A211225 -id:A211225 - cp's OEIS frontend

A211223 Numbers k for which sigma(k) = sigma(x) + sigma(y), where k = x + y.

3, 8, 9, 10, 15, 20, 21, 30, 32, 33, 39, 40, 49, 51, 55, 56, 57, 62, 63, 69, 70, 75, 85, 87, 88, 90, 92, 93, 94, 96, 99, 104, 105, 108, 110, 111, 114, 116, 117, 123, 125, 126, 128, 129, 130, 134, 135, 136, 140, 141, 145, 147, 150, 152, 153, 155, 158, 159, 160

Offset: 1

Paolo P. Lava, Apr 27 2012

A211225(a(n)) > 0. - Reinhard Zumkeller, Jan 06 2013

			sigma(49) = sigma(8) + sigma(41) that is 57 = 15 + 42.
sigma(93) = sigma(31) + sigma(62) that is 128 = 32 + 96.
In more than one way: sigma(117) = sigma(41) + sigma(76) = sigma(52) + sigma(65) = sigma(56) + sigma(61) that is 182 = 42 + 140 = 98 + 84 = 120 + 62.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A083207, A204830, A204831, A211224, A211225.
Cf. A218852.
Cf. A000203.

a211223 n = a211223_list !! (n-1)
a211223_list = map (+ 1) $ findIndices (> 0) a211225_list
-- Reinhard Zumkeller, Jan 06 2013

with(numtheory);
A211223:=proc(q)
local i,n;
for n from 1 to q do
  for i from 1 to trunc(n/2) do
    if sigma(i)+sigma(n-i)=sigma(n) then print(n); break; fi;
od; od; end:
A211223(10000);

sigmaPartitionQ[n_] := With[{s = DivisorSigma[1, n], ip = IntegerPartitions[ n, {2}]}, MemberQ[ip, {x_, y_} /; s == DivisorSigma[ 1, x] + DivisorSigma[ 1, y]]]; Select[Range[160], sigmaPartitionQ] (* Jean-François Alcover, Aug 19 2013 *)

is(n)=my(t=sigma(n));for(i=1,n\2,if(sigma(i)+sigma(n-i)==t, return(1))) \\ Charles R Greathouse IV, May 04 2012

A211224 Least k with precisely n partitions k = x + y satisfying sigma(k) = sigma(x) + sigma(y).

Original entry on oeis.org

3, 32, 117, 183, 393, 728, 933, 2193, 2528, 1173, 6136, 2990, 4070, 8211, 11488, 12616, 6112, 22287, 20584, 37468, 38675, 35245, 41416, 55825, 43616, 66385, 56810, 94040, 88736, 93975, 90068, 174515, 169376, 146965, 139196, 210453, 140576, 177248

Offset: 1

Paolo P. Lava, May 04 2012

nonn

Subset of A211223.

			a(7)=933; 933 has 7 partitions of two numbers, x and y, for which sigma(933) = sigma(x) + sigma(y) = 1248. In fact:
sigma(311) + sigma(622) = 312 + 936 = 1248;
sigma(322) + sigma(611) = 576 + 672 = 1248;
sigma(370) + sigma(563) = 684 + 564 = 1248;
sigma(391) + sigma(542) = 432 + 816 = 1248;
sigma(398) + sigma(535) = 600 + 648 = 1248;
sigma(407) + sigma(526) = 456 + 792 = 1248;
sigma(442) + sigma(491) = 756 + 492 = 1248;
Furthermore 933 is the minimum number to have this property.

Donovan Johnson, Table of n, a(n) for n = 1..100

Cf. A083207, A204830, A204831, A211223, A211225.

with(numtheory);
A211224:=proc(q)
local a,b,i,j,n,v;
v:=array(1..10000); for n from 1 to 10000 do v[n]:=0; od;
a:=0;
for n from 1 to q do
  b:=0;
  for i from 1 to trunc(n/2) do
    if sigma(i)+sigma(n-i)=sigma(n) then b:=b+1; fi; od;
  if b=a+1 then a:=b; print(n); j:=1;
     while v[b+j]>0 do a:=b+j; print(v[b+j]); j:=j+1; od;
  else if b>a+1 then if v[b]=0 then v[b]:=n; fi; fi; fi;
od; end:
A211224(1000);

ct(n)=my(t=sigma(n));sum(i=1,n\2,sigma(i)+sigma(n-i)==t)
v=vector(100);for(n=1,1e5,t=ct(n);if(t&&t<=#v&&!v[t],v[t]=n));v
\\ Charles R Greathouse IV, May 04 2012

A212662 Numbers k for which k' = x' + y', where x > 0, k = x + y, and k', x', y' are the arithmetic derivatives of k, x, y.

Original entry on oeis.org

3, 6, 9, 12, 15, 18, 21, 24, 25, 27, 30, 33, 36, 39, 42, 45, 48, 50, 51, 54, 55, 57, 60, 63, 66, 69, 72, 75, 78, 81, 82, 84, 85, 87, 90, 93, 95, 96, 99, 100, 102, 105, 108, 110, 111, 114, 116, 117, 120, 121, 123, 125, 126, 129, 132, 135, 138, 141, 144, 145

Offset: 1

Paolo P. Lava, May 23 2012

nonn

			k=24, x=8, y=16 and 24=8+16; k'=44, x'=12, y'=32 and 44=12+32.
In more than one way:
k=39, x=4, y=35 and 39=4+35; k'=16, x'=4, y'=12 and 16=4+12;
k=39, x=13, y=26 and 39=13+26; k'=16, x’=1, y'=15 and 16=1+15.
k=255, x=54, y=201 and 255=54+201; k'=151, x'=81, y'=70 and 16=4+12;
k=255, x=85, y=170 and 255=85+170; k'=151, x'=22, y'=129 and 16=1+15;
k=255, x=114, y=141 and 39=13+26; k'=151, x'=101, y'=50 and 16=1+15.

Paolo P. Lava, Table of n, a(n) for n = 1..5000

Cf. A003415, A212663, A212664.

Cf. A211223, A211224, A211225.

with(numtheory);
A212662:=proc(q)
local a,b,c,i,n,p,pfs;
for n from 1 to q do
  pfs:=ifactors(n)[2]; a:=n*add(op(2,p)/op(1,p),p=pfs);
  for i from 1 to trunc(n/2) do
    pfs:=ifactors(i)[2]; b:=i*add(op(2,p)/op(1,p),p=pfs);
    pfs:=ifactors(n-i)[2]; c:=(n-i)*add(op(2,p)/op(1,p),p=pfs);
    if a=b+c then print(n); break; fi;
  od;
od; end:
A212662(1000);

ard(n)=vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(m) = for (k=1, m\2, if (ard(m-k)+ard(k) == ard(m), return(1))); \\ Michel Marcus, Aug 27 2022

A212663 Number of ways to represent n’ as x’ + y’, where x+y = n, x > 0, and n’, x’, y’ are the arithmetic derivatives of n, x, y.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 1, 1, 0, 1, 2, 0, 1

Offset: 1

Paolo P. Lava, May 23 2012

nonn

Paolo P. Lava, Table of n, a(n) for n = 1..5000

Cf. A003415, A211223-A211225, A212662, A212664.

with(numtheory);
A212663:=proc(q)
local a,b,c,i,n,p,pfs,t;
for n from 1 to q do
  pfs:=ifactors(n)[2]; a:=n*add(op(2,p)/op(1,p),p=pfs); t:=0;
  for i from 1 to trunc(n/2) do
   pfs:=ifactors(i)[2]; b:=i*add(op(2,p)/op(1,p),p=pfs);
   pfs:=ifactors(n-i)[2]; c:=(n-i)*add(op(2,p)/op(1,p),p=pfs);
   if a=b+c then t:=t+1; fi;
  od;
  print(t);
od; end:
A212663(1000);

A212664 Least k with precisely n partitions k = x + y satisfying x > 0 and k’ = x’ + y’, where k’, x’, y’ are the arithmetic derivatives of k, x, y.

Original entry on oeis.org

3, 39, 213, 903, 2379, 2343, 6545, 12325, 15015, 16107, 45045, 134225, 80535, 142545, 205205, 255255, 346035, 533715, 615615, 645645, 997815, 1601145, 1369095, 1936935

Offset: 1

Paolo P. Lava, May 23 2012

			n=2343, x=162, y=2181 and 2343=162+2181; n’=1027, x’=297, y’=730 and 1027=297+730.
n=2343, x=308, y=2035 and 2343=308+2035; n’=1027, x’=380, y’=647 and 1027=+380+647.
n=2343, x=377, y=1966 and 2343=377+1966; n’=1027, x’=42, y’=985 and 1027=42+985.
n=2343, x=484, y=1859 and 2343=484+1859; n’=1027, x’=572, y’=455 and 1027=572+455.
n=2343, x=505, y=1838 and 2343=505+1838; n’=1027, x’=106, y’=921 and 1027=106+921.
n=2343, x=781, y=1562 and 2343=781+1562; n’=1027, x’=82, y’=945 and 1027=82+945.

Cf. A003415, A211223-A211225, A212662, A212663.

with(numtheory);
A212664:=proc(q)
local a, b, c, d, f, i, j, n, p, pfs, v;
v:=array(1..100); for n from 1 to 100 do v[n]:=0; od;
a:=0;
for n from 1 to q do
  pfs:=ifactors(n)[2]; c:=n*add(op(2,p)/op(1,p),p=pfs); b:=0;
  for i from 1 to trunc(n/2) do
    pfs:=ifactors(i)[2]; d:=i*add(op(2,p)/op(1,p),p=pfs);
    pfs:=ifactors(n-i)[2]; f:=(n-i)*add(op(2,p)/op(1,p),p=pfs);
    if c=d+f then b:=b+1; fi; od;
    if b=a+1 then a:=b; print(b,n); j:=1;
       while v[b+j]>0 do a:=b+j; print(b,v[b+j]); j:=j+1; od;
    else if b>a+1 then if v[b]=0 then v[b]:=n; fi; fi; fi;
od; end:
A212664(100000);

a(12)-a(24) from Donovan Johnson, May 25 2012

A210732 Numbers n for which sigma(n)=sigma(x)+sigma(y), where n=x+y and sigma(n) is the sum of the anti-divisors of n.

Original entry on oeis.org

6, 9, 15, 18, 21, 24, 27, 30, 31, 33, 37, 39, 43, 44, 46, 47, 53, 56, 57, 62, 65, 66, 70, 73, 74, 75, 76, 78, 81, 83, 86, 88, 90, 91, 92, 93, 97, 99, 102, 103, 106, 107, 109, 110, 114, 116, 117, 118, 119, 121, 122, 123, 125, 126, 127, 129, 131, 133, 135, 136

Offset: 3

Paolo P. Lava, May 10 2012

nonn

Similar to A211223 but using anti-divisors.

			sigma*(127)=sigma*(45)+sigma*(82) that is 212=86+126.
In more than one way:
sigma*(133)=sigma*(50)+sigma*(83)=sigma*(52)+sigma*(81) that is
204=80+124=94+110.

Cf. A066272, A066417, A083207, A204830, A204831, A211223-A211225.

with(numtheory);
A210732:=proc(q)
local a,b,c,i,j,k,n;
for n from 3 to q do
  a:=0;
  for k from 2 to n-1 do if abs((n mod k)-k/2)<1 then a:=a+k; fi; od;
  for i from 1 to trunc(n/2) do
   b:=0; c:=0;
   for k from 2 to i-1 do if abs((i mod k)-k/2)<1 then b:=b+k; fi; od;
   for k from 2 to n-i-1 do if abs(((n-i) mod k)-k/2)<1 then c:=c+k; fi; od;
   if a=b+c then print(n); break; fi;
  od;
od; end:
A210732(10000);

A375154 Least k with exactly n partitions k = x + y + z satisfying sigma(k) = sigma(x) + sigma(y) + sigma(z).

Original entry on oeis.org

5, 25, 33, 39, 38, 58, 65, 86, 123, 85, 82, 92, 98, 152, 99, 158, 106, 135, 153, 145, 215, 186, 142, 189, 235, 178, 185, 165, 147, 315, 274, 231, 214, 305, 171, 332, 207, 290, 310, 344, 266, 358, 583, 297, 261, 278, 285, 488, 255, 334, 302, 369, 309, 2888, 284

Offset: 1

Paolo P. Lava, Aug 01 2024

nonn

No other prime apart from initial 5.

			a(7) = 65 and 65 has 7 partitions of three numbers, x, y and z, for which sigma(65) = sigma(x) + sigma(y) + sigma(z) = 84. In fact:
sigma(2) + sigma(14) + sigma(49) = 3 + 24 + 57 = 84;
sigma(3) + sigma(7) + sigma(55) = 4 + 8 + 72 = 84;
sigma(3) + sigma(23) + sigma(39) = 4 + 24 + 56 = 84;
sigma(5) + sigma(5) + sigma(55) = 6 + 6 + 72 = 84;
sigma(7) + sigma(19) + sigma(39) = 8 + 20 + 56 = 84;
sigma(10) + sigma(14) + sigma(41) = 18 + 24 + 42 = 84;
sigma(13) + sigma(13) + sigma(39) = 14 + 14 + 56 = 84;
Furthermore 65 is the minimum number to have this property.

David A. Corneth, Table of n, a(n) for n = 1..764

Cf. A000203, A211223, A211224, A211225, A373047.

f[n_] := Count[IntegerPartitions[n, {3}], ?(Total[DivisorSigma[1, #]] == DivisorSigma[1, n] &)]; With[{s = Array[f, 600]}, TakeWhile[FirstPosition[s, #] & /@ Range[Max[s]] // Flatten, !MissingQ[#] &]] (* _Amiram Eldar, Aug 02 2024 *)

A381747 a(n) is the number of solutions to tau(x) + tau(n-x) = tau(n) where 1 <= x <= floor(n/2).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 0, 3, 0, 1, 0, 2, 1, 2, 0, 2, 0, 1, 1, 3, 0, 1, 0, 4, 0, 3, 0, 3, 0, 3, 1, 4, 0, 1, 0, 2, 1, 2, 0, 3, 0, 5, 6, 4, 0, 0, 0, 5, 0, 5, 0, 5, 1, 4, 0, 4, 0, 2, 0, 3, 6, 4, 0, 5, 0, 8, 1, 5, 0, 3, 0, 5, 8, 8, 0, 5, 0, 3, 0, 5, 0, 3, 1, 5, 0, 6

Offset: 1

Felix Huber, Mar 30 2025

Observation: For even positive multiples of 48, k <= 17000, a(k) = 0 only for k = 1*48, 2304 = 48*48 and 3600 = 75*48. The next numbers k which are multiples of 48 and for which a(k) = 0 are 46656, 63504, 233280, 513216, 793152, 2286144, 3111696.

			a(10) = 3 because tau(x) + tau(10-x) = tau(10) has 3 solutions for 0 <= x <= 5:
  x = 1: tau(1) + tau(9) = 1 + 3 = 4 = tau(10);
  x = 3: tau(3) + tau(7) = 2 + 2 = 4 = tau(10);
  x = 5: tau(5) + tau(5) = 2 + 2 = 4 = tau(10).

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A000005, A000430, A211225, A382074.

with(NumberTheory):
A381747:=proc(n)
    local a,x;
    a:=0;
    for x to n/2 do
        if tau(x)+tau(n-x)=tau(n) then
            a:=a+1
        fi
    od;
    return a
end proc;
seq(A381747(n),n=1..88);

a(n) = my(nd=numdiv(n)); sum(x=1, n\2, numdiv(x)+numdiv(n-x) == nd); \\ Michel Marcus, Apr 26 2025

a(A000430(n)) = 0 for n > 3.

A218011 Numbers n for which n’ = x’*y’, where x>0, y>0, n = x + y and n’, x’, y’ are the arithmetic derivatives of n, x, y.

Original entry on oeis.org

5, 7, 13, 19, 31, 43, 48, 55, 61, 73, 74, 87, 103, 106, 109, 117, 139, 146, 151, 159, 160, 178, 181, 193, 199, 202, 208, 212, 225, 229, 236, 241, 252, 267, 268, 271, 283, 285, 298, 313, 349, 357, 362, 386, 403, 411, 421, 433, 455, 463, 496, 511, 519, 523, 535

Offset: 1

Paolo P. Lava, Oct 18 2012

nonn

The greatest prime in a twin primes couple is in the sequence. In fact if the twin primes are a and b, with a

			n= 612, x=85,  y=527; n’=1056, x’=22, y’=48 and 1056=22*48.
n= 752, x=361, y=391; n’=1520, x’=38, y’=40 and 1520=38*40.
n= 779, x=36,  y=743; n’=60,   x’=60, y’=1  and 60=60*1.

Antti Karttunen, Table of n, a(n) for n = 1..12822 (first 250 terms from Paolo P. Lava)

Cf. A003415, A211223, A211224, A211225, A212662, A212663, A212664.

Subsequences: A006512 (primes in this sequence), A370126 (k with a solution where both x and y are composite).

with(numtheory);
A218011:= proc(i)
local a,b,c,n,p,pfs,q;
for n from 1 to i do
for q from 1 to trunc(n/2) do
  a:=q*add(op(2,p)/op(1,p),p= ifactors(q)[2]);
  b:=(n-q)*add(op(2,p)/op(1,p),p= ifactors(n-q)[2]);
  c:=n*add(op(2,p)/op(1,p),p= ifactors(n)[2]);
  if c=a*b then lprint(n,q,n-q); break; fi;
od; od;
end:
A218011(1000000);

dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; f[n_] := Select[Range[n/2], dn[#]*dn[n - #] == dn[n] &]; Select[Range[535], Length[f[#]] > 0 &] (* T. D. Noe, Oct 18 2012 *)

up_to = 2^18;
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
v003415 = vector(up_to,n,A003415(n));
isA218011(n) = { my(z=v003415[n]); for(x=2,ceil(n/2),if(!(z%v003415[x]), if(z==v003415[x]*v003415[n-x], return(1)))); (0); }; \\ Antti Karttunen, Feb 22 2024

A373047 Least k with exactly n partitions k = x + y + z satisfying sigma(k) = sigma(x) + sigma(y) + sigma(z), where sigma*(k) is the sum of the anti-divisors of k.

Original entry on oeis.org

11, 33, 16, 20, 26, 37, 40, 19, 43, 46, 93, 91, 80, 76, 39, 78, 155, 103, 74, 135, 128, 152, 116, 117, 190, 104, 187, 138, 168, 147, 160, 223, 208, 403, 281, 173, 163, 170, 250, 243, 272, 257, 258, 232, 222, 278, 266, 245, 352, 253, 279, 256, 288, 295, 231, 291

Offset: 1

Paolo P. Lava, Aug 02 2024

			a(7) = 40 and 40 has 7 partitions of three numbers, x, y and
z, for which sigma*(65) = sigma*(x) + sigma*(y) + sigma*(z) = 55. In fact:
sigma*(1) + sigma*+(4) + sigma*(35) = 0 + 3 + 52 = 55;
sigma*(1) + sigma*(12) + sigma*(27) = 0 + 13 + 42 = 55;
sigma*(1) + sigma*(14) + sigma*(25) = 0 + 16 + 39 = 55;
sigma*(4) + sigma*(14) + sigma*(22) = 3 + 16 + 36 = 55;
sigma*(5) + sigma*(8) + sigma*(27) = 5 + 8 + 42 = 55;
sigma*(9) + sigma*(13) + sigma*(18) = 8 + 19 + 28 = 55;
sigma*(10) + sigma*(12) + sigma*(18) = 14 + 13 + 28 = 55;
Furthermore 40 is the minimum number to have this property.

Cf. A066417, A210732, A211223, A211224, A211225, A375154.

Showing 1-10 of 12 results. Next

cp's OEIS Frontend

A211223 Numbers k for which sigma(k) = sigma(x) + sigma(y), where k = x + y.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

A211224 Least k with precisely n partitions k = x + y satisfying sigma(k) = sigma(x) + sigma(y).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

A212662 Numbers k for which k' = x' + y', where x > 0, k = x + y, and k', x', y' are the arithmetic derivatives of k, x, y.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

A212663 Number of ways to represent n’ as x’ + y’, where x+y = n, x > 0, and n’, x’, y’ are the arithmetic derivatives of n, x, y.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A212664 Least k with precisely n partitions k = x + y satisfying x > 0 and k’ = x’ + y’, where k’, x’, y’ are the arithmetic derivatives of k, x, y.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Extensions

A210732 Numbers n for which sigma*(n)=sigma*(x)+sigma*(y), where n=x+y and sigma*(n) is the sum of the anti-divisors of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

A375154 Least k with exactly n partitions k = x + y + z satisfying sigma(k) = sigma(x) + sigma(y) + sigma(z).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A381747 a(n) is the number of solutions to tau(x) + tau(n-x) = tau(n) where 1 <= x <= floor(n/2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A210732 Numbers n for which sigma(n)=sigma(x)+sigma(y), where n=x+y and sigma(n) is the sum of the anti-divisors of n.

A373047 Least k with exactly n partitions k = x + y + z satisfying sigma(k) = sigma(x) + sigma(y) + sigma(z), where sigma*(k) is the sum of the anti-divisors of k.