A212663 -id:A212663 - cp's OEIS frontend

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.

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

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

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

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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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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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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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