1,60

First differences of A262065.

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

Eric Weisstein's World of Mathematics, Palindromic Number

Eric Weisstein's World of Mathematics, Sexagesimal

Wikipedia, Palindromic_number

Wikipedia, Sexagesimal

Index entries for sequences related to palindromes

a(n) = 1 for n = 1..59, as the first 60 sexagesimal palindromes are 0..59;

a(60) = (1*60^1 + 1*60^0) - 59*(60^0) = 61 - 59 = 2;

a(61) = (2*60^1 + 2*60^0) - (1*60^1+1*60^0) = 122 - 61 = 61.

(Haskell)

a262079 n = a262079_list !! (n-1)

a262079_list = zipWith (-) (tail a262065_list) a262065_list

Cf. A262065, A086862.

nonn,base,new

Reinhard Zumkeller, Sep 10 2015

approved

1,3

Table of n, a(n) for n=1..23.

Eric Weisstein's World of Mathematics, Palindromic Number

Eric Weisstein's World of Mathematics, Sexagesimal

Wikipedia, Palindromic_number

Wikipedia, Sexagesimal

Index entries for sequences related to palindromes

n = 22: 41*60^2 + 20*60^1 + 41*60^0 = A262065(2541) = A002113(1148) = 148841 = a(22);

n = 23: 2*60^4 + 1*60^3 + 27*60^2 + 1*60^1 + 2*60^0 = A262065(7348) = A002113(12623) = 26233262 = a(23).

(Haskell)

-- import Data.List.Ordered (isect)

a262069 n = a262069_list !! (n-1)

a262069_list = isect a002113_list a262065_list

Intersection of A002113 and a262065.

nonn,base,more,new

Reinhard Zumkeller, Sep 10 2015

approved

1,3

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

Eric Weisstein's World of Mathematics, Palindromic Number

Eric Weisstein's World of Mathematics, Sexagesimal

Wikipedia, Palindromic_number

Wikipedia, Sexagesimal

Index entries for sequences related to palindromes

.      n | a(n) |  base 60          n |  a(n) |  base 60

.   -----+------+-----------    ------+-------+--------------

.    100 | 2440 | [40, 40]       1000 | 56355 | [15, 39, 15]

.    101 | 2501 | [41, 41]       1001 | 56415 | [15, 40, 15]

.    102 | 2562 | [42, 42]       1002 | 56475 | [15, 41, 15]

.    103 | 2623 | [43, 43]       1003 | 56535 | [15, 42, 15]

.    104 | 2684 | [44, 44]       1004 | 56595 | [15, 43, 15]

.    105 | 2745 | [45, 45]       1005 | 56655 | [15, 44, 15]

.    106 | 2806 | [46, 46]       1006 | 56715 | [15, 45, 15]

.    107 | 2867 | [47, 47]       1007 | 56775 | [15, 46, 15]

.    108 | 2928 | [48, 48]       1008 | 56835 | [15, 47, 15]

.    109 | 2989 | [49, 49]       1009 | 56895 | [15, 48, 15]

.    110 | 3050 | [50, 50]       1010 | 56955 | [15, 49, 15]

.    111 | 3111 | [51, 51]       1011 | 57015 | [15, 50, 15]

.    112 | 3172 | [52, 52]       1012 | 57075 | [15, 51, 15]

.    113 | 3233 | [53, 53]       1013 | 57135 | [15, 52, 15]

.    114 | 3294 | [54, 54]       1014 | 57195 | [15, 53, 15]

.    115 | 3355 | [55, 55]       1015 | 57255 | [15, 54, 15]

.    116 | 3416 | [56, 56]       1016 | 57315 | [15, 55, 15]

.    117 | 3477 | [57, 57]       1017 | 57375 | [15, 56, 15]

.    118 | 3538 | [58, 58]       1018 | 57435 | [15, 57, 15]

.    119 | 3599 | [59, 59]       1019 | 57495 | [15, 58, 15]

.    120 | 3601 | [1, 0, 1]      1020 | 57555 | [15, 59, 15]

.    121 | 3660 | [1, 1, 0]      1021 | 57616 | [16, 0, 16]

.    122 | 3661 | [1, 1, 1]      1022 | 57676 | [16, 1, 16]

.    123 | 3721 | [1, 2, 1]      1023 | 57736 | [16, 2, 16]

.    124 | 3781 | [1, 3, 1]      1024 | 57796 | [16, 3, 16]

.    125 | 3841 | [1, 4, 1]      1025 | 57856 | [16, 4, 16]  .

(Haskell)

import Data.List.Ordered (union)

a262065 n = a262065_list !! (n-1)

a262065_list = union us vs where

   us = [val60 $ bs ++ reverse bs | bs <- bss]

   vs = [0..59] ++ [val60 $ bs ++ cs ++ reverse bs |

          bs <- tail bss, cs <- take 60 bss]

   bss = iterate s [0] where

         s [] = [1]; s (59:ds) = 0 : s ds; s (d:ds) = (d + 1) : ds

   val60 = foldr (\b v -> 60 * v + b) 0

Cf. A262079 (first differences).

Intersection with A002113: A262069.

Corresponding sequences for bases 2 through 12: A006995, A014190, A014192, A029952, A029953, A029954, A029803, A029955, A002113, A029956, A029957.

nonn,base,look,new

Reinhard Zumkeller, Sep 10 2015

approved

0,3

In contrast to A262039, here we "round down" to the next smaller palindrome A261423(n) if it is at the same distance or closer, else we "round up" to the next larger palindrome A262038(n).

Table of n, a(n) for n=0..73.

a(10) = 9 since we round down if the next larger palindrome (here 11) is at the same distance, both 9 and 11 are here at distance 1 from n = 10.

a(16) = 11 since |16 - 11| = 5 is smaller than |16 - 22| = 6.

a(17) = 22 since |17 - 22| = 5 is smaller than |17 - 11| = 6.

a(27) = 22 since |22 - 27| = 5 is smaller than |27 - 33| = 6.

a(28) = 33 since |33 - 28| = 5 is smaller than |22 - 28| = 6, and so on.

a(100) = 99 because we round down in this case, where 99 and 101 both are at distance 1 from n = 100.

palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d];

f[n_] := Block[{k = n}, While[Nand[palQ@ k, k > -1], k--]; k];

g[n_] := Block[{k = n}, While[! palQ@ k, k++]; k];

h[n_] := Block[{a = f@ n, b = g@ n}, Which[palQ@ n, n, (b - n) - (n - a) >= 0, a, (b - n) - (n - a) < 0, b]]; Table[h@ n, {n, 0, 73}] (* Michael De Vlieger, Sep 09 2015 *)

Cf. A002113, A261423, A262037, A262038, A262039.

nonn,base,new

M. F. Hasler, Sep 08 2015

approved

0,3

In analogy to the numerical "round" function, we "round up" to the next larger palindrome A262038(n) if it is at the same distance or closer, else we "round down" to the next smaller palindrome A261423(n). See A262040 for a variant where the next smaller palindrome is chosen in case of equal distance.

Table of n, a(n) for n=0..73.

a(10) = 11 since we round up if the next smaller palindrome (here 9) is at the same distance, both 9 and 11 are here at distance 1 from n = 10.

a(16) = 11 since |16 - 11| = 5 is smaller than |16 - 22| = 6.

a(17) = 22 since |17 - 22| = 5 is smaller than |17 - 11| = 6.

a(27) = 22 since |22 - 27| = 5 is smaller than |27 - 33| = 6.

a(28) = 33 since |33 - 28| = 5 is smaller than |22 - 28| = 6, and so on.

a(100) = 101 because we round up again in this case, where 99 and 101 both are at distance 1 from n = 100.

palQ[n_] := Block[{d = IntegerDigits@ n}, d == Reverse@ d];

f[n_] := Block[{k = n}, While[Nand[palQ@ k, k > -1], k--]; k];

g[n_] := Block[{k = n}, While[! palQ@ k, k++]; k];

h[n_] := Block[{a = f@ n, b = g@ n}, Which[palQ@ n, n, (b - n) - (n - a) > 0, a, (b - n) - (n - a) <= 0, b]]; Table[h@ n, {n, 0, 73}] (* Michael De Vlieger, Sep 09 2015 *)

Cf. A002113, A261423, A262037, A262038, A262040.

nonn,base,new

M. F. Hasler, Sep 08 2015

approved

0,2

a(n) is the denominator of sigma(m)/m when m is A243512(n), the least integer i such that sigma(i)/i = (k+n)/k for some k.

Table of n, a(n) for n=0..62.

a(n) = A017665(A243512(n)) - n.

For n=2, A243512(2) is 120 with sigma(120)/120=3/1 and 3/1=(2+1)/1 so a(2)=1.

For n=3, A243512(3) is 4 with sigma(4)/4=7/4 and 7/4=(4+3)/4 so a(3)=4.

f[n_] := Block[{r = DivisorSigma[1, n]/n}, Numerator[r] - Denominator@ r]; Denominator[DivisorSigma[-1, #]] & /@ Table[i = 1; While[f@ i != n, i++]; i, {n, 0, 62}] (* Michael De Vlieger, Sep 10 2015 *)

(PARI) oksk(n, k) = {my(ab = sigma(k, -1)); numerator(ab) == denominator(ab)+n; }

a(n) = {my(k=1); while(!oksk(n, k), k++); denominator(sigma(k, -1)); }

Cf. A017665, A017666, A243473, A243512.

nonn,new

Michel Marcus, Sep 10 2015

approved

0,4

Alois P. Heinz, Columns k = 0..36, flattened

: 1;

:    1;

:       1;

:       3,  1;

:           4,     1;

:          10,     5,    1;

:          60,    15,    6,    1;

:                140,   21,    7,   1;

:                280,  224,   28,   8,  1;

:               1260,  630,  336,  36,  9,  1;

:              12600, 3780, 1050, 480, 45, 10, 1;

b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, 1,

       b(n, i-1) +`if`(i>n, 0, binomial(n, i)*b(n-i, i-1))))

    end:

T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):

seq(seq(T(n, k), n=k..k*(k+1)/2), k=0..7);

Row sums give A007837.

Column sums give A262073.

Cf. A000217, A002024, A262071, A262072 (same read by rows).

nonn,tabf,new

Alois P. Heinz, Sep 10 2015

approved

0,3

Alois P. Heinz, Table of n, a(n) for n = 0..36

a(n) = Sum_{k=n..n*(n+1)/2} A262072(k,n).

b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, 1,

       b(n, i-1) +`if`(i>n, 0, binomial(n, i)*b(n-i, i-1))))

    end:

T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):

a:= n-> add(T(k, n), k=n..n*(n+1)/2):

seq(a(n), n=0..14);

Column sums of A262072 or A262078.

Cf. A000217.

nonn,new

Alois P. Heinz, Sep 10 2015

approved

0,4

Alois P. Heinz, Rows n = 0..200, flattened

Triangle T(n,k) begins:

: 1;

:    1;

:       1;

:       3,  1;

:           4,     1;

:          10,     5,    1;

:          60,    15,    6,    1;

:                140,   21,    7,   1;

:                280,  224,   28,   8,  1;

:               1260,  630,  336,  36,  9,  1;

:              12600, 3780, 1050, 480, 45, 10, 1;

b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, 1,

       b(n, i-1) +`if`(i>n, 0, binomial(n, i)*b(n-i, i-1))))

    end:

T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):

seq(seq(T(n, k), k=ceil((sqrt(1+8*n)-1)/2)..n), n=0..14);

Row sums give A007837.

Column sums give A262073.

Cf. A002024, A262071, A262078 (same read by columns).

nonn,tabf,new

Alois P. Heinz, Sep 10 2015

approved

2,3

A lower bound on the number of weighings which suffice to determine the counterfeit (heavier) coins in a set of n coins given a balance scale and the information that there are exactly two heavier coins present.

Records occur at n=2, 3, 4, 5, 8, 14, 23, 39, 67, 116, 199, 345, 596,...

Table of n, a(n) for n=2..120.

Anping Li, Three counterfeit coins problem, J. Comb. Theory A 66 (1994) 93-101 eq. (3).

Anping Li, On the conjecture at two counterfeit coins, Discr. Math. 133 (1-3) (1994) 301-306

Wen An Liu, Qi Min Zhang, Zan Kan Nie, Optimal search procedure on coin-weighing problem, J. Statl. Plan. Inf. 136 (2006) 4419-4435.

R. Tosic, Two counterfeit coins, Discr. Math. 46 (3) (1993) 295-298, eq. (2).

seq(ceil(log[3](binomial(n, 2))), n=2..120) ;

(PARI) first(m)=vector(m, i, i++; ceil(log(binomial(i, 2))/log(3))) \\ Anders Hellström, Sep 10 2015

(MAGMA) [Ceiling(Log(3, Binomial(n, 2))): n in [2..120]]; // Bruno Berselli, Sep 10 2015

Cf. A080342 (single counterfeit coin).

nonn,easy,new

R. J. Mathar, Sep 10 2015

approved