A211224 -id:A211224 - 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

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

Original entry on oeis.org

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

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

A271384 Least k with precisely n partitions k = x + y satisfying phi(k) = phi(x) + phi(y), where phi(k) is the Euler totient function of k.

Original entry on oeis.org

3, 14, 20, 28, 44, 92, 112, 224, 266, 260, 404, 380, 476, 552, 558, 696, 860, 984, 846, 1062, 1388, 1128, 1278, 1752, 1494, 1422, 2034, 1926, 1704, 1992, 2358, 2466, 2712, 2424, 2718, 3222, 3006, 3258, 4924, 3288, 3582, 4296, 3798, 4008, 4518, 5688, 5094, 5352

Offset: 1

Paolo P. Lava, Apr 06 2016

			phi(28) = phi(6) + phi(22) = phi(8) + phi(20) = phi(12) + phi(16) = phi(14) + phi(14) = 12 and 28 is the least number with 4 partitions of two numbers with this property: therefore a(4) = 28;
phi(112) = phi(14) + phi(98) = phi(24) + phi(88) = phi(30) + phi(82) = phi(32) + phi(80) = phi(36) + phi(76) = phi(48) + phi(64) = phi(56) + phi(56) = 48 and 112 is the least number with 7 partitions of two numbers with this property: therefore a(7) = 112.

Paolo P. Lava, Table of n, a(n) for n = 1..50

Cf. A000005, A211224, A271382.

with(numtheory): P:=proc(q) local a,h,k,n; for h from 1 to q do
for n from 2*h to q do a:=0; for k from 1 to trunc(n/2) do if phi(n)=phi(k)+phi(n-k) then a:=a+1; fi; od;
if a=h then print(n); break; fi; od; od; end: P(10^9);

Table[SelectFirst[Range[10 + 5 n^2], Function[k, With[{e = EulerPhi@ k},
Count[Transpose@ {Range[k - 1, Ceiling[k/2], -1], Range@ Floor[k/2]}, x_ /; Total@ EulerPhi@ x == e] == n]]], {n, 25}] (* Michael De Vlieger, Apr 06 2016, Version 10 *)

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

A271382 Least k with precisely n partitions k = x + y satisfying d(k) = d(x) + d(y), where d(k) is the number of divisors of k.

Original entry on oeis.org

2, 14, 10, 26, 44, 45, 126, 68, 99, 104, 162, 117, 98, 124, 232, 164, 148, 200, 260, 333, 231, 244, 248, 297, 273, 284, 315, 406, 332, 345, 385, 430, 344, 399, 388, 436, 429, 488, 465, 495, 472, 525, 561, 555, 621, 556, 632, 604, 652, 712, 536, 693, 735, 675

Offset: 1

Paolo P. Lava, Apr 06 2016

			d(10) = d(1) + d(9) = d(3) + d(7) = d(5) + d(5) = 4 and 10 is the least number with 3 partitions of two numbers with this property: therefore a(3) = 10;
d(126) = d(21) + d(105) = d(22) + d(104) = d(28) + d(98) = d(38) + d(33) = d(40) + d(86) = d(50) + d(76) = d(63) + d(63) = 12 and 126 is the least number with 7 partitions of two numbers with this property: therefore a(7) = 126.

Paolo P. Lava, Table of n, a(n) for n = 1..1200

Cf. A000005, A211224, A271384.

with(numtheory): P:=proc(q) local a,h,k,n; for h from 1 to q do for n from 2*h to q do
a:=0; for k from 1 to trunc(n/2) do if tau(n)=tau(k)+tau(n-k) then a:=a+1; fi; od;
if a=h then print(n); break; fi; od; od; end: P(10^6);

nn = 10^3; Table[SelectFirst[Range@ nn, Function[k, With[{e = DivisorSigma[0, k]}, Count[Transpose@ {Range[k - 1, Ceiling[k/2], -1], Range@ Floor[k/2]}, x_ /; Total@ DivisorSigma[0, x] == e] == n]]], {n, 54}] (* Michael De Vlieger, Apr 06 2016 *)

isok(k, n) = {my(nb = 0, tau = numdiv(k)); for (j=1, k\2, if (numdiv(j)+numdiv(k-j) == tau, nb++); if (nb > n, return (0));); nb == n;}
a(n) = {k=2; while (!isok(k, n), k++); k;} \\ Michel Marcus, Apr 07 2016

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 1-8 of 8 results.

cp's OEIS Frontend

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

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

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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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A271384 Least k with precisely n partitions k = x + y satisfying phi(k) = phi(x) + phi(y), where phi(k) is the Euler totient function of k.

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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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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.

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A271382 Least k with precisely n partitions k = x + y satisfying d(k) = d(x) + d(y), where d(k) is the number of divisors of k.

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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.