A069942 -id:A069942 - cp's OEIS frontend

A072228 Numbers k such that k equals the sum of the reverses of the proper divisors of k.

6, 244, 285, 133857, 141635817, 273722772124

Offset: 1

Joseph L. Pe, Jul 05 2002

I call these numbers "anti-perfect" (compare with the picture-perfect numbers A069942).

a(7) > 10^12. - Giovanni Resta, Apr 07 2017

			The proper divisors of 285 are 1, 3, 5, 15, 19, 57, 95, with sum of reverses = 1 + 3 + 5 + 51 + 91 + 75 + 59 = 285. Therefore 285 is anti-perfect.

Cf. A069942, A101701.

f[n_] := FromDigits[Reverse[IntegerDigits[n]]]; Do[ If[n == Apply[Plus, Map[f, Drop[Divisors[n], -1]]], Print[n]], {n, 2, 10^7}]
Select[Range[15*10^7],Total[IntegerReverse[Most[Divisors[#]]]]==#&] (* Requires Mathematica version 10 or later *) (* The program takes a long time to run *) (* Harvey P. Dale, Aug 31 2016 *)

One more term from Farideh Firoozbakht, Dec 23 2004

a(6) from Giovanni Resta, Apr 07 2017

A072234 Numbers k such that reverse(k) = sum of the proper divisors of k.

Original entry on oeis.org

6, 498906, 20671542, 41673714, 73687923, 4158499614, 922964834547

Offset: 1

Joseph L. Pe, Jul 05 2002

Mark Ganson conjectures that all terms are divisible by 3.
Jens Kruse Andersen discovered that 4158499614 is in the sequence (although he did not rule out the possibility that there were missing terms below this - that was established by Giovanni Resta).
a(8) > 10^13. - Giovanni Resta, Dec 12 2013

			The proper divisors of 498906 are 1, 2, 3, 6, 9, 18, 27, 54, 9239, 18478, 27717, 55434, 83151, 166302, 249453, which sum to 609894, the reverse of 498906; hence 498906 is a term of the sequence.

Cf. A001065, A004086, A069942.

f = IntegerReverse; Do[ If[f[n] == Apply[Plus, Drop[Divisors[n], -1]], Print[n]], {n, 2, 10^9}]
Select[Range[500000],IntegerReverse[#]==Total[Most[Divisors[#]]]&] (* The program generates the first 2 terms of the sequence. To generate more, increase the Range constant but the program may take a long time to run. *) (* Harvey P. Dale, Dec 30 2024 *)

for(n=1,10^9,if(sigma(n)-n==eval(concat(Vecrev(Str(n)))),print1(n,","))) \\ Edward Jiang, Sep 10 2014

a(6) confirmed and a(7) discovered by Giovanni Resta, Dec 12 2013

A075130 Primes with decimal representation 140{{0}10{9}89}.

Original entry on oeis.org

140001089, 1401099989, 14000109989, 140108910989, 1401089109989, 14001089001089, 14010890109989, 140000010891089, 140000109999989, 140010890010989, 140010989001089, 140010989109989, 140108900109989

Offset: 1

Jens Kruse Andersen, Sep 04 2002

57 times one of these primes is a picture-perfect number (see A075131).

			a(4)=140108910989 because this is the 4th prime of the required form.

Sean A. Irvine, Table of n, a(n) for n = 1..1000
Discussion forum, Systematic Undertaking to Find Picture-Perfect Numbers
Joseph Pe, Picture-Perfect Numbers and Other Digit-Reversal Diversions
Rulthan P. Sumicad, On the Picture-Perfect Number, J. Math. Stat. Studies (2023).

A075131(n)=57*a(n)

Cf. A075131, A069942.

A080716 Numbers n such that sum of the divisors of n equals the sum of the reversals of the divisors of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 30, 33, 42, 44, 55, 66, 77, 88, 99, 101, 121, 131, 151, 181, 191, 202, 242, 262, 303, 313, 330, 353, 363, 373, 383, 393, 404, 462, 484, 505, 606, 626, 681, 707, 727, 757, 772, 787, 797, 808, 824, 890, 909, 919, 929, 939, 989, 1111

Offset: 1

Joseph L. Pe, Mar 05 2003

			Sum of divisors of 30: 1+2+3+5+6+10+15+30=72; sum of reversals of divisors of 30: 1+2+3+5+6+1+51+3=72. Therefore 30 belongs to the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..8811 (terms below 10^10, terms 1..300 from Paolo P. Lava)

Cf. A000203, A069192, A069942.

isA080716 := proc(n)
    simplify(A069192(n) = numtheory[sigma](n)) ;
end proc:
for n from 1 to 1000 do
    if isA080716(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Sep 09 2015

rev[n_] := FromDigits[Reverse[IntegerDigits[n]]]; Select[Range[10^4], Apply[Plus, Map[rev, Divisors[ # ]]] == DivisorSigma[1, # ] &]
Select[Range[1200],Total[IntegerReverse/@Divisors[#]]==DivisorSigma[1,#]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 07 2020 *)

A195144 Reversal of n equals the sum of the reversals of the anti-divisors of n.

Original entry on oeis.org

5, 8, 895799

Offset: 1

Paolo P. Lava and Donovan Johnson, Sep 12 2011

Like A069942 but using anti-divisors. No other terms up to 3*10^10.

			The first two terms are banal cases: anti-divisors of 5 are 2 and 3 and their reversals are again 5, 2 and 3 and 2+3 = 5. The same for 8: 3+5 = 8. Anti-divisors of 895799 are 2, 3, 199, 597, 3001, 9003, 597199 and 2+3+991+795+1003+3009+991795 = 997598.

Cf. A066272, A069942.

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(j)
local h,m,n,r;
for m from 3 to j  do
  h:=0;
  for r from 2 to m-1 do
    if abs((m mod r)-r/2)<1 then h:=h+Rev(r); print(r); fi;
  od;
  if h=Rev(m) then print(m); fi;
od;
end:
P(1000000);

A196677 Numbers n such that sum of the divisors of n equals the sum of the reversals of the divisors of n. Numbers with all palindrome divisors are not in the sequence.

Original entry on oeis.org

30, 42, 330, 462, 681, 772, 824, 890, 989, 2180, 3030, 4242, 4542, 4722, 8074, 9775, 17331, 23980, 33330, 35823, 36213, 43263, 46662, 47324, 55805, 62121, 62421, 65301, 65451, 66441, 66741, 68181, 68331, 68631, 68781, 69171, 71215, 71452, 73565, 74391, 74417, 74572, 74972

Offset: 1

Paolo P. Lava, Oct 05 2011

Subset of A080716.
The numbers that are not considered here belong to A062687, numbers all of whose divisors are palindromic. - Michel Marcus, Oct 10 2014
The sequence contains the terms palindromic numbers: 989, 97079, 98789, 99299, 1226221, 1794971, 13488431,…. Divisors(97079) = {1, 193, 503, 97079} and 193 + 503 = 696 = 391 + 305. Divisors(1794971) = {1, 1031, 1741, 1794971} and 1031 + 1741 = 2772 = 1301 + 1471. - Marius A. Burtea, Nov 20 2019

			Divisors of 989 are 1, 23, 43, 989 and 1+23+43+989=1+32+34+989=1056.
Divisors of 8074 are 1, 2, 11, 22, 367, 734, 4037, 8074 and 1+2+11+22+367+734+4037+8074=1+2+11+22+763+437+7304+4708=13248.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (terms 1..57 from R. J. Mathar, 58..1000 from Amiram Eldar)

Cf. A000203, A069192, A069942, A080716.

f:=func; g:=func; [k:k in [1..80000]| g(k) and not forall{d:d in Divisors(k)|f(d)}]; // Marius A. Burtea, Nov 20 2019

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(j)
local h,m,n,ok,p,r,t;
for m from 1 to j  do
  p:=divisors(m); t:=0; ok:=0;
  for r from 1 to nops(p) do t:=t+Rev(p[r]); if p[r]<>Rev(p[r]) then ok:=1; fi;     od;
  if ok=1 and sigma(m)=t then print(m); fi;
od;
end:
P(100000);
# alternative
isA196677 := proc(n)
    isA080716(n) and not isA062687(n) ;
end proc:
n := 1;
for i from 1 do
    if isA196677(i) then
        printf("%d %d\n",n,i) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Sep 09 2015

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

Paolo P. Lava, Jan 20 2012

A066466 is a subsequence of this sequence.

			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.

Dumitru Damian, Table of n, a(n) for n = 1..235 (terms up to 10^9)

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

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

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

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

A075131 Picture-perfect numbers of form 57*p for p in A075130. The decimal reversal is equal to the sum of the reversed proper divisors.

Original entry on oeis.org

7980062073, 79862699373, 798006269373, 7986207926373, 79862079269373, 798062073062073, 798620736269373, 7980000620792073, 7980006269999373, 7980620730626373, 7980626373062073, 7980626379269373, 7986207306269373

Offset: 1

Jens Kruse Andersen, Sep 04 2002

All known picture-perfect numbers are of this form, except the first 4 in A069942. - Jens Kruse Andersen, May 06 2008

			a(4)=57*140108910989=7986207926373.

Discussion forum, Systematic Undertaking to Find Picture-Perfect Numbers
Joseph Pe, Picture-Perfect Numbers and Other Digit-Reversal Diversions

a(n) = 57*A075130(n)

Cf. A075130, A069942.

Edited by Jens Kruse Andersen, May 06 2008

A101701 Numbers n such that n = sum of the reversals of divisors of n.

Original entry on oeis.org

1, 207321, 890827, 7591023, 18368601, 4885292403

Offset: 1

Farideh Firoozbakht, Dec 23 2004

a(7) > 10^11. - Donovan Johnson, Dec 27 2013

			18368601 is in the sequence because divisors of 18368601 are 1, 3, 6122867, 18368601 and 18368601 = 1 + 3 + 7682216 + 10686381.

Cf. A069942, A072228.

Do[h = Divisors[n]; l = Length[h]; If[n == Sum[ FromDigits[Reverse[IntegerDigits[h[[k]]]]], {k, l}], Print[n]], {n, 370000000}]

from sympy import divisors
A101701_list = [n for n in range(1,10**6) if n == sum([int(d) for d in (str(x)[::-1] for x in divisors(n))])]
# Chai Wah Wu, Dec 06 2014

a(6) from Donovan Johnson, Dec 07 2008

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

Paolo P. Lava, Jan 20 2012

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

			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.

Cf. A069942, A195144, A203616.

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);

Select[Range[33*10^8],Total[IntegerReverse/@Divisors[#]] == IntegerReverse[ DivisorSigma[ 1,#]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 09 2018 *)

a(13)-a(16) from Donovan Johnson, Jan 29 2012

a(17)-a(19) from Giovanni Resta, Aug 30 2018

cp's OEIS Frontend

A072228 Numbers k such that k equals the sum of the reverses of the proper divisors of k.

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A072234 Numbers k such that reverse(k) = sum of the proper divisors of k.

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A075130 Primes with decimal representation 140{{0}10{9}89}.

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A080716 Numbers n such that sum of the divisors of n equals the sum of the reversals of the divisors of n.

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Maple

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A195144 Reversal of n equals the sum of the reversals of the anti-divisors of n.

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A196677 Numbers n such that sum of the divisors of n equals the sum of the reversals of the divisors of n. Numbers with all palindrome divisors are not in the sequence.

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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.

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A075131 Picture-perfect numbers of form 57*p for p in A075130. The decimal reversal is equal to the sum of the reversed proper divisors.

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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.