A181595 -id:A181595 - cp's OEIS frontend

A303357 Unitary near-perfect numbers: unitary abundant numbers n such that usigma(n) - 2n is a unitary divisor of n, where usigma(n) is the sum of unitary divisors of n (A034448).

295680, 13278720, 363095040, 454755840, 675333120, 694256640, 845053440, 1038428160, 2274455040, 2357921280, 3099048960, 5021076480, 6114339840, 9643096320, 9817328640, 14495416320, 17121377280, 23787294720, 30583418880, 36277463040, 45129477120, 114499338240, 211380879360

Offset: 1

Amiram Eldar, Apr 22 2018

nonn

The unitary version of A181595.

All the terms up to a(23) are divisible by 2^8 * 3 * 5. - Giovanni Resta, Apr 26 2018

			295680 is in the sequence since usigma(295680) - 2*295680 = 592128 - 591360 = 768 and 768 is a unitary divisor of 295680.

Cf. A034448, A181595, A303356.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; aQ[n_] :=
Module[{d}, d = usigma[n] - 2 n; If[d <= 0, False, Divisible[n, d] && GCD[d, n/d] == 1]]; n = 1; seq={}; Do[ If[aQ[n], AppendTo[seq,n]]; n++, {k, 1, 300000}]; seq

a(9)-a(23) from Giovanni Resta, Apr 26 2018

A362969 Nonunitary near-perfect numbers: k such that nusigma(k) = k + d where d is a nonunitary divisor of k.

Original entry on oeis.org

48, 80, 96, 160, 224, 352, 416, 480, 896, 1472, 1476, 1856, 2688, 3968, 6016, 7552, 7808, 8550, 8700, 10332, 17010, 20300, 22496, 36448, 44384, 54944, 63488, 65024, 71264, 73710, 97300, 97792, 114176, 122368, 128512, 310976, 392192, 490496, 515072, 521216, 549990

Offset: 1

Jenaro Tomaszewski, May 10 2023

nonn

The nonunitary version of near-perfect numbers (A181595).

			For k = 352, nusigma(352) = 360. 360 - 352 = 8, which is a nonunitary divisor of 352.

Amiram Eldar, Table of n, a(n) for n = 1..250

Cf. A048146 (nusigma), A181595.

q[n_] := Module[{d = Select[Divisors[n], ! CoprimeQ[#, n/#] &], s}, s = Total[d]; AnyTrue[d, n + # == s &]]; Select[Range[10^4], q] (* Amiram Eldar, May 11 2023 *)

nusigma(n) = {my(f = factor(n)); sigma(f) - prod(i = 1, #f~, f[i, 1]^f[i, 2] + 1);}
is(n) = {my(d = nusigma(n) - n); d > 0 && !(n%d) && gcd(d, n/d) > 1; } \\ Amiram Eldar, May 20 2023

A383148 k-facile numbers: Numbers m such that the sum of the divisors of m is equal to 2*m+s where s is a product of distinct divisors of m.

Original entry on oeis.org

12, 18, 20, 24, 30, 40, 42, 54, 56, 60, 66, 78, 84, 88, 90, 102, 104, 114, 120, 132, 138, 140, 168, 174, 186, 196, 204, 222, 224, 234, 246, 252, 258, 264, 270, 280, 282, 308, 312, 318, 348, 354, 360, 364, 366, 368, 380, 402, 414, 420, 426, 438, 440, 456, 464, 468, 474, 476

Offset: 1

Joshua Zelinsky, Apr 17 2025

nonn

Subsequence of A005101 but seem to be much rarer.

			The sum of the divisors of 60 is 168, and 168 = 2*60 + 48, and 48 = 4*12 and 4 and 12 are divisors of 60, so 60 is in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
S. Flora Jeba, Anirban Roy, and Manjil P. Saikia, On k-Facile Perfect Numbers, Algebra and Its Applications (ICAA-2023), Springer Proc. Math. Stat., Vol. 474 (2025), 111-121.

Subsequence of A005101.

Cf. A000203, A000396, A181595.

q[m_] := Module[{d = Divisors[m], ab}, ab = Total[d] - 2*m; ab > 0 && AnyTrue[Subsets[d], Times @@ # == ab &]]; Select[Range[500], q] (* Amiram Eldar, Apr 18 2025 *)

prodDistinctDiv(n, k, f=factor(n))=my(D=divisors([n,f])); helper(D[2..#D], k)
helper(v,k)=if(k==1, return(1)); v=select(d->k%d==0, v); if(#v<3, if(#v==2, return(v[2]==k || vecprod(v)==k)); return(#v && v[1]==k)); my(u=v[1..#v-1]); helper(u,k) || helper(u,k/v[#v])
is(n,f=factor(n))=my(t=sigma([n,f])-2*n); t>1 && prodDistinctDiv(n, t, f) \\ Charles R Greathouse IV, Apr 24 2025

def facile_candidates(n):
    from itertools import combinations
    divs = divisors(n)
    sigma_n = sigma(n, 1)
    candidates = set()
    # Generate all products of distinct combinations of divisors
    for r in range(2, len(divs)+1):  # start from 2-element products to avoid m=n
        for combo in combinations(divs, r):
            product = prod(combo)
            if product < sigma_n:
                candidates.add(product)
    return sorted(candidates)
def find_facile_perfects(x):
    result = []
    for n in range(1, x+1):
        sig = sigma(n, 1)
        if sig < 2*n:
            continue
        candidates = facile_candidates(n)
        for m in candidates:
            if sig == 2*n + m:
                print(n,m)
                result.append(n)
                break
    return result

A181707 Numbers of the form m=2^(t-1)*(2^t-33), where 2^t-33 is prime.

Original entry on oeis.org

992, 28544, 122624, 507392, 34355412992, 8796023816192, 140737211531264, 144115179217485824, 9671406556844465630216192, 162259276829213066154002603835392, 11417981541647679048463794346093005918389141504

Offset: 1

Vladimir Shevelev, Nov 06 2010

nonn

Generated by t= 6, 8, 9, 10, 18, 22, 24, 29, 42, 54, 77, 90, 102, 137,...

A subsequence of A181595 because the abundance of m is 32, and 32 divides 2^(t-1) and therefore divides m.

Cf. A181595, A181701, A000396, A181703, A181704, A181705, A181706, A175989.

Definition simplified and more terms added by R. J. Mathar, Nov 18 2010

A181711 Numbers of the form m(2^k-1), where m = 2^(k-1)(2^k-1) is a perfect number (A000396).

Original entry on oeis.org

18, 196, 15376, 1032256, 274810802176, 1125882727038976, 72057319160283136, 4951760152529835082242850816, 6129982163463555428116476125461573244012649752219877376

Offset: 1

Vladimir Shevelev, Nov 07 2010

nonn

The associated exponents k are in A000043: 2, 3, 5, 7, 13, 17, 19 ,31, 61, ...

One can prove that, if m = 2^(k-1)*(2^k-1) is a perfect number, then m*2^k and m*(2^k-1) are both in A181595. Thus every even term in A000396 is a difference of two terms in A181595.

			With k=3, m = 2^(k-1)*(2^k - 1) = 2^2*(8 - 1) = 28 is a perfect number (A000396), so m*(2^k - 1) = 28*7 = 196 is in the sequence. - _Michael B. Porter_, Jul 19 2016

Cf. A000043, A000396, A181595, A181596, A181701, A181710.

If odd perfect numbers do not exist, then a(n) = A181710(n) - A000396(n).

a(n) = A019279(n)*(A000668(n))^2 if there are no odd superperfect numbers. - César Aguilera, Jun 13 2017

Definition condensed by R. J. Mathar, Dec 05 2010

A289056 Near 3-perfect numbers of the form 2^ap^tq, where a >= 1, t = 1 or 2, p < q are both primes.

Original entry on oeis.org

180, 240, 360, 1344, 1872, 2688, 3744, 5376, 6048, 6496

Offset: 1

Vladimir Shevelev, Jun 23 2017

A positive number m is called a near perfect number if the sum of its divisors (A000203) is 3*m+d, where d is a proper divisor of m. Recently [Das and Saikia] proved that there exist only 10 such numbers with the restriction in the name.

			For m=240, d=24, A000203(m) = 744 = 3*240 + d. So 240 is a member.

B. Das and H. K. Saikia, On near 3-perfect numbers, Sohag Journal of Math., Vol. 4, No. 1 (2017), 1-5.
Paul Pollack and Vladimir Shevelev, On perfect and near-perfect numbers, J. Number Theory 132 (2012), pp. 3037-3046.

Cf. A000203, A181595.

cp's OEIS Frontend

A303357 Unitary near-perfect numbers: unitary abundant numbers n such that usigma(n) - 2n is a unitary divisor of n, where usigma(n) is the sum of unitary divisors of n (A034448).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A362969 Nonunitary near-perfect numbers: k such that nusigma(k) = k + d where d is a nonunitary divisor of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A383148 k-facile numbers: Numbers m such that the sum of the divisors of m is equal to 2*m+s where s is a product of distinct divisors of m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

A181707 Numbers of the form m=2^(t-1)*(2^t-33), where 2^t-33 is prime.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A181711 Numbers of the form m*(2^k-1), where m = 2^(k-1)*(2^k-1) is a perfect number (A000396).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A289056 Near 3-perfect numbers of the form 2^a*p^t*q, where a >= 1, t = 1 or 2, p < q are both primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A181711 Numbers of the form m(2^k-1), where m = 2^(k-1)(2^k-1) is a perfect number (A000396).

A289056 Near 3-perfect numbers of the form 2^ap^tq, where a >= 1, t = 1 or 2, p < q are both primes.