A048146 -id:A048146 - cp's OEIS frontend

A064115 Numbers k such that k and k+1 have the same sum of non-unitary divisors (A048146), for A048146(k) > 0.

188, 1484, 3915, 14750, 19196, 20150, 79947, 164996, 190484, 219375, 253827, 639387, 718011, 835515, 1172374, 1380483, 2026323, 2064249, 3611708, 5507540, 6128108, 6374403, 6872984, 10073132, 10558250, 11360547, 12770450, 13000635, 14458364, 16366292, 19127907

Offset: 1

Jason Earls, Sep 09 2001

nonn

The sequence snud(a(n)) = snud(1 + a(n)) is A103846(n). - Emeric Deutsch, Feb 17 2005

			snud(1484) = 864, snud(1485) = 864.

Giovanni Resta, Table of n, a(n) for n = 1..344 (terms < 10^11, first 30 terms from Harry J. Smith)

Cf. A048146, A103846.

nusigma[1]=0; nusigma[n_] := DivisorSigma[1, n] - Times @@ (1 + Power @@@ FactorInteger[n]); seq={}; s1=0; Do[s2=nusigma[n]; If[s1>0 && s2==s1, AppendTo[seq, n-1]]; s1=s2, {n, 1, 10^6}]; seq (* Amiram Eldar, Jun 10 2019 *)

snud(n)= { sumdiv(n, d, if(gcd(d, n/d)!=1, d)) }
{ n=0; for (m=1, 10^9, s=snud(m); if (s>0 && s==snud(m + 1), write("b064115.txt", n++, " ", m); if (n==30, break)) ) } \\ Harry J. Smith, Sep 07 2009

More terms from Emeric Deutsch, Feb 17 2005

A325813 a(n) = gcd(A034448(n)-n, n-A048146(n)), where A034448 and A048146 are respectively the sum of unitary and non-unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 12, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 4, 1, 7, 3, 6, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 3, 1, 2, 1, 12, 1, 6, 1, 2, 1, 2, 1, 4, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 1, 21, 1, 1, 6, 1, 2, 3

Offset: 1

Antti Karttunen, May 23 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A034460, A323166, A325812, A325814.

Cf. also A325385.

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));

a(n) = gcd(A034460(n), A325814(n)).

A325814 a(n) = n - A048146(n), where A048146 is the sum of non-unitary divisors of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 6, 10, 11, 4, 13, 14, 15, 2, 17, 9, 19, 8, 21, 22, 23, 0, 20, 26, 15, 12, 29, 30, 31, 2, 33, 34, 35, -5, 37, 38, 39, 4, 41, 42, 43, 20, 27, 46, 47, -8, 42, 35, 51, 24, 53, 18, 55, 8, 57, 58, 59, 12, 61, 62, 39, 2, 65, 66, 67, 32, 69, 70, 71, -33, 73, 74, 55, 36, 77, 78, 79, -4, 42, 82, 83, 20

Offset: 1

Antti Karttunen, May 23 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A033879, A034448, A034460, A048146, A064591 (positions of zeros), A228058, A325812, A325813, A325824.
Cf. also A325314.
Cf. A002117, A013661.

Table[n - DivisorSum[n, # &, ! CoprimeQ[#, n/#] &], {n, 84}] (* Michael De Vlieger, May 27 2019 *)

A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));

a(n) = n - A048146(n).
a(n) = A033879(n) + A034460(n).
a(A228058(n)) = A325824(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1/2 - zeta(2) * (1 - 1/zeta(3)) / 2 = 0.3617493553... . - Amiram Eldar, Feb 22 2024

A325974 Arithmetic mean of {sum of non-unitary divisors} and {sum of nonsquarefree divisors}: a(n) = (1/2)*(A048146(n) + A162296(n)).

Original entry on oeis.org

0, 0, 0, 3, 0, 0, 0, 9, 6, 0, 0, 12, 0, 0, 0, 21, 0, 18, 0, 18, 0, 0, 0, 36, 15, 0, 24, 24, 0, 0, 0, 45, 0, 0, 0, 60, 0, 0, 0, 54, 0, 0, 0, 36, 36, 0, 0, 84, 28, 45, 0, 42, 0, 72, 0, 72, 0, 0, 0, 72, 0, 0, 48, 93, 0, 0, 0, 54, 0, 0, 0, 144, 0, 0, 60, 60, 0, 0, 0, 126, 78, 0, 0, 96, 0, 0, 0, 108, 0, 108, 0, 72, 0, 0, 0, 180, 0, 84, 72

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

			For n = 36, its divisors are 1, 2, 3, 4, 6, 9, 12, 18, 36. Of these, non-unitary divisors are 2, 3, 6, 12 and 18 so A048146(36) = 2+3+6+12+18 = 41, while the nonsquarefree divisors are 4, 9, 12, 18 and 36, so A162296(36) = 4+9+12+18+36 = 79, thus a(36) = (41+79)/2 = 60.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A005117, A034448, A048146, A162296, A325973, A325975, A325978.

Cf. A002117, A013661.

A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A048146(n) = (sigma(n)-A034448(n));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325974(n) = ((A048146(n)+A162296(n))/2);

a(n) = (1/2)*(A048146(n) + A162296(n)).
a(n) = A000203(n) - A325973(n).
a(n) = n - A325978(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2)*(1/2 - 1/(4*zeta(3))) - 1/4 = 0.2303588390... . - Amiram Eldar, Feb 22 2024

A329882 Nonunitary superabundant numbers: numbers m such that nusigma(m)/m > nusigma(k)/k for all k < m, where nusigma(m) is the sum of nonunitary divisors of m (A048146).

Original entry on oeis.org

1, 4, 8, 16, 24, 36, 48, 72, 144, 288, 360, 432, 720, 1440, 1800, 2160, 3600, 7200, 10800, 15120, 21600, 25200, 50400, 75600, 151200, 302400, 453600, 529200, 831600, 1058400, 1663200, 2116800, 3175200, 3326400, 4989600, 5821200, 9979200, 11642400, 21621600

Offset: 1

Amiram Eldar, Nov 23 2019

nonn

The nonunitary version of A004394.

Cf. A034448, A048146, A064597, A309141.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); nusigma[n_] := DivisorSigma[1, n] - usigma[n]; rm = -1; s = {}; Do[r = nusigma[n]/n; If[r > rm, rm = r; AppendTo[s, n]], {n, 1, 10000}]; s

A329883 Nonunitary highly abundant numbers: numbers m such that nusigma(m) > nusigma(k) for all k < m, where s(n) is the sum of nonunitary divisors of n (A048146).

Original entry on oeis.org

1, 4, 8, 12, 16, 24, 32, 36, 48, 64, 72, 96, 108, 120, 144, 180, 192, 216, 288, 360, 432, 504, 576, 648, 720, 864, 1008, 1080, 1296, 1440, 1728, 1800, 2016, 2160, 2520, 2880, 3024, 3240, 3456, 3528, 3600, 4320, 5040, 5400, 5760, 6048, 6480, 7056, 7200, 8640

Offset: 1

Amiram Eldar, Nov 23 2019

nonn

The corresponding record values are 0, 2, 6, 8, 14, 24, 30, 41, 56, 62, 105, 120, 140, 144, 233, 246, 248, 348, 489, 630, 764, 840, ...

Amiram Eldar, Table of n, a(n) for n = 1..500

The nonunitary version of A002093.

Cf. A048146, A064597, A309141, A329882.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); nusigma[n_] := DivisorSigma[1, n] - usigma[n]; num = -1; s = {}; Do[nu = nusigma[n]; If[nu > num, num = nu; AppendTo[s, n]], {n, 1, 10^4}]; 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)));

A329879 Numbers k such that k and nusigma(k) have the same set of prime divisors, where nusigma(k) is the sum of nonunitary divisors of k (A048146).

Original entry on oeis.org

4, 9, 24, 25, 49, 54, 112, 121, 150, 169, 289, 294, 361, 480, 529, 726, 750, 841, 961, 1014, 1369, 1681, 1734, 1849, 1984, 2058, 2166, 2209, 2430, 2520, 2688, 2809, 3174, 3481, 3721, 3780, 4489, 5041, 5046, 5329, 5760, 5766, 6241, 6889, 7921, 7986, 8214, 8700

Offset: 1

Amiram Eldar, Nov 23 2019

nonn

Numbers k such that rad(nusigma(k)) = rad(k), where rad(k) is the squarefree kernel of k (A007947).

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

Cf. A007947, A027598, A048146, A329858.

rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); nusigma[n_] := DivisorSigma[1, n] - usigma[n]; Select[Range[10^4], rad[nusigma[#]] == rad[#] &]

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[#]] == # &]

A063885 z(sigma(n)) = 2n, where z(n) = A048146.

Original entry on oeis.org

24, 1536, 1631, 47360, 82458

Offset: 1

Jason Earls, Aug 28 2001

Cf. A000203, A034448, A048146, A019284.

u(n) = sumdiv(n,d, if(gcd(d,n/d)==1,d));
z(n) = sigma(n)-u(n);
for(n=1,10^7, if(z(sigma(n))==2*n,print1(n, ", ")))

cp's OEIS Frontend

A064115 Numbers k such that k and k+1 have the same sum of non-unitary divisors (A048146), for A048146(k) > 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A325813 a(n) = gcd(A034448(n)-n, n-A048146(n)), where A034448 and A048146 are respectively the sum of unitary and non-unitary divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A325814 a(n) = n - A048146(n), where A048146 is the sum of non-unitary divisors of n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A325974 Arithmetic mean of {sum of non-unitary divisors} and {sum of nonsquarefree divisors}: a(n) = (1/2)*(A048146(n) + A162296(n)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A329882 Nonunitary superabundant numbers: numbers m such that nusigma(m)/m > nusigma(k)/k for all k < m, where nusigma(m) is the sum of nonunitary divisors of m (A048146).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A329883 Nonunitary highly abundant numbers: numbers m such that nusigma(m) > nusigma(k) for all k < m, where s(n) is the sum of nonunitary divisors of n (A048146).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A329879 Numbers k such that k and nusigma(k) have the same set of prime divisors, where nusigma(k) is the sum of nonunitary divisors of k (A048146).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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).

Views

Author

Keywords