A050782 -id:A050782 - cp's OEIS frontend

A050810 From sequence A050782, the n's corresponding to the first occurrence of m (or 0 if not defined).

0, 1, 106, 37, 53, 103, 42, 23, 29, 19, 0, 56, 21, 38, 18, 35, 17, 16, 14, 26, 0, 12, 96, 43, 913, 2081, 812, 703, 754, 24, 0, 165, 726, 64, 603, 15, 592, 27, 13, 479, 0, 261, 541, 277, 48, 1129, 437, 649, 524, 538, 0, 291, 406, 1359, 533, 93, 377, 466, 1052, 279, 0

Offset: 0

Patrick De Geest, Oct 15 1999

Cf. A002113, A050782.

nmax = 60; Clear[f]; f[max_] := f[max] = (r = Reap[For[n = 1, n <= max, n++, k = 1; If[IntegerQ[n/10], m = 0, While[k <= max && Reverse[id = IntegerDigits[k*n]] != id, k++]; m = k]; Sow[{n, m}]]][[2, 1]]; a[0] = 0; a[n_] := (s = Select[r, #[[2]] == n &, 1]; If[s == {}, 0, s[[1, 1]]]); Table[a[n], {n, 0, nmax}]); f[nmax]; f[max = 2 nmax]; While[Print["max = ", max]; f[max] != f[max/2], max = 2 max]; A050810 = f[max] (* Jean-François Alcover, Dec 31 2015 *)

A307278 Record values in A050782.

Original entry on oeis.org

0, 1, 21, 38, 66, 518, 845, 12345679, 172839506, 445372913, 1173450678278939, 111222333444555666777889

Offset: 1

N. J. A. Sloane, Apr 05 2019

Cf. A020485, A050782, A307279.

a(11)-a(12) from Giovanni Resta, Oct 15 2022

A307279 Indices of record values in A050782.

Original entry on oeis.org

0, 1, 12, 13, 32, 54, 71, 81, 162, 8081, 8181, 8991

Offset: 1

N. J. A. Sloane, Apr 05 2019

Cf. A020485, A050782, A307278.

a(11)-a(12) from Giovanni Resta, Oct 15 2022

A020485 Least positive palindromic multiple of n, or 0 if none exists.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 252, 494, 252, 525, 272, 272, 252, 171, 0, 252, 22, 161, 696, 525, 494, 999, 252, 232, 0, 434, 2112, 33, 272, 525, 252, 111, 494, 585, 0, 656, 252, 989, 44, 585, 414, 141, 2112, 343, 0, 969, 676, 212, 27972, 55, 616, 171, 232, 767, 0, 26962

Offset: 0

David W. Wilson

Smallest positive palindrome divisible by n, or 0 if no such palindrome exists (which happens iff n is a multiple of 10). - N. J. A. Sloane, Apr 04 2019

The existence of palindromic multiples is a corollary of the theorem that an arithmetic progression with initial term c and a positive common difference d contains infinitely many palindromic numbers unless both of these numbers are multiples of 10. - M. Harminc (harminc(AT)duro.science.upjs.sk), Jul 14 2000

Giovanni Resta, Table of n, a(n) for n = 0..10000 (first 8181 terms from N. J. A. Sloane)
Ely Golden, Python program for generating terms of this sequence
M. Harminc and R. Sotak, Palindromic numbers in arithmetic progressions, Fibonacci Quarterly Journal, Jun-Jul (1998), pp. 259-262.

Cf. A002113, A050782.

a(n) = n*A050782(n). - Michel Marcus, Jan 22 2019

a(0)=0 added by N. J. A. Sloane, Apr 04 2019

A061816 Obtain m by omitting trailing zeros from n (cf. A004151); a(n) = smallest multiple k*m which is a palindrome.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 252, 494, 252, 525, 272, 272, 252, 171, 2, 252, 22, 161, 696, 525, 494, 999, 252, 232, 3, 434, 2112, 33, 272, 525, 252, 111, 494, 585, 4, 656, 252, 989, 44, 585, 414, 141, 2112, 343, 5, 969, 676, 212, 27972, 55, 616, 171, 232

Offset: 0

Klaus Brockhaus, Jun 25 2001

Every positive integer is a factor of a palindrome, unless it is a multiple of 10 (D. G. Radcliffe, see links).

Every integer n has a multiple of the form 99...9900...00. To see that n has a multiple that's a palindrome (allowing 0's on the left) with even digits, let 9n divide 99...9900...00; then n divides 22...2200...00. - Dean Hickerson, Jun 29 2001

			For n = 30 we have m = 3, 1*m = 3 is a palindrome, so a(30) = 3. For n = m = 12 the smallest palindromic multiple is 21*m = 252, so a(12) = 252.

P. De Geest, Smallest multipliers to make a number palindromic.

Cf. A050782, A062293, A061915, A061916. Values of k are given in A061906.

stop := 200000; for n := 0 to maxarg do k := 1; test := true; while test and k < stop do mp := omit_trailzeros(n)*k; if test := mp <> int_reverse(mp) then inc(k); end; end; if k < stop then write(mp," "); else write(-1," "); end; end;

A062279 Smallest multiple k*n of n which is a palindrome or becomes a palindrome when 0's are added on the left (e.g. 10 becomes 010 which is a palindrome).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 60, 494, 70, 30, 80, 272, 90, 171, 20, 252, 22, 161, 600, 50, 494, 999, 252, 232, 30, 434, 800, 33, 272, 70, 252, 111, 494, 585, 40, 656, 252, 989, 44, 90, 414, 141, 2112, 343, 50, 969, 676, 212, 9990, 55, 616, 171, 232, 767

Offset: 0

Amarnath Murthy, Jun 17 2001

Every positive integer is a factor of a palindrome, unless it is a multiple of 10 (D. G. Radcliffe, see Links).

			a(13) = 494 is the smallest multiple of 13 which is a palindrome.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
P. De Geest, Smallest multipliers to make a number palindromic.

Cf. A050782, A062293. Values of k are given in A061674.

Cf. A141709.

: maxarg := 60; stop := 200000; for n := 0 to maxarg do k := 1; test := true; while test and k < stop do m := omit_trailzeros(n*k); if test := m <> int_reverse(m) then inc(k); end; end; if k < stop then write(n*k," "); else write(-1," "); end; end;

a062279 0 = 0
a062279 n = until ((== 1) . a136522 . a004151) (+ n) n
-- Reinhard Zumkeller, May 06 2013

A136522(A004151(a(n))) = 1. - Reinhard Zumkeller, May 06 2013

Corrected and extended by Larry Reeves (larryr(AT)acm.org) and Klaus Brockhaus, Jun 18 2001

A061906 Obtain m by omitting trailing zeros from n; a(n) = smallest k such that k*m is a palindrome.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 21, 38, 18, 35, 17, 16, 14, 9, 1, 12, 1, 7, 29, 21, 19, 37, 9, 8, 1, 14, 66, 1, 8, 15, 7, 3, 13, 15, 1, 16, 6, 23, 1, 13, 9, 3, 44, 7, 1, 19, 13, 4, 518, 1, 11, 3, 4, 13, 1, 442, 7, 4, 33, 9, 1, 11, 4, 6, 1, 845, 88, 4, 3, 7, 287, 1, 11, 6, 1, 12345679, 8

Offset: 0

Klaus Brockhaus, Jun 25 2001

Every positive integer is a factor of a palindrome, unless it is a multiple of 10 (D. G. Radcliffe, see Links).

Every integer n has a multiple of the form 99...9900...00. To see that n has a multiple that's a palindrome (allowing 0's on the left) with even digits, let 9n divide 99...9900...00; then n divides 22...2200...00. - Dean Hickerson, Jun 29 2001

			For n = 30 we have m = 3, 1*m = 3 is a palindrome, so a(30) = 1. For n = m = 12 the smallest palindromic multiple is 21*m = 252, so a(12) = 21.

Chai Wah Wu, Table of n, a(n) for n = 0..8180
P. De Geest, Smallest multipliers to make a number palindromic.

Cf. A050782, A062293, A061915, A061916, A061816. Values of k*m are given in A061906.

stop := 20000000; for n := 0 to maxarg do k := 1; test := true; while test and k < stop do mp := omit_trailzeros(n)*k; if test := mp <> int_reverse(mp) then inc(k); end; end; if k < stop then write(k," "); else write(-1," "); end; end;

skp[n_]:=Module[{m=n/10^IntegerExponent[n,10],k=1},While[!PalindromeQ[k*m],k++];k]; Array[ skp,90,0] (* Harvey P. Dale, Jul 04 2024 *)

from _future_ import division
def palgen(l, b=10): # generator of palindromes in base b of length <= 2*l
    if l > 0:
        yield 0
        for x in range(1, l+1):
            n = b**(x-1)
            n2 = n*b
            for y in range(n, n2):
                k, m = y//b, 0
                while k >= b:
                    k, r = divmod(k, b)
                    m = b*m + r
                yield y*n + b*m + k
            for y in range(n, n2):
                k, m = y, 0
                while k >= b:
                    k, r = divmod(k, b)
                    m = b*m + r
                yield y*n2 + b*m + k
def A050782(n, l=10):
    if n % 10:
        x = palgen(l)
        next(x)  # replace with x.next() in Python 2.x
        for i in x:
            q, r = divmod(i, n)
            if not r:
                return q
        else:
            return 'search limit reached.'
    else:
        return 0
def A061906(n, l=10):
    return A050782(int(str(n).rstrip('0')),l) if n > 0 else 1
# Chai Wah Wu, Dec 30 2014

A141707 Least k>0 such that (2n-1)k is palindromic in base 2.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 5, 1, 1, 27, 1, 89, 13, 1, 49, 1, 1, 13, 69, 5, 25, 3, 1, 103, 29, 1, 63, 3, 9, 103, 7, 1, 1, 19, 37, 147, 1, 13, 3, 19, 11, 45, 1, 37, 23, 3, 1, 27, 61, 1, 233, 47, 13, 1, 21, 23, 59, 525, 5, 1, 93, 23, 41, 1, 1, 49, 27, 13, 187, 87, 269, 15, 111, 13, 29, 7, 1, 13, 3

Offset: 1

M. F. Hasler, Jul 17 2008

Even numbers cannot be palindromic in base 2 (unless leading zeros are considered), that's why we search only for odd numbers 2n-1 the k-values such that k(2n-1) is palindromic in base 2. Obviously they are necessarily also odd.

a(A044051(n)) = 1. - Reinhard Zumkeller, Apr 20 2015

			a(1..5)=1 since 1,3,5,7,9 are already palindromic in base 2.
a(6)=3 since 2*6-1=11 and 2*11=22 are not palindromic in base 2, but 3*11=33 is.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A050782, A141708, A178225, A044051.

a141707 n = head [k | k <- [1, 3 ..], a178225 (k * (2 * n - 1)) == 1]
-- Reinhard Zumkeller, Apr 20 2015

lkp[n_]:=Module[{k=1,n2=2n-1},While[IntegerDigits[k*n2,2]!= Reverse[ IntegerDigits[ k*n2,2]],k++];k]; Array[lkp,80] (* Harvey P. Dale, Mar 19 2016 *)

A141707(n,L=10^9)={ n=2*n-1; forstep(k=1,L,2, binary(k*n)-vecextract(binary(k*n),"-1..1") || return(k))}

def binpal(n): b = bin(n)[2:]; return b == b[::-1]
def a(n):
    m = 2*n - 1
    km = m
    while not binpal(km): km += m
    return km//m
print([a(n) for n in range(1, 80)]) # Michael S. Branicky, Mar 20 2022

A141709 Least positive multiple of n which is palindromic in base 2, allowing for leading zeros (or: ignoring trailing zeros).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 33, 12, 65, 14, 15, 16, 17, 18, 513, 20, 21, 66, 2047, 24, 325, 130, 27, 28, 1421, 30, 31, 32, 33, 34, 455, 36, 2553, 1026, 195, 40, 1025, 42, 129, 132, 45, 4094, 4841, 48, 1421, 650, 51, 260, 3339, 54, 165, 56, 513, 2842, 6077, 60, 427, 62

Offset: 1

M. F. Hasler, Jul 17 2008

Even numbers cannot be palindromic in base 2, unless leading zeros are considered (or, equivalently, resp. more precisely, trailing zeros are discarded). This is done in this version of A141708, which therefore does not need to be restricted to odd n as it has been done for A141707 and A141708.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A050782, A141707-A141708, A062279, A203070, A000265, A178225.

a141709 n = until ((== 1) . a178225 . a000265) (+ n) n
-- Reinhard Zumkeller, Nov 06 2012

notpalbinQ[i_]:=Module[{id=IntegerDigits[i,2]},While[Last[id]==0,id=Most[id]];id!= Reverse[id]]; lm[n_]:=Module[{k=1},While[notpalbinQ[k n],k++];k n]; Array[lm,70] (* Harvey P. Dale, Dec 28 2011 *)

A141709(n)=forstep(k=n,10^9,n,vecextract(t=binary(k>>valuation(k,2)),"-1..1")-t || return(k))

A178225(A000265(a(n))) = 1. - Reinhard Zumkeller, Nov 06 2012

A190301 Smallest number h such that n*h is a repunit (A002275), or 0 if no such h exists.

Original entry on oeis.org

1, 0, 37, 0, 0, 0, 15873, 0, 12345679, 0, 1, 0, 8547, 0, 0, 0, 65359477124183, 0, 5847953216374269, 0, 5291, 0, 48309178743961352657, 0, 0, 0, 4115226337448559670781893, 0, 38314176245210727969348659, 0, 3584229390681, 0, 3367, 0, 0, 0, 3, 0, 2849, 0, 271, 0

Offset: 1

Jaroslav Krizek, May 07 2011

nonn

			For n = 7: a(7) = 15873 because 7 * 15873 = 111111. Repunit 111111 is the smallest repunit with prime factor 7.

Robert G. Wilson v, Table of n, a(n) for n = 1..1001 (first 300 terms from T. D. Noe)

Cf. A084681 (repunit length), A216479 (the repunit).
Cf. A050782 = the smallest number h such that n*h is palindromic number, A083117 = the smallest number h such that n*h is repdigit number.
Cf. A004290, A079339, A181060, A181061.

Table[If[GCD[n, 10] > 1, 0, k = MultiplicativeOrder[10, 9*n]; (10^k - 1)/(9*n)], {n, 100}] (* T. D. Noe, May 08 2011 *)

a(n)=if(gcd(n,10)>1, 0, (10^znorder(Mod(10,9*n))-1)/9/n) \\ Charles R Greathouse IV, Aug 28 2016

cp's OEIS Frontend

A050810 From sequence A050782, the n's corresponding to the first occurrence of m (or 0 if not defined).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A307278 Record values in A050782.

Views

Author

Keywords

Crossrefs

Extensions

A307279 Indices of record values in A050782.

Views

Author

Keywords

Crossrefs

Extensions

A020485 Least positive palindromic multiple of n, or 0 if none exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A061816 Obtain m by omitting trailing zeros from n (cf. A004151); a(n) = smallest multiple k*m which is a palindrome.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

ARIBAS

A062279 Smallest multiple k*n of n which is a palindrome or becomes a palindrome when 0's are added on the left (e.g. 10 becomes 010 which is a palindrome).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

ARIBAS

Haskell

Formula

Extensions

A061906 Obtain m by omitting trailing zeros from n; a(n) = smallest k such that k*m is a palindrome.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

ARIBAS

Mathematica

Python

A141707 Least k>0 such that (2n-1)k is palindromic in base 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Python

A141709 Least positive multiple of n which is palindromic in base 2, allowing for leading zeros (or: ignoring trailing zeros).

Views

Author