A265737 -id:A265737 - cp's OEIS frontend

A281555 Consider any concatenation of the type n = concat(a,b). Sequence lists numbers whose reverses are the sum of the products of some of such couples a and b.

351, 621, 886, 1641, 1901, 2401, 4122, 4322, 5101, 6568, 23913, 30591, 46649, 60291, 90877, 199001, 216322, 223471, 237391, 297498, 394391, 405278, 420552, 425322, 430762, 456478, 470149, 546649, 861226, 910001, 920781, 1740821, 2008541, 2329876, 2348812, 2414722

Offset: 1

Paolo P. Lava, Jan 24 2017

			3*51 = 153;
6*21 = 126;
8*86 = 688;
1*641 + 16*41 + 164*1 = 1461;
1*901 + 190*1 = 1091.

Cf. A265737, A267939.

with(combinat): with(numtheory): T:=proc(w) local x, y, z; x:=w; y:=0;
for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end:
P:=proc(q) local a, j, k,n; for n from 1 to q do a:={};
for k from 1 to ilog10(n) do a:=a union {(n mod 10^k)*trunc(n/10^k)}; od; a:=choose(a);
for k from 2 to nops(a) do if T(n)=add(a[k][j], j=1..nops(a[k])) then print(n); break; fi; od;
od; end: P(10^9);

Select[Range[10^5], Function[w, MemberQ[Total /@ Rest@ Subsets@ Map[Times @@ Map[FromDigits@ # &, TakeDrop[w, #]] &, Range[Length@ w - 1]], FromDigits@ Reverse@ w]]@ IntegerDigits@ # &] (* Michael De Vlieger, Jan 24 2017, Version 10.3 *)

A363422 Numbers k which satisfy k = concat(a,b,...) and ab... = reverse(k), for some two or more a,b,...

Original entry on oeis.org

351, 621, 886, 5931, 86673, 97533, 425322, 430762, 920781, 3524751, 4495491, 4834872, 5594151, 5941971, 6218001, 6801381, 6916671, 8630841, 32331001, 44235301, 57982563, 67968432, 68577483, 69617484, 71673981, 88873491, 89943354, 119910901, 338752611

Offset: 1

David L. Reens, Jun 01 2023

k > reverse(k) for all terms, sometimes narrowly, see a(28) = 119910901.
This is easily shown: c=concat(a,b), c/a > (c-b)/a = 10^(#digits of b) > b; c > b*a.
Follows for triple or higher concatenations by induction.
Of the first 39 terms, 12 arise due to concatenations of only two numbers and are therefore also present in A281555.
No terms yet found with a product of more than five numbers.
Sometimes a term B relates to an earlier term A via a particular number N for which B=concat(A,N) and reverse(B)=reverse(A)*reverse(N). This is true of B=a(15), A=a(2), and N=8001 for example.

			153 = 3*51.
1395 = 5*9*31.
1945944 = 44*9*54*91.
1008126 = 6*21*8001.
171548496 = 6*94*84*51*71.

Cf. A267939, A281555, A265737.

A267939 is contained in the intersection of this sequence and A281555.

# Find numbers with a de-concatenation that multiplies to their reverse.
import math
def digits(x):
    y = []
    while x>0:
        y = [x%10] + y
        x//=10
    return y
def check(x):
    xx = digits(x)
    if xx[0] < xx[-1]:
        return
    for i in range(1,2**(len(xx)-1)):
        for dnum,digit in enumerate(xx):
            if dnum==0:
                thisProd = [xx[0]]
            elif i&(2**(dnum-1)):
                if digit==0:
                    break
                thisProd += [digit]
            else:
                thisProd[-1] = thisProd[-1]*10+digit
        answer = math.prod(thisProd)
        if not answer%10==xx[0]:
            continue
        if digits(answer)[-1::-1]==xx:
            print('\r'+str(thisProd).replace(', ','x')[1:-1])
            return
    return
i=0
while True:
    i += 1
    if not i%10000:
        print('\r'+str(i),end='')
    check(i)

cp's OEIS Frontend

A281555 Consider any concatenation of the type n = concat(a,b). Sequence lists numbers whose reverses are the sum of the products of some of such couples a and b.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

A363422 Numbers k which satisfy k = concat(a,b,...) and ab... = reverse(k), for some two or more a,b,...

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

cp's OEIS Frontend

A281555 Consider any concatenation of the type n = concat(a,b). Sequence lists numbers whose reverses are the sum of the products of some of such couples a and b.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

A363422 Numbers k which satisfy k = concat(a,b,...) and a*b*... = reverse(k), for some two or more a,b,...

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python

A363422 Numbers k which satisfy k = concat(a,b,...) and ab... = reverse(k), for some two or more a,b,...