A076750 -id:A076750 - cp's OEIS frontend

A325148 Squares which can be expressed as the product of a number and its reversal.

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 400, 484, 900, 1089, 1600, 1936, 2500, 3025, 3600, 4356, 4900, 5929, 6400, 7744, 8100, 9801, 10000, 10201, 12100, 12321, 14641, 17161, 19881, 22801, 25921, 29241, 32761, 36481, 40000, 40804, 44944, 48400, 49284, 53824, 58564, 63504, 68644, 73984, 79524, 85264

Offset: 1

Bernard Schott, Apr 03 2019

The numbers k such that k * rev(k) is a square are in A306273.
The squares of palindromes of A014186 are a subsequence.
The square roots of the first 65 terms of this sequence (from 0 to 160000) are exactly the first 65 terms of A061917. Then a(66) = 162409 = 403^2 and the non-palindrome 403 is the first term of another sequence A325151.

			Zero ways: 169 = 13^2 cannot be equal to k * rev(k).
One way: 400 = 200 * 2; 10201 = 101 * 101; 162409 = 169 * 961.
Two ways: 7683984 = 2772 * 2772 = 1584 * 4851.
Three ways: 6350400 = 14400 * 441 = 25200 * 252 = 44100 * 144.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms for n = 1..231 from R. J. Mathar)
Bernard Schott, The different ways

Equals A325149 Union A083408.
Cf. A325149 (only one way), A083408 (at least two ways). A325150 (exactly two ways), A307019 (exactly three ways).
Subsequences: A014186 (square of palindromes), A076750 (product of a non-palindrome and its reversal, where leading zeros are not allowed).
Cf. A061917, A325151 (some square roots of this sequence).

isA305231 := proc(n)
    local d;
    for d in numtheory[divisors](n) do
        if d = digrev(n/d) then
            return true ;
        end if;
    end do:
    false ;
end proc:
n := 1;
for i from 0 to 4000 do
    i2 := i^2 ;
    if isA305231(i2) then
        printf("%d %d\n",n,i2) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 09 2019

{0}~Join~Select[Range[10^3]^2,(d1=Select[Divisors[n=#],#<=Sqrt@n&];Or@@Table[d1[[k]]==(IntegerReverse/@(n/d1))[[k]],{k,Length@d1}])&] (* Giorgos Kalogeropoulos, Jun 09 2021 *)

from sympy import divisors
A325148_list = [0]
for n in range(10**6):
    n2 = n**2
    for m in divisors(n2):
        if m > n:
            break
        if m == int(str(n2//m)[::-1]):
            A325148_list.append(n2)
            break # Chai Wah Wu, Jun 09 2021

Intersection of A305231 and A000290. - R. J. Mathar, Aug 09 2019

Definition corrected by N. J. A. Sloane, Aug 01 2019

A325149 Squares which can be expressed as the product of a number and its reverse in exactly one way.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 400, 484, 900, 1089, 1600, 1936, 2500, 3025, 3600, 4356, 4900, 5929, 6400, 7744, 8100, 9801, 10000, 10201, 12100, 12321, 14641, 17161, 19881, 22801, 25921, 29241, 32761, 36481, 40000, 40804, 44944, 48400, 49284, 53824, 58564, 68644, 73984, 79524, 85264, 90000

Offset: 1

Bernard Schott, Apr 03 2019

The first 47 terms of this sequence (from 0 to 58564) are identical to the first 47 terms of A325148. The square 63504 is not present because it can be expressed in two ways: 63504 = 252 * 252 = 144 * 441.
There are three families of squares in this sequence:
1) Squares of palindromes in A002113\A117281.
2) Squares of non-palindromes which form the sequence A325151.
These squares are a subsequence of A076750.
3) Squares of (m*10^q) with q >= 1 and m palindrome in A002113\A117281.

			For each family:
1) Square of palindromes: 53824 = 232^2 = 232 * 232.
2) Square of non-palindromes m^2 = k*rev(k) with k and rev(k) which have the same number of digits: 162409 = 403^2 = 169 * 961.
3) Square ends with zeros: 48400 = 220^2 = 2200 * 22.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition, p. 168.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A325148 (at least one way), A083408 (at least two ways), A325150 (exactly two ways), A307019 (exactly three ways).

Cf. A014186 (squares of palindromes), A076750.

a(52) corrected by Chai Wah Wu, Apr 11 2019

Definition corrected by N. J. A. Sloane, Aug 01 2019

A325151 Non-palindromes numbers not ending in 0 whose square is the product of a number and its reverse in only one way.

Original entry on oeis.org

403, 504, 816, 3267, 4356, 22932, 31003, 32967, 35143, 41004, 43956, 45864, 48616, 55242, 58422, 66976, 75525, 329967, 341033, 403403, 439956, 451044, 504504, 784806, 816816, 1341331, 2341332, 2452442, 2480742, 2570652, 2749572, 3010003, 3141313, 3299967, 3321133, 3462543, 4010004, 4030403, 4035297, 4252424, 4399956

Offset: 1

Bernard Schott, Apr 05 2019

The squares of these non-palindromes numbers form the second family of the terms of A325149, they form a subsequence of A076750.

The squares of these non-palindromes numbers generate also a family of terms of A325150 by this way: 4030^2 = 16900 * 961 = 96100 * 169.

			403^2 = 169 * 961.

Chai Wah Wu, Table of n, a(n) for n = 1..576

Subsequence of A207373.

Definition and terms corrected by Chai Wah Wu, Apr 12 2019

A072443 Nonsquares which are the product of two numbers with the same digits (leading zeros are forbidden).

Original entry on oeis.org

252, 403, 574, 736, 765, 976, 1008, 1207, 1300, 1458, 1462, 1612, 1729, 1855, 1944, 2268, 2296, 2430, 2668, 2701, 2944, 3154, 3478, 3627, 3640, 4032, 4275, 4606, 4930, 5092, 5605, 5848, 6624, 6786, 7663, 8722, 11110, 12240, 13390, 13552, 14560, 14803, 15750, 16074

Offset: 1

N. J. A. Sloane, Nov 11 2002

			12*21 = 252 = 12*21, 403 = 13*31, 574 = 14*41, etc

P. Vaderlind, R. K. Guy and L. C. Larsen, The Inquisitive Problem Solver, Math. Assoc. Am., 2002, Problem P185.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

A077760 is a subsequence.

Cf. A076750, A129623.

{for(n=100,15000,k=floor(log(n)/log(100)); f=divisors(n); v=[]; for(h=1,matsize(f)[2], if(10^k1, w=[]; for(i=1,b,s=[]; a=v[i]; while(a>0,d=divrem(a,10); a=d[1]; s=concat(d[2],s)); w=concat(w,[vecsort(s)])); c=0; for(i=1,b-1, for(j=i+1,b,if(c<1&&w[i]==w[j],if(v[i]*v[j]==n,print1(n,","); c=1))))))}

from math import isqrt
from sympy import divisors
def ok(n): return isqrt(n)**2Michael S. Branicky, Sep 08 2024

Extended by Klaus Brockhaus, Nov 12 2002

a(42) and beyond from Michael S. Branicky, Sep 08 2024

A129623 Numbers which are the product of a non-palindrome and its reversal, where leading zeros are not allowed.

Original entry on oeis.org

252, 403, 574, 736, 765, 976, 1008, 1207, 1300, 1458, 1462, 1612, 1729, 1855, 1944, 2268, 2296, 2430, 2668, 2701, 2944, 3154, 3478, 3627, 3640, 4032, 4275, 4606, 4930, 5092, 5605, 5848, 6624, 6786, 7663, 8722, 20502, 23632, 26962, 30492, 31003, 34222

Offset: 1

Tanya Khovanova, May 30 2007

The smallest square in this sequence is 63504 = 252*252 = 144*441.

			252 = 12*21.

Michael S. Branicky, Table of n, a(n) for n = 1..11195 (terms < 10^9)

Cf. A072443, A076750.

Take[Union[ Transpose[ Select[Table[{n, n* FromDigits[Reverse[IntegerDigits[n]]]}, {n, 1000}], Mod[ #[[1]], 10] != 0 && #[[1]] != FromDigits[Reverse[IntegerDigits[ #[[1]]]]] &]][[2]]], 100]
upto2ndigits@n_ := Union@(If[(i = IntegerReverse@#) > #, i*#, Nothing] & /@Range@(10^n - 1)); upto2ndigits@3 (* Hans Rudolf Widmer, Sep 06 2024 *)

from sympy import divisors
def ok(n): return any(n==d*int(s[::-1]) for d in divisors(n)[1:-1] if (s:=str(d))!=s[::-1] and s[-1]!="0")
print([k for k in range(36000) if ok(k)]) # Michael S. Branicky, Sep 07 2024

# instantly generates 44185 terms with n = 5
def aupto2ndigits(n): return(sorted(set(i*int(s[::-1]) for i in range(12, 10**n) if i%10 != 0 and (s:=str(i)) != s[::-1])))
print(aupto2ndigits(2))
# Michael S. Branicky, Sep 08 2024 after Hans Rudolf Widmer

Offset corrected by Stefano Spezia, Sep 07 2024

A207373 Numbers whose square is the product of a number and its reverse.

Original entry on oeis.org

252, 403, 504, 660, 816, 2772, 3267, 4356, 20502, 22932, 23632, 25452, 26962, 27972, 31003, 32967, 35143, 41004, 43956, 45864, 48384, 48616, 55242, 58422, 66976, 75525, 225522, 252252, 259952, 279972, 329967, 341033, 403403, 439956, 451044, 504504, 619916

Offset: 1

Carmine Suriano, Feb 17 2012

Number and its reverse must have the same number of digits.

Number squared cannot be a palindrome. - Harvey P. Dale, Mar 12 2017

			35143^2 = 96721*12769.

Giovanni Resta, Table of n, a(n) for n = 1..1169 (terms < 10^12, first 90 terms from Harvey P. Dale)

Cf. A076750.

nir[n_]:=If[PalindromeQ[n]||Divisible[n,10],0,n IntegerReverse[n]]; Sqrt[#] &/@ Select[Array[nir,500000],#!=0&&IntegerQ[Sqrt[#]]&]//Union (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 12 2017 *)

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A325148 Squares which can be expressed as the product of a number and its reversal.

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A325149 Squares which can be expressed as the product of a number and its reverse in exactly one way.

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A325151 Non-palindromes numbers not ending in 0 whose square is the product of a number and its reverse in only one way.

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A072443 Nonsquares which are the product of two numbers with the same digits (leading zeros are forbidden).

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A129623 Numbers which are the product of a non-palindrome and its reversal, where leading zeros are not allowed.

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A207373 Numbers whose square is the product of a number and its reverse.

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