A064591 -id:A064591 - cp's OEIS frontend

A064598 Nonunitary deficient numbers: the sum of the nonunitary divisors of n is less than n; i.e., sigma(n) - usigma(n) < n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76

Offset: 1

Dean Hickerson, Sep 25 2001

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A048146, A064591, A064597.

nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); For[ n=1, True, n++, If[ nusigma[ n ]

usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) } { n=0; for (m=1, 10^9, if (sigma(m) - usigma(m) < m, write("b064598.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 19 2009

A327633 Noninfinitary perfect numbers: numbers k whose sum of noninfinitary divisors equals k.

Original entry on oeis.org

112, 1344, 32512, 390144, 483840, 5930176, 2952609792

Offset: 1

Amiram Eldar, Sep 20 2019

Numbers k such that sigma(k) - isigma(k) = A000203(k) - A049417(k) = k.

No more terms below 3 * 10^10.

			112 is in the sequence since its noninfinitary divisors are {2, 4, 8, 14, 28, 56} whose sum is 112.

Cf. A000203, A007357, A049417, A064591, A077609, A282900, A282940.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); nisigma[1] = 0; nisigma[n] := DivisorSigma[1, n] - Times @@ (Flatten @ (f @@@ FactorInteger[n]) + 1); Select[Range[500000], nisigma[#] == # &]

A327945 Nonunitary pseudoperfect numbers: numbers that are equal to the sum of a subset of their nonunitary divisors.

Original entry on oeis.org

24, 36, 48, 72, 80, 96, 108, 112, 120, 144, 160, 168, 180, 192, 200, 216, 224, 240, 252, 264, 288, 300, 312, 320, 324, 336, 352, 360, 384, 392, 396, 400, 408, 416, 432, 448, 456, 468, 480, 504, 528, 540, 552, 560, 576, 588, 600, 612, 624, 640, 648, 672, 684

Offset: 1

Amiram Eldar, Sep 30 2019

nonn

The nonunitary version of A005835.

			36 is in the sequence since its nonunitary divisors are 2, 3, 6, 12, 18 and 36 = 6 + 12 + 18.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Supersequence of A064591.

Cf. A005835, A064597, A293188, A292985, A306983.

nudiv[n_] := Module[{d = Divisors[n]}, Select[d, GCD[#, n/#] > 1 &]]; s = {}; Do[d = nudiv[n]; If[Total[d] < n, Continue[]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c > 0, AppendTo[s, n]], {n, 1, 700}]; s

A325812 Numbers k such that gcd(A034448(k)-k, k-A048146(k)) is equal to abs(k-A048146(k)).

Original entry on oeis.org

1, 6, 12, 28, 56, 60, 108, 120, 132, 168, 264, 280, 312, 408, 420, 440, 456, 496, 528, 540, 552, 696, 700, 728, 744, 756, 760, 888, 984, 992, 1032, 1128, 1140, 1188, 1272, 1404, 1416, 1456, 1464, 1608, 1704, 1710, 1752, 1836, 1896, 1992, 2052, 2136, 2328, 2424, 2472, 2484, 2568, 2616, 2646, 2712, 3048, 3132, 3144, 3288, 3336, 3344

Offset: 1

Antti Karttunen, May 23 2019

nonn

Numbers k for which A325813(k) is equal to abs(A325814(k)).
Numbers k such that A325814(k) is not zero (not in A064591) and divides A034460(k).
Conjecture: after the initial one all other terms are even. If this holds then there are no odd perfect numbers.

Antti Karttunen, Table of n, a(n) for n = 1..25000
Index entries for sequences where any odd perfect numbers must occur

Cf. A034448, A034460, A048146, A064591, A325813, A325814, A325822.

Cf. A000396 (a subsequence).

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325813(n) = gcd(A034460(n), A325814(n));
isA325812(n) = (A325813(n)==abs(A325814(n)));
\\ Alternatively:
isA325812(n) = (A325814(n) && !(A034460(n)%A325814(n)));

A330824 Numbers of the form 2^(2*p), where p is a Mersenne exponent, A000043.

Original entry on oeis.org

16, 64, 1024, 16384, 67108864, 17179869184, 274877906944, 4611686018427387904, 5316911983139663491615228241121378304

Offset: 1

Walter Kehowski, Jan 06 2020

nonn

Also the second element of the power-spectral basis of A064591. The first element of the power-spectral basis of A064591 is A133049.

			a(1) = 2^(2*2) = 16. Also A133049(1) = 3^2 = 9, and the spectral basis of A064591(1) = 24 is {9, 16}, consisting of primes and powers.

Amiram Eldar, Table of n, a(n) for n = 1..15

Cf. A000043, A000668, A064591, A132794, A133049.

a := proc(n) if isprime(2^n-1) then return 2^(2*n) fi; end;
[seq(a(n),n=1..31)]; # ithprime(31) = 127

2^(2*MersennePrimeExponent[Range[10]]) (* Harvey P. Dale, Jun 27 2023 *)

forprime(p=1,99,isprime(2^p-1)&&print1(4^p",")) \\ or better: {A330824(n)=4^A000043(n)}. - M. F. Hasler, Feb 07 2020

a(n) = 2^(2*A000043(n)) = 4^A000043(n).

A064599 The sum of the nonunitary divisors of n is a divisor of n; i.e., sigma(n) - usigma(n) divides n.

Original entry on oeis.org

4, 9, 18, 24, 25, 49, 112, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 1984, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569

Offset: 1

Dean Hickerson, Sep 25 2001

The sequence consists of the nonunitary perfect numbers (A064591), squares of primes (A001248) and 18.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A048146, A064591.

nusigma[ n_ ] := DivisorSigma[ 1, n ]-Times@@(1+Power@@#&/@FactorInteger[ n ]); For[ n=1, True, n++, v=nusigma[ n ]; If[ v>0&&Mod[ n, v ]==0, Print[ n ] ] ]

usigma(n)= { local(f,s=1); f=factor(n); for(i=1, matsize(f)[1], s*=1 + f[i, 1]^f[i, 2]); return(s) }
{ n=0; for (m=1, 10^9, v=sigma(m) - usigma(m); if (v>0 && m%v == 0, write("b064599.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 19 2009

A333927 Recursive perfect numbers: numbers k such that A333926(k) = 2*k.

Original entry on oeis.org

6, 28, 264, 1104, 3360, 75840, 151062912, 606557952, 2171581440

Offset: 1

Amiram Eldar, Apr 10 2020

Since a recursive divisor is also a (1+e)-divisor (see A049599), then the first 6 terms and other terms of this sequence coincide with those of A049603.

			264 is a term since the sum of its recursive divisors is 1 + 2 + 3 + 6 + 8 + 11 + 22 + 24 + 33 + 66 + 88 + 264 = 528 = 2 * 264.

Cf. A049603, A282446, A333926.

Analogous sequences: A000396, A002827 (unitary), A007357 (infinitary), A054979 (exponential), A064591 (nonunitary).

recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[10^5], recDivSum[#] == 2*# &]

A329884 Nonunitary superperfect numbers: numbers k such that nusigma(nusigma(k)) = k, where nusigma(k) = sigma(k) - usigma(k) is the sum of nonunitary divisors of k (A048146).

Original entry on oeis.org

24, 48, 56, 112, 192, 248, 252, 328, 448, 496, 768, 1016, 1792, 1984, 2032, 3240, 6462, 7936, 8128, 11616, 11808, 17412, 20538, 32512, 49152, 65528, 114688, 131056, 507904, 524224, 786432, 1048568, 1835008, 2080768, 2096896, 2097136, 3145728, 4194296, 7340032

Offset: 1

Amiram Eldar, Nov 23 2019

nonn

Analogous to superperfect numbers (A019279) as nonunitary perfect numbers (A064591) is analogous to perfect numbers (A000396).

Amiram Eldar, Table of n, a(n) for n = 1..53 (terms below 10^10)

Cf. A000396, A019279, A034448, A038843, A048146, A064591, A329881.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); nusigma[n_] := DivisorSigma[1, n] - usigma[n]; Select[Range[10^6], nusigma[nusigma[#]] == # &]

A335141 Numbers that are both unitary pseudoperfect (A293188) and nonunitary pseudoperfect (A327945).

Original entry on oeis.org

840, 2940, 7260, 9240, 10140, 10920, 13860, 14280, 15960, 16380, 17160, 18480, 19320, 20580, 21420, 21840, 22440, 23100, 23940, 24024, 24360, 25080, 26040, 26520, 27300, 28560, 29640, 30360, 30870, 31080, 31920, 32340, 34440, 34650, 35700, 35880, 36120, 36960

Offset: 1

Amiram Eldar, May 25 2020

nonn

All the terms are nonsquarefree (since squarefree numbers do not have nonunitary divisors).

All the terms are either 3-abundant numbers (A068403) or 3-perfect numbers (A005820). None of the 6 known 3-perfect numbers are terms of this sequence. If there is a term that is 3-perfect, it is also a unitary perfect (A002827) and a nonunitary perfect (A064591).

			840 is a term since its aliquot unitary divisors are {1, 3, 5, 7, 8, 15, 21, 24, 35, 40, 56, 105, 120, 168, 280} and 1 + 5 + 7 + 8 + 15 + 35 + 40 + 56 + 105 + 120 + 168 + 280 = 840, and its nonunitary divisors are {2, 4, 6, 10, 12, 14, 20, 28, 30, 42, 60, 70, 84, 140, 210, 420} and 70 + 140 + 210 + 420 = 840.

Amiram Eldar, Table of n, a(n) for n = 1..400

Intersection of A293188 and A327945.
Subsequence of A335140.
Cf. A002827, A005820, A064591, A068403.

pspQ[n_] := Module[{d = Divisors[n], ud, nd, x}, ud = Select[d, CoprimeQ[#, n/#] &]; nd = Complement[d, ud]; ud = Most[ud]; Plus @@ ud >= n && Plus @@ nd >= n && SeriesCoefficient[Series[Product[1 + x^ud[[i]], {i, Length[ud]}], {x, 0, n}], n] > 0 && SeriesCoefficient[Series[Product[1 + x^nd[[i]], {i, Length[nd]}], {x, 0, n}], n] > 0]; Select[Range[10^4], pspQ]

A063870 Numbers n such that sigma(n) - usigma(n) = 3n/2.

Original entry on oeis.org

480, 2688, 2095104, 16854816, 41055200, 1839272960, 5905219584, 204004720640

Offset: 1

Jason Earls, Aug 27 2001

nonn

Cf. A034448, A000203, A048146, A063846, A064591.

u(n) = sumdiv(n,d, if(gcd(d,n/d)==1,d)); for(n=1,10000, if(sigma(n)-u(n)==3*n/2,print(n)))

More terms from Dean Hickerson, Sep 25 2001

There are no others less than 1.5*10^13, but here's a larger one: 948990933336933380096. - Dean Hickerson, Sep 25 2001

cp's OEIS Frontend

A064598 Nonunitary deficient numbers: the sum of the nonunitary divisors of n is less than n; i.e., sigma(n) - usigma(n) < n.

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A327633 Noninfinitary perfect numbers: numbers k whose sum of noninfinitary divisors equals k.

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A327945 Nonunitary pseudoperfect numbers: numbers that are equal to the sum of a subset of their nonunitary divisors.

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A325812 Numbers k such that gcd(A034448(k)-k, k-A048146(k)) is equal to abs(k-A048146(k)).

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A330824 Numbers of the form 2^(2*p), where p is a Mersenne exponent, A000043.

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A064599 The sum of the nonunitary divisors of n is a divisor of n; i.e., sigma(n) - usigma(n) divides n.

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A333927 Recursive perfect numbers: numbers k such that A333926(k) = 2*k.

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A329884 Nonunitary superperfect numbers: numbers k such that nusigma(nusigma(k)) = k, where nusigma(k) = sigma(k) - usigma(k) is the sum of nonunitary divisors of k (A048146).

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A335141 Numbers that are both unitary pseudoperfect (A293188) and nonunitary pseudoperfect (A327945).