655, 47, 2456999, 311, 2267, 2239, 104399, 59, 5499647, 63, 151469, 3263, 3779, 3455, 76767599, 75, 4679, 4479, 15343019, 83, 19071359, 87, 372599, 353279, 6767

Offset: 1

Jaroslav Krizek, Sep 20 2013

nonn

Number of nonempty subsets of divisors of n = A100587(n).

			For n = 2^2 = 4; divisors of 4: {1, 2, 4}; nonempty subsets of divisors of n: {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; sum of products of elements of subsets = 1 + 2 + 4 + 2 + 4 + 8 + 8 = 29 = (1+1) * (2+1) * (4+1) - 1.

Cf. A229335 (sum of sums of elements of nonempty subsets of divisors of n), A229336 (product of sums of elements of nonempty subsets of divisors of n), A229338 (product of products of elements of nonempty subsets of divisors of n).

Let a, b, c, ..., k be all divisors of n; a(n) = (a+1) * (b+1) * ... * (k+1) - 1.
a(p) = 2p+1, a(p^2) = 2(p+1)(p^2+1) - 1.
a(n) = A020696(n) - 1.

A276714 Numbers n such that n and n+3 have the same number and sum of divisors (A000005 and A000203).

Original entry on oeis.org

42677635, 276742235, 6439057062, 7512673242, 43592652562, 48847956255, 48880963215, 55018687182, 60184185702, 91484515395, 100774916235, 101926379835, 111886551315, 122388340095, 133012188855, 137978601142, 247631352255, 263171068875, 293467635615, 305946896255

Offset: 1

Jaroslav Krizek, Sep 16 2016

nonn

Intersection of A015861 and A276713.

Also numbers n such that A229335(n) = A229335(n+3).

			42677635 is in sequence because tau(42677635) = tau(42677638) = 32 and sigma(42677635) = sigma(42677638) = 68769792.

Martin Ehrenstein, Table of n, a(n) for n = 1..30

Cf. A015861, A229335, A276713, A276715.

Cf. Similar sequences with numbers n such that n and n+k have the same number and sum of divisors for k=1: A054004, for k=2: A229254.

[n: n in [A015861(k)] | NumberOfDivisors(n) eq  NumberOfDivisors(n+3) and SumOfDivisors(n) eq  SumOfDivisors(n+3)]

More terms from Martin Ehrenstein, Jul 12 2024

A291901 Numbers n such that the sum of sums of elements of subsets of divisors of n is a perfect number (A000396).

Original entry on oeis.org

2, 4, 13, 16, 64, 4096, 65536, 262144, 3145341, 932181397, 1073741824, 1152921504606846976, 309485009821345068724781056, 81129638414606681695789005144064, 85070591730234615865843651857942052864, 75603657215035519123837860069507929970384679

Offset: 1

Jaroslav Krizek, Nov 02 2017

nonn

Numbers n such that A229335(n) is in the sequence of perfect numbers, A000396.
Corresponding values of perfect numbers: 6, 28, 28, 496, 8128, 33550336, 8589869056, 137438691328, 33550336, ...
All even superperfect numbers A061652(n) are terms in this sequence.
Primes q of the form 2^(p-2) * (2^p - 1) - 1 where p is a Mersenne exponent (A000043) are terms: 2, 13, ...

			Divisors of 4: {1, 2, 4}; nonempty subsets of divisors of n: {1}, {2}, {4}, {1, 2}, {1, 4}, {2, 4}, {1, 2, 4}; sum of sums of elements of subsets = 1 + 2 + 4 + 3 + 5 + 6 + 7 = 28 (perfect number).
sigma(16) * 2^(tau(16) - 1) = 31 * 16 = 496 (perfect number).

Max Alekseyev, Table of n, a(n) for n = 1..71

Cf. A000043, A000396, A061652, A090748, A229335.

[n: n in [1..10^6]  | SumOfDivisors(SumOfDivisors(n)* (2^(NumberOfDivisors(n)-1))) eq 2*(SumOfDivisors(n)* (2^(NumberOfDivisors(n)-1)))];

isA000396 := proc(n)
    numtheory[sigma](n)=2*n ;
    simplify(%) ;
end proc:
for n from 1 do
    if isA000396(A229335(n)) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2017

Select[Range[2^20], DivisorSigma[1, DivisorSigma[1, #] 2^(DivisorSigma[0, #] - 1)] == 2 (DivisorSigma[1, #] 2^(DivisorSigma[0, #] - 1)) &] (* Michael De Vlieger, Nov 02 2017 *)

Terms a(10) onward added by Max Alekseyev, Sep 18 2024

cp's OEIS Frontend

A316225 Numbers k that divide the sum of sums of elements of all subsets of divisors of k (A229335).

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A061652 Even superperfect numbers: 2^(p-1) where 2^p-1 is a Mersenne prime (A000668).

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A237290 Sum of positive numbers k <= sigma(n) that are a sum of any subset of distinct divisors of n.

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A229254 Numbers k such that k and k+2 have the same number (A000005) and sum of divisors (A000203).

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A229338 Product of products of elements of subsets of divisors of n.

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A237289 Sum of positive numbers k <= sigma(n) that are not a sum of any subset of distinct divisors of n.

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A229336 Product of sums of elements of nonempty subsets of divisors of n.

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