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.

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A203616 Numbers k such that the reversal of sigma*(k) equals the sum of the reversals of the anti-divisors of k, where sigma*(k) is the sum of the anti-divisors of k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 20, 63, 96, 97, 317, 596, 1473, 3934, 26777, 27684, 50867, 51767, 62417, 322001, 393216, 1308775, 1420260, 1851474, 2651867, 2659067, 3040656, 3227267, 3289277, 3376007, 4626917, 4639067, 5378507, 6054521, 6227027, 6239839, 6439067, 6581929
Offset: 1

Views

Author

Paolo P. Lava, Jan 20 2012

Keywords

Comments

A066466 is a subsequence of this sequence.

Examples

			n=317. Anti-divisors: 2, 3, 5, 127, 211.
Sum of the reversals of the anti-divisors: 2+3+5+721+112=843.
Sigma*(317)=348 and its reversal is 843.
n=1473. Anti-divisors: 2, 5, 6, 7, 19, 31, 95, 155, 421, 589, 982.
Sum of the reversals of the anti-divisors:
2+5+6+7+91+13+59+551+124+985+289=2132.
Sigma*(1473)=2312 and its reversal is 2132.

Links

Crossrefs

Cf. A004086, A066272, A066466, A069942, A195144, A203615.

Programs

  • Maple
    isA203616:=proc(j) local a,b,c;   a:=0; b:=0;
   for c from 2 to j-1 do
     if abs((j mod c)-c/2)<1 then a:=a+A004086(c); b:=b+c; fi;
   od;
evalb(A004086(b)=a) end: # simplified by M. F. Hasler, Jan 29 2012
for n to 10^7 do if isA203616(n) then lprint(n) fi od: # simplified by M. F. Hasler, Jan 29 2012
  • Python
    from itertools import count, islice
from sympy.ntheory.factor_ import antidivisors
def a203616():
    isa = lambda n: str(sum((a:=antidivisors(n))))[::-1]==str(sum(map(int, (str()[::-1] for  in a))))
    yield from (n for n in count(1) if isa(n))
a203616_list = [*islice(a203616(), 20)] # Dumitru Damian, Feb 12 2024

Extensions

a(22)-a(40) from Dumitru Damian, Feb 12 2024

A203615 Reversal of sigma(n) equals the sum of the reversals of the divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 21, 938, 17797, 44045, 87001, 454085, 2217425, 8156450, 8475789, 3293216050, 11130063842, 44662814795, 77084972662
Offset: 1

Views

Author

Paolo P. Lava, Jan 20 2012

Keywords

Comments

a(20) > 2.34*10^12. - Giovanni Resta, Aug 30 2018

Examples

			n=17797. Divisors: 1, 13, 37, 481, 1369, 17797.
Sum of the reversals of the divisors: 1+31+73+184+9631+79771=89691.
Sigma(17797)=19698 and its reversal is 89691.
n=454085. Divisors: 1, 5, 197, 461, 985, 2305, 90817, 454085.
Sum of the reversals of the divisors: 1+5+791+164+589+5032+71809+580454=658845.
Sigma(454085)=548856 and its reversal is 658845.

Crossrefs

Cf. A069942, A195144, A203616.

Programs

  • Maple
    with(numtheory);
Rev:=proc(n)
local a, i, k;
  i:=convert(n,base,10); a:=0;
  for k from 1 to nops(i) do a:=a*10+i[k]; od;
  a;
end:
P:=proc(s)
local a, b, c, j, pfs;
for j from 1 to s do
  b:=divisors(j); a:=0;
  for c from 1 to nops(b) do a:=a+Rev(b[c]); od;
  if Rev(sigma(j))=a then print(j); fi;
od;
end:
P(10000000);
  • Mathematica
    Select[Range[33*10^8],Total[IntegerReverse/@Divisors[#]] == IntegerReverse[ DivisorSigma[ 1,#]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 09 2018 *)

Extensions

a(13)-a(16) from Donovan Johnson, Jan 29 2012
a(17)-a(19) from Giovanni Resta, Aug 30 2018
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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