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 1-4 of 4 results.

A247012 Consider the aliquot parts, in ascending order, of a composite number. Take their sum and repeat the process deleting the minimum number and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to the reverse of themselves.

Original entry on oeis.org

6, 133, 172, 841, 1005, 1603, 4258, 5299, 192901, 498906, 1633303, 5307589, 16333303, 20671542, 41673714, 42999958, 73687923
Offset: 1

Views

Author

Paolo P. Lava, Sep 09 2014

Keywords

Comments

A072234 is a subset of this sequence.
a(18) > 2*10^8. - Tyler Busby, Mar 19 2023

Examples

			Aliquot parts of 1005 are 1, 3, 5, 15, 67, 201 and 335:
  1 + 3 + 5 + 15 + 67 + 201 + 335 = 627;
  3 + 5 + 15 + 67 + 201 + 335 + 627 = 1253;
  5 + 15 + 67 + 201 + 335 + 627 + 1253 = 2503;
  15 + 67 + 201 + 335 + 627 + 1253 + 2503 = 5001 that is the reverse of 1005.
Aliquot parts of 1603 are 1, 7 and 229:
  1 + 7 + 229 = 237;
  7 + 229 + 237 = 473;
  229 + 237 + 473 = 939;
  237 + 473 + 939 = 1649;
  473 + 939 + 1649 = 3061 that is the reverse of 1603;

Crossrefs

Cf. A072234, A097090, A246544, A247013.

Programs

  • Maple
    with(numtheory): R:=proc(w) local x,y; x:=w; y:=0;
while x>0 do y:=10*y+(x mod 10); x:=trunc(x/10); od: y; end:
P:=proc(q,h) local a,b,c,k,n,t,v; v:=array(1..h);
for n from 2 to q do if not isprime(n) then
a:=sort([op(divisors(n))]); b:=nops(a)-1; c:=ilog10(n)+1;
for k from 1 to b do v[k]:=a[k]; od;
t:=b+1; v[t]:=add(v[k], k=1..b);
if R(v[t])=n then print(n); else
while ilog10(v[t])+1<=c do t:=t+1; v[t]:=add(v[k], k=t-b..t-1);
if R(v[t])=n then print(n); break; fi; od; fi; fi; od;
end: P(10^9, 1000);
  • Mathematica
    A247012 = {};
For[n = 4, n <= 1000000, n++,
 If[PrimeQ[n], Continue[]];
 r = IntegerReverse[n];
 a = Most[Divisors[n]];
 sum = Total[a];
 While[sum < r, sum = Total[a = Join[Rest[a], {sum}]]];
 If[sum == r, AppendTo[A247012, n]];
]; A247012 (* Robert Price, Sep 08 2019 *)
  • Python
    from sympy import isprime, divisors
A247012_list = []
for n in range(2,10**9):
    m = int(str(n)[::-1])
    if not isprime(n):
        x = divisors(n)
        x.pop()
        y = sum(x)
        while y < m:
            x, y = x[1:]+[y], 2*y-x[0]
        if y == m:
            A247012_list.append(n) # Chai Wah Wu, Sep 12 2014

Extensions

a(9), a(11)-a(17) from Chai Wah Wu, Sep 13 2014

A248134 Consider a number x as a concatenation of two integers, a and b: x = concat(a,b). Take their sum and repeat the process deleting the minimum number and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

Original entry on oeis.org

14, 19, 21, 28, 42, 47, 63, 84, 105, 126, 147, 149, 168, 189, 199, 298, 323, 497, 646, 795, 911, 969, 1292, 1499, 1822, 1999, 2087, 2733, 2998, 3089, 3248, 3379, 3644, 4555, 4997, 5411, 5466, 6178, 6377, 6496, 7288, 7995, 8199, 9161, 9267, 9744, 10822, 12356
Offset: 1

Views

Author

Paolo P. Lava, Oct 02 2014

Keywords

Comments

If the number x is rewritten as concat(a,b), the problem is to find a value of y such that x = a*F(y) + b*F(y+1), if a < b, or x = b*F(y) + a*F(y+1), if a > b, where F(y) is a Fibonacci number (see values of x, a, b, y, for 1
Similar to A130792 but here the minimum number is deleted since the beginning.
All the listed numbers admit only one concatenation, concat(a,b), that, through the addition process, leads to themselves. Is there any number that admit more than one single concatenation?
Sequence is infinite. Let us consider the numbers 19, 199, 1999, 19...9 and let us divide them as concat(1,9), concat(1,99), concat(1,999), concat(1,9...9). In two steps we have the initial numbers back: 1 + 9 = 10 and 9 + 10 = 19; 1 + 99 = 100 and 99 + 100 = 199, etc.

Examples

			Let us rewrite 5411 as 54 U 11. Then:
11 + 54 = 65;
54 + 65 =  119;
65 + 119 = 184;
119 + 184 = 303;
184 + 303 = 487;
303 + 487 = 790;
487 + 790 = 1277;
790 + 1277 = 2067;
1277 + 2067 = 3344;
2067 + 3344 = 5411, that is 11*F(10) + 54*F(11) = 11*55 + 54*89 = 605 + 4806 = 5411.

Links

Crossrefs

Cf. A000045, A007629, A130792, A246544, A247012, A247013.

Programs

  • Maple
    P:=proc(q,h) local a,b,k,n,t,v; v:=array(1..h);
for n from 1 to q do for k from 1 to ilog10(n) do
a:=n mod 10^k; b:=trunc(n/10^k); if a

A258142 Consider the unitary aliquot parts, in ascending order, of a composite number. Take their sum and repeat the process deleting the minimum number and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

Original entry on oeis.org

6, 21, 60, 85, 90, 261, 976, 2009, 87360, 97273, 4948133, 68353213
Offset: 1

Views

Author

Paolo P. Lava, May 22 2015

Keywords

Comments

A002827 is a subset of this sequence.
No more terms below 10^8. - Amiram Eldar, Jan 12 2019

Examples

			Divisors of 85 are 1, 5, 17, 85. Unitary aliquot parts are 1, 5, 17.
We have:
1 + 5 + 17 = 23;
5 + 17 + 23 = 45;
17 + 23 + 45 = 85.
Divisors of 2009 are 1, 7, 41, 49, 287, 2009.
Unitary aliquot parts are 1, 41, 49. We have:
1 + 41 + 49 = 91;
41 + 49 + 91 = 181;
49 + 91 + 181 = 321;
91 + 181 + 321 = 593;
181 + 321 + 593 = 1095;
321 + 593 + 1095 = 2009.

Links

Crossrefs

Cf. A002827, A034444, A246544, A247012, A247013, A248134.

Programs

  • Maple
    with(numtheory):P:=proc(q,h) local a,b,k,n,t,v; v:=array(1..h);
for n from 1 to q do if not isprime(n) then b:=sort([op(divisors(n))]); a:=[];
for k from 1 to nops(b)-1 do if gcd(b[k],n/b[k])=1 then a:=[op(a),b[k]]; fi; od;
a:=sort(a); b:=nops(a); if b>1 then for k from 1 to b do v[k]:=a[k]; od;
t:=b+1; v[t]:=add(v[k], k=1..b); while v[t]
  • Mathematica
    aQ[n_] := Module[{s = Most[Select[Divisors[n], GCD[#, n/#] == 1 &]]}, If[Length[s] == 1, False, While[Total[s] < n, AppendTo[s, Total[s]]; s = Rest[s]]; Total[s] == n]]; Select[Range[2, 10^8], aQ] (* Amiram Eldar, Jan 12 2019 *)

Extensions

a(11)-a(12) from Amiram Eldar, Jan 12 2019

A307859 Consider the non-unitary aliquot parts, in ascending order, of a composite number. Take their sum and repeat the process deleting the minimum number and adding the previous sum. The sequence lists the numbers that after some iterations reach a sum equal to themselves.

Original entry on oeis.org

24, 112, 189, 578, 1984, 2125, 3993, 5043, 9583, 19197, 32512, 126445, 149565, 175689, 225578, 236883, 1589949, 1862935, 1928125, 3171174, 5860526, 6149405, 11442047, 16731741, 60634549, 75062535, 134201344, 177816209, 1162143369, 4474779517, 10369035821
Offset: 1

Views

Author

Paolo P. Lava, May 02 2019

Keywords

Examples

			Divisors of 578 are 1, 2, 17, 34, 289, 578. Non-unitary aliquot parts are 17 and 34.
We have:
   17 +  34 =  51;
   34 +  51 =  85;
   51 +  85 = 136;
   85 + 136 = 221;
  136 + 221 = 357;
  221 + 357 = 578.

Links

Crossrefs

Cf. A002827, A034444, A246544, A247012, A247013, A248134, A258142.

Programs

  • Maple
    with(numtheory):P:=proc(q,h) local a,b,c,k,n,t,v; v:=array(1..h);
for n from 1 to q do if not isprime(n) then b:=sort([op(divisors(n))]);
a:=[]; for k from 2 to nops(b)-1 do if gcd(b[k],n/b[k])>1 then
a:=[op(a),b[k]]; fi; od; b:=nops(a); if b>1 then c:=0;
for k from 1 to b do v[k]:=a[k]; c:=c+a[k]: od;
t:=b+1; v[t]:=c; while v[t]
  • Mathematica
    aQ[n_] := CompositeQ[n] && Module[{s = Select[Divisors[n], GCD[#, n/#] != 1 &]}, If[Length[s] < 2, False, While[Total[s] < n, AppendTo[s, Total[s]]; s = Rest[s]]; Total[s] == n]]; Select[Range[10^4], aQ] (* Amiram Eldar, May 07 2019 *)

Extensions

a(20)-a(31) from Amiram Eldar, May 07 2019
Showing 1-4 of 4 results.

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