A262743 -id:A262743 - cp's OEIS frontend

A262873 Predestined numbers A262743 in which every term is generated by at least one pair of products where all (and only those) first product's factor's digits are, in reverse order, the same as those of the second two factors.

504, 756, 806, 868, 1008, 1148, 1176, 1209, 1472, 1475, 1512, 1638, 1708, 2016, 2184, 2208, 2418, 2548, 2730, 2772, 2924, 3024, 3388, 4416

Offset: 1

Francesco Di Matteo, Oct 03 2015

In this sequence, the position of the multiplication sign in the reversed order is irrelevant, so, e.g., 11088 (48*231 and 132*84), 1176 (4*294 and 49*24) and 2548 (4*637 and 7*364) are in the sequence.

			504  = 12*42 = 24*21;
756  = 12*63 = 36*21;
806  = 13*62 = 26*31;
868  = 4*217 = 7*124;
1008 = 12*84 = 48*21;
1148 = 14*82 = 28*41;
1176 = 4*294 = 49*24, etc.

Francesco Di Matteo, Sequenze ludiche, Game Edizioni (2015), page 34.

Francesco Di Matteo, Table of n, a(n) for n = 1..132
Algebra.com, Question 265287
A. Marchini and F. Di Matteo, All the first 132 terms calculated
Math Forum at Drexel, Reversing the Digits

Subsequence of A262743.

Cf. A228164 (contains only symmetrical digits' factors)

A262989 Predestined numbers A262743 generated from at least a pair of products in which, for each product, all digits 0 through 9 are used, and each digit appears exactly once.

Original entry on oeis.org

248665082, 248695370, 249063875, 253674980, 256175640, 257930648, 257938064, 260577504, 260817480, 263987504, 264713960, 267766632, 267953048, 269037548, 269045192, 269174192, 269307584, 269735900, 269937500

Offset: 1

Francesco Di Matteo, Oct 06 2015

Sequence obtained using the A050278 sequence of pandigitals numbers "over" the A262743 sequence of predestined numbers.
Pandigital numbers are numbers containing the digits 0 through 9 (in this case Version 1: each digit appears exactly once).
This is a finite sequence: first term is 248665082 (106*2345897 and 2378*104569) and last term is 8282993378 (853*9710426 and 8503*974126).
The sequence contains 95009 terms. - Giovanni Resta, Oct 07 2015

			248665082 = 106*2345897 and 2378*104569;
248695370 = 10*24869537 and 1045*237986, 1045*237986 and 1*248695370;
249063875 = 2375*104869 and 1*249063875;
...
8270423667 = 87*95062341 and 957*8642031;
8271362484 = 957*8643012 and 8526*970134;
8282993378 = 853*9710426 and 8503*974126.

Francesco Di Matteo, Sequenze ludiche, Game Edizioni (2015), page 37.

Francesco Di Matteo, Table of n, a(n) for n = 1..19
Andrea Marchini, The first 19 terms calculated
Andrea Marchini, The last 23 terms calculated
Andrea Marchini, All the 95009 terms with product pairs, Oct 10 2015.

Cf. A050278, A262743.

good[w_]:=Block[{L={}}, Do[If[ Length[ Select[ Join[w[[i]], w[[j]]], Mod[#,10]==0&]]<=1,AppendTo[L, {w[[i]], w[[j]]}]], {i, Length@w}, {j, i-1}]; L]; f[w_]:=Select[ Table[ FromDigits/@ {Take[w, i], Take[w, i-10]}, {i, 5}], #[[1]] <= #[[2]] && IntegerLength[#[[1]]] + IntegerLength[ #[[2]]] == 10&]; p = Select[ Permutations@ Range[0, 9], First[#] > 0&]; t = SplitBy[ Sort[{ Times@@ #, #} &/@ Flatten[ f/@ p, 1]], First]; u = Select[ (Last/@ #) &/@ Select[t, Length[#] > 1&], good[#] != {} &]; seq = Union[ Times @@@ Flatten[u, 1]]; Length@ seq (* Giovanni Resta, Oct 07 2015 *)

A228164 Numbers n having at least two distinct symmetrical pairs of divisors (a, b) and (b', a') such that n = ab = b'a' with a' = reverse(a) and b' = reverse(b).

Original entry on oeis.org

504, 756, 806, 1008, 1148, 1209, 1472, 1512, 2016, 2208, 2418, 2772, 2924, 3024, 4416, 4433, 5544, 6314, 8096, 8316, 8415, 8866, 10736, 11088, 12628, 13277, 13299, 14300, 16038, 16082, 16192, 16632, 17732, 20405, 21384, 22176, 24288, 24948, 25452, 26598, 26730

Offset: 1

Michel Lagneau, Aug 17 2013

A pair of integers (a, b) is symmetrical for multiplication when the product a*b is the same as the product b'*a' where a' = reverse(a) and b' = reverse(b). A double pair shows a symmetrical structure, for example:
23*64 = 46*32;
42*36 = 63*24;
21*36 = 63*12;
21*48 = 84*12;
31*26 = 62*13.
Because it is possible to obtain a number of double pairs equal to 1, 2, 3, ... we introduce the notion of "symmetrical order" denoted So(n) for each number n of the sequence corresponding to the number of double pairs.
The numbers of the sequence n = 50904, 55944, 76356, 81406, 83916, ... generate two double pairs of the form (a, b) and (b', a'), (c, d) and (d', c') such that n = a*b = b'*a' with a' = reverse(a) and b' = reverse(b) and n = c*d = d'*c' with c' = reverse(c) and d' = reverse(d). Hence So(50904) = 2, So(55944) = 2, ...
The number n = 101808 implies So(n) = 3 because this number generates 3 double couples (see the example below).
The sequence shows primitive and nonprimitive values: for example n = 504, 756, 806, ... are primitive values, but n = 1008 = 2*504, 1512 = 2*756, 2016 = 4*504, ... are not primitive values. A primitive number contains a couple of divisors (a, b) where a (and/or) b has decimal digits less than 5.

			504 is in the sequence because the two pairs of divisors (42, 12) and (21, 24) have the property 42*12 = 21*24 = 504 with 42 = reverse(24) and 12 = reverse(21).
50904 is in the sequence because we obtain two double pairs of divisors: (12, 4242) and (2424, 21), (42, 1212) and (2121, 24);
101808 is in the sequence because we obtain three double pairs of divisors: (12, 8484) and (4848, 21), (24, 4242) and (2424, 42), (48, 2121) and (1212, 84).
From _Michael De Vlieger_, Sep 15 2017: (Start)
First positions of numbers k of symmetrical pairs that appear for a(n) <= 10^7.
k     n     a(n)
----------------
2     1      504
3     4     1008
4    17     5544
6    98   101808
8   274   559944
(End)

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, 2nd Ed. (1997), p. 142.

Michael De Vlieger, Table of n, a(n) for n = 1..1000
Michael De Vlieger, Symmetrical divisor pairs for numbers m in A228164 with 1 <= m <= 10^7.

Cf. A262873 (a subsequence of predestined numbers A262743).

with(numtheory):for n from 2 to 50000 do:x:=divisors(n):n1:=nops(x):ii:=0:for a from 2 to n1-1 while(ii=0) do:m:=n/x[a]:m1:=convert(m,base,10):nn1:=nops(m1): m2:=convert(x[a],base,10):nn2:=nops(m2): s1:=sum('m1[nn1-i+1]*10^(i-1)', 'i'=1..nn1): s2:=sum('m2[nn2-i+1]*10^(i-1)', 'i'=1..nn2):for b from a+1 to n1-1 while(ii=0) do:q:=n/x[b]:if s1=q and s2=x[b] and m<>x[b] then ii:=1:printf(`%d, `,n):else fi:od:od:od:

Select[Range[10^7], Function[n, Count[Rest@ Select[Divisors@ n, # <= Sqrt@ n &], ?(And[IntegerReverse@ # != #, IntegerReverse@ # IntegerReverse[n/#] == n] &)] > 1]] (* _Michael De Vlieger, Oct 09 2015, updated Sep 15 2017 *)

A245386 Numbers in A245385 where P, Q, R, and S are all distinct.

Original entry on oeis.org

164, 195, 265, 498, 1664, 1995, 2665, 4847, 4998, 6545, 7424, 16664, 19995, 21775, 24996, 26665, 43243, 49998, 86486, 148480, 166664, 175150, 199995, 217775, 249996, 266665, 368180, 484847, 499998, 654545, 742424, 1001001, 1081075, 1216216, 1249992, 1297290, 1451850, 1471468, 1481477

Offset: 1

Derek Orr, Jul 20 2014

This sequence does not contain any repdigits, unlike A245385.

			4*84847 = 48484*7 = 339388. Thus 484847 is a member of this sequence.

Cf. A245385, A245364.

Cf. A262743 (predestined numbers).

for n in range(1,10**7):
  s = str(n)
  count = 0
  for i in range(1,len(s)):
    if i != len(s) - i:
      if int(s[:i]) != int(s[len(s)-i:]):
        num = int(s[:i])*int(s[i:])
        if num != 0:
          if num == int(s[:len(s)-i])*int(s[len(s)-i:]):
            count += 1
            break
  if count > 0:
    print(n,end=', ')

cp's OEIS Frontend

A262873 Predestined numbers A262743 in which every term is generated by at least one pair of products where all (and only those) first product's factor's digits are, in reverse order, the same as those of the second two factors.

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A262989 Predestined numbers A262743 generated from at least a pair of products in which, for each product, all digits 0 through 9 are used, and each digit appears exactly once.

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Mathematica

A228164 Numbers n having at least two distinct symmetrical pairs of divisors (a, b) and (b', a') such that n = a*b = b'*a' with a' = reverse(a) and b' = reverse(b).

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Maple

Mathematica

A245386 Numbers in A245385 where P, Q, R, and S are all distinct.

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Python

A228164 Numbers n having at least two distinct symmetrical pairs of divisors (a, b) and (b', a') such that n = ab = b'a' with a' = reverse(a) and b' = reverse(b).