A211223 -id:A211223 - cp's OEIS frontend

A211225 Number of ways to represent sigma(n) as sigma(x) + sigma(y) where x+y = n.

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

Offset: 1

Paolo P. Lava, May 07 2012

nonn

From an idea of Charles R Greathouse IV.

a(A211223(n)) > 0. - Reinhard Zumkeller, Jan 06 2013

			a(3)=1 because sigma(3)=sigma(1)+sigma(2)=4;
a(32)=2 because sigma(32)=sigma(4)+sigma(28)=sigma(14)+sigma(18)=63;
a(117)=3 because sigma(117)=sigma(41)+sigma(76)=sigma(52)+sigma(65)=sigma(56)+sigma(61)=182; etc.

Paolo P. Lava, Table of n, a(n) for n = 1..10000

Cf. A083207, A204830, A204831, A211223, A211224.

Cf. A000203.

a211225 n = length $ filter (== a000203 n) $ zipWith (+) us' vs where
   (us,vs@(v:_)) = splitAt (fromInteger $ (n - 1) `div` 2) a000203_list
   us' = if even n then v : reverse us else reverse us
-- Reinhard Zumkeller, Jan 06 2013

with(numtheory);
A211225:=proc(q)
local b,i,n;
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;
  print(b)
od; end:
A211225(1000);

a[n_] := With[{s = DivisorSigma[1, n]}, Sum[Boole[s == DivisorSigma[1, x] + DivisorSigma[1, n-x]], {x, 1, Quotient[n, 2]}]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 04 2023 *)

a(n)=my(t=sigma(n)); sum(i=1, n\2, sigma(i)+sigma(n-i)==t) \\ Charles R Greathouse IV, May 07 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

A218852 Numbers n for which sigma(n) = sigma(x) + sigma(y) + sigma(z), where n = x + y + z, with x, y, z all positive.

Original entry on oeis.org

5, 7, 10, 13, 14, 15, 16, 19, 20, 21, 25, 26, 27, 28, 31, 32, 33, 34, 35, 38, 39, 40, 42, 43, 44, 45, 46, 49, 50, 51, 52, 54, 55, 56, 57, 58, 61, 62, 63, 64, 65, 66, 68, 69, 70, 73, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96

Offset: 1

Jon Perry, Nov 07 2012

nonn

Contains the greater of every twin prime pair.

			sigma(1) + sigma(1) + sigma(3) = sigma(5) = 6.
sigma(2) + sigma(2) + sigma(6) = sigma(10) = 18.
*sigma(2) + sigma(8) + sigma(30) = sigma(40) = 90.
*sigma(6) + sigma(10) + sigma(24) = sigma(40) = 90.
sigma(8) + sigma(8) + sigma(24) = sigma(40) = 90.
Hence, 5, 10 and 40 are in the sequence.
Note that (*) means that (x+y+z) divides xyz as well.

Harvey P. Dale, Table of n, a(n) for n = 1..400

Cf. A000203, A211223, A218980.

isA218852 := proc(n)
    local x,y,z ;
    for x from 1 to n-2 do
        for y from x to n-x-1 do
            z := n-x-y ;
            if numtheory[sigma](x)+numtheory[sigma](y)+numtheory[sigma](z) = numtheory[sigma](n) then
                return true;
            end if;
        end do:
    end do:
    return false;
end proc:
for n from 3 to 120 do
    if isA218852(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Nov 07 2012

xyzQ[n_]:=Module[{ips=Total/@(DivisorSigma[1,#]&/@IntegerPartitions[n,{3}])},Total[Boole[DivisorSigma[1,n]==#&/@ips]]>0]; Select[Range[ 100], xyzQ] (* Harvey P. Dale, Jun 22 2020 *)

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 *)

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 13 results. Next

cp's OEIS Frontend

A211225 Number of ways to represent sigma(n) as sigma(x) + sigma(y) where x+y = n.

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

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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.

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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.

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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.

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A218852 Numbers n for which sigma(n) = sigma(x) + sigma(y) + sigma(z), where n = x + y + z, with x, y, z all positive.

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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.

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

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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.