cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Paolo P. Lava, May 23 2012

Keywords

Links

Crossrefs

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

Programs

  • Maple
    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

Views

Author

Paolo P. Lava, May 23 2012

Keywords

Examples

			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.

Crossrefs

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

Programs

  • Maple
    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);

Extensions

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

Views

Author

Paolo P. Lava, Oct 18 2012

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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).

Programs

  • Maple
    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);
  • Mathematica
    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 *)
  • PARI
    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

A356594 Numbers k for which there exists at least one pair of positive integers (x,y) such that k = x + y and k' = x' + y', and every such pair is coprime.

Original entry on oeis.org

3, 25, 55, 82, 85, 95, 116, 121, 145, 194, 226, 245, 253, 289, 295, 301, 305, 332, 335, 343, 362, 391, 407, 418, 422, 446, 455, 493, 529, 535, 548, 583, 611, 671, 731, 745, 749, 754, 778, 779, 781, 785, 799, 805, 815, 817, 818, 833, 838, 845, 866, 869, 899, 917, 931, 943, 955, 959, 985, 995, 998
Offset: 1

Views

Author

Giosuè Cavallo, Aug 14 2022

Keywords

Comments

Subsequence of A212662.
a(1)=3 is the only prime term.
It is not known if this sequence is finite.
Every term in A212662 is a multiple of a term in this sequence (this could be considered its primitive sequence), and no term in this sequence divides another term of this sequence.
In general, we do not have (i+j)'=i'+j'; this is in contrast with the normal derivative, because the derivative of the sum of two functions is equal to the sum of the derivatives of the individual functions. The terms in this sequence are the ones for which there exist positive integers i and j with the properties illustrated above, thus building another common point between the arithmetic derivative and the normal derivative.

Crossrefs

Cf. A003415 (arithmetic derivative), A212662.

Programs

  • PARI
    D(n)={x=factor(n);n*sum(i=1,matsize(x)[1],x[i,2]/x[i,1])}
Base(n)={for(i=1,n\2,if(D(n-i)+D(i)==D(n)&&gcd(i,n-i)==1,return(1)))}
Impure(n)={for(i=1,n\2,if(D(n-i)+D(i)==D(n)&&gcd(i,n-i)!=1,return(1)))}
IsTerm(n)={Base(n)&&!Impure(n)}
Bp(n)={for(i=2,n,if(IsTerm(i),print1(i, ", ")))}
Bp(1000)
  • PARI
    ard(n)=vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok1(m) = for (k=1, m\2, if (ard(m-k)+ard(k) == ard(m), return(1))); \\ A212662
isok(m) = if (isok1(m), my(d=divisors(m)); for (k=1, #d, if((d[k]!=m) && isok1(d[k]), return(0))); return(1)); \\ Michel Marcus, Aug 28 2022

A375238 Least k with exactly n partitions k = x + y + z satisfying k' = x' + y' + z', where k' is the arithmetic derivative of k.

Original entry on oeis.org

5, 9, 22, 35, 65, 63, 70, 62, 82, 110, 75, 143, 130, 169, 142, 186, 170, 194, 230, 284, 234, 195, 147, 345, 238, 245, 323, 290, 286, 294, 285, 334, 430, 534, 458, 255, 385, 434, 390, 418, 374, 399, 441, 526, 518, 382, 748, 598, 578, 454, 455, 585, 507, 435, 582
Offset: 1

Views

Author

Paolo P. Lava, Aug 06 2024

Keywords

Examples

			a(7) = 70 and 70 has 7 partitions of three numbers, x, y and z, for which 70' = x' + y' + z' = 59. In fact:
5' + 21' + 44' = 1 + 10 + 48 = 59;
6' + 14' + 50' = 5 + 9 + 45 = 59;
6' + 22' + 42' = 5 + 13 + 41 = 59;
10' + 10' + 50' = 7 + 7 + 45 = 59;
13' + 24' + 33' = 1 + 44 + 14 = 59;
13' + 27' + 30' = 1 + 27 + 31 = 59;
14' + 14' + 42' = 9 + 9 + 41 = 59.
Furthermore 70 is the minimum number to have this property.

Crossrefs

Cf. A003415, A212662, A373047, A375154.
Showing 1-5 of 5 results.

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