Displaying 1-10 of 13 results found.
In base 2, one 'Reverse and Add' step is needed to reach a palindrome.
+10
12
2, 4, 6, 8, 10, 12, 14, 16, 18, 24, 30, 32, 34, 36, 38, 40, 42, 48, 52, 56, 62, 64, 66, 68, 70, 80, 82, 96, 100, 102, 112, 114, 126, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150, 160, 162, 168, 170, 176, 178, 192, 196, 198, 200, 204, 208, 212, 224
COMMENTS
The analog of A065206 in base 2. The number of steps starts at 0, so palindromes (cf. A006995) are excluded.
PROG
(ARIBAS): function b2revadd_steps(k, stop: integer); var c, n, m, steps, rev: integer; begin n := 0; c := 0; while c < stop do m := n; rev := b2reverse(m); steps := 0; while steps < k and m <> rev do m := m + rev; rev := b2reverse(m); inc(steps); end; if steps = k and m = rev then write(n, " "); inc(c); end; inc(n); end; end; b2revadd_steps(1, 66).
(PARI) Rev(x)= { local(d, r=0); while (x>0, d=x%10; x\=10; r=r*10 + d); return(r) } digitsIn(x)= { local(d); if (x==0, return(1)); d=1 + log(x)\log(10); if (10^d == x, d++, if (10^(d-1) > x, d--)); return(d) } Palin(x)= { local(d, e, f, i, t, y); if (x==0, return(1)); y=x; d=digitsIn(x); t=10^(d - 1); for (i=1, d\2, f=y-10*(y\10); y\=10; e=x\t; x-=t*e; t/=10; if (e!=f, return(0)) ); return(1) } baseE(x, b)= { local(d, e=0, f=1); while (x>0, d=x-b*(x\b); x\=b; e+=d*f; f*=10); return(e) } baseI(x, b)= { local(d, e=0, f=1); while (x>0, d=x-10*(x\10); x\=10; e+=d*f; f*=b); return(e) } { n=0; for (m=1, 10^9, b=baseE(m, 2); r=Rev(b); d=baseI(r, 2) + m; b=baseE(d, 2); if (Palin(b), write("b066122.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Feb 01 2010
One iteration of Reverse and Add is needed to reach a palindrome.
+10
6
1, 2, 3, 4, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 38, 40, 41, 42, 43, 44, 45, 47, 50, 51, 52, 53, 54, 56, 60, 61, 62, 63, 65, 70, 71, 72, 74, 80, 81, 83, 90, 92, 100, 101, 102, 103, 104, 105, 106
COMMENTS
The number of iterations starts at 1, so palindromes (cf. A002113) are not excluded. The corresponding sequence excluding palindromes is A065206.
MATHEMATICA
rev[n_] := FromDigits[Reverse[IntegerDigits[n]]]; Select[Range[106], rev[x = # + rev[#]] == x &] (* Jayanta Basu, Jul 24 2013 *) Select[Range[120], PalindromeQ[#+IntegerReverse[#]]&] (* Harvey P. Dale, Jul 04 2022 *)
PROG
(Haskell)
a015976 n = a015976_list !! (n-1)
a015976_list = filter ((== 1) . a136522 . a056964) [1..]
Two 'Reverse and Add' steps are needed to reach a palindrome.
+10
4
19, 28, 37, 39, 46, 48, 49, 57, 58, 64, 67, 73, 75, 76, 82, 84, 85, 91, 93, 94, 109, 119, 129, 139, 149, 150, 152, 153, 154, 159, 160, 162, 163, 169, 170, 172, 173, 179, 189, 208, 218, 219, 228, 229, 238, 239, 248, 250, 251, 253, 258, 259, 260, 261, 268, 269
COMMENTS
The number of steps starts at 0, so palindromes (cf. A002113) are excluded.
MATHEMATICA
trasQ[n_]:=Length[NestWhileList[IntegerReverse[#]+#&, n, !PalindromeQ[ #]&, 1, 5]] ==3; Select[Range[300], trasQ] (* Harvey P. Dale, Apr 13 2022 *)
PROG
(ARIBAS): revadd_steps(2, 58). For the definition of function revadd_steps see A065206. (PARI)
Rev(x)= { local(d, r=0); while (x>0, d=x-10*(x\10); x\=10; r=r*10 + d); return(r) }
digitsIn(x)= { local(d); if (x==0, return(1)); d=1 + log(x)\log(10); if (10^d == x, d++, if (10^(d-1) > x, d--)); return(d) }
Palin(x)= { local(d, e, f, i, t, y); if (x==0, return(1)); y=x; d=digitsIn(x); t=10^(d - 1); for (i=1, d\2, f=y-10*(y\10); y\=10; e=x\t; x-=t*e; t/=10; if (e!=f, return(0)) ); return(1) }
{ n=0; for (m = 0, 10^9, p=m; b=1; for (i=1, 2, if (Palin(p), b=0; break); p=Rev(p) + p); if (b && Palin(p), write("b065207.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 14 2009 (Python)
def ra(n): s = str(n); return int(s) + int(s[::-1])
def ispal(n): s = str(n); return s == s[::-1]
def aupto(limit):
alst = []
for k in range(limit+1):
if ispal(k): continue
k2 = ra(k)
if ispal(k2): continue
if ispal(ra(k2)): alst.append(k)
return alst
Three 'Reverse and Add' steps are needed to reach a palindrome.
+10
3
59, 68, 86, 95, 155, 156, 157, 158, 164, 165, 168, 178, 180, 184, 185, 186, 194, 199, 249, 254, 255, 256, 257, 263, 264, 267, 277, 283, 284, 285, 293, 298, 299, 348, 349, 354, 355, 356, 362, 366, 376, 382, 384, 389, 392, 397, 398, 399, 439, 447, 448, 449, 452, 453, 455, 461, 462, 465, 475, 481, 482, 483, 488, 489, 491, 496, 497, 498, 499, 538, 546, 547, 548, 551, 552, 553, 554, 560, 561, 564, 574, 580, 581, 582, 587, 588, 590, 596, 597, 598, 637, 645, 647, 650, 651, 652, 653, 660, 663, 673, 680, 681, 687, 694, 695, 697, 736, 744, 745, 746, 750
COMMENTS
The number of steps starts at 0, so palindromes (cf. A002113) are excluded.
MATHEMATICA
palQ[n_]:=Module[{idn=IntegerDigits[n]}, idn==Reverse[idn]]
tst[n_]:=palQ/@NestList[#+FromDigits[Reverse[IntegerDigits[#]]]&, n, 3]=={False, False, False, True}
PROG
(ARIBAS): revadd_steps(3, 54). For the definition of function revadd_steps see A065206. (PARI) Rev(x)= { local(d, r=0); while (x>0, d=x-10*(x\10); x\=10; r=r*10 + d); return(r) } digitsIn(x)= { local(d); if (x==0, return(1)); d=1 + log(x)\log(10); if (10^d == x, d++, if (10^(d-1) > x, d--)); return(d) } Palin(x)= { local(d, e, f, i, t, y); if (x==0, return(1)); y=x; d=digitsIn(x); t=10^(d - 1); for (i=1, d\2, f=y-10*(y\10); y\=10; e=x\t; x-=t*e; t/=10; if (e!=f, return(0)) ); return(1) } { n=0; for (m = 0, 10^9, p=m; b=1; for (i=1, 3, if (Palin(p), b=0; break); p=Rev(p) + p); if (b && Palin(p), write("b065208.txt", n++, " ", m); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 14 2009
Five 'Reverse and Add' steps are needed to reach a palindrome.
+10
3
166, 176, 198, 265, 275, 297, 364, 374, 396, 463, 473, 495, 549, 562, 572, 594, 639, 648, 661, 671, 693, 738, 760, 770, 792, 837, 846, 891, 936, 945, 990, 1396, 1486, 1576, 1666, 1697, 1756, 1787, 1791, 1793, 1796, 1846, 1877, 1883, 1886, 1890, 1936
COMMENTS
The number of steps starts at 0, so palindromes (cf. A002113) are excluded.
MATHEMATICA
palQ[k_]:=IntegerDigits[k]==Reverse[IntegerDigits[k]]; fraQ[n_]:= Module[ {ras=NestList[#+FromDigits[Reverse[IntegerDigits[#]]]&, n, 5]}, palQ/@ ras=={False, False, False, False, False, True}]; Select[Range[2000], fraQ] (* Harvey P. Dale, Sep 28 2015 *)
Seven 'Reverse and Add' steps are needed to reach a palindrome.
+10
3
188, 190, 197, 287, 296, 386, 395, 485, 584, 593, 683, 692, 782, 791, 881, 890, 980, 1394, 1484, 1574, 1664, 1754, 1844, 1898, 1934, 1988, 1992, 1994, 1999, 2393, 2483, 2573, 2663, 2753, 2843, 2897, 2933, 2987, 2991, 2993, 2998, 3089, 3179, 3269, 3359
COMMENTS
The number of steps starts at 0, so palindromes (cf. A002113) are excluded.
MATHEMATICA
lenQ[n_]:= Length[NestWhileList[# + FromDigits[Reverse[IntegerDigits[#]]]&, n, #!= FromDigits[Reverse[IntegerDigits[#]]]&, 1, 10]] == 8; Select[Range[500], lenQ] (* Vincenzo Librandi, Sep 24 2013 *)
Eight 'Reverse and Add' steps are needed to reach a palindrome.
+10
3
193, 391, 490, 589, 688, 886, 985, 1993, 1995, 2994, 3991, 4990, 4992, 5991, 6990, 8059, 8149, 8239, 8329, 8419, 8509, 8599, 8689, 8779, 8869, 8959, 9058, 9069, 9089, 9148, 9159, 9179, 9238, 9249, 9269, 9328, 9359, 9418, 9429, 9508, 9519, 9539, 9598
COMMENTS
The number of steps starts at 0, so palindromes (cf. A002113) are excluded.
MATHEMATICA
lenQ[n_]:=Length[NestWhileList[#+FromDigits[Reverse[IntegerDigits[#]]]&, n, #!=FromDigits[Reverse[IntegerDigits[#]]]&, 1, 10]]==9; Select[Range[ 10000], lenQ] (* Harvey P. Dale, Aug 09 2013 *)
Numbers n such that nine 'Reverse and Add' steps are needed to reach a palindrome.
+10
3
1397, 1487, 1577, 1667, 1757, 1847, 1937, 2396, 2486, 2576, 2666, 2756, 2846, 2936, 2999, 3395, 3485, 3575, 3665, 3755, 3845, 3935, 3998, 4394, 4484, 4574, 4754, 4844, 4934, 4997, 5393, 5483, 5573, 5663, 5753, 5843, 5933, 5996, 6392, 6482, 6572
COMMENTS
The number of steps starts at 0, so palindromes (cf. A002113) are excluded.
MATHEMATICA
lenQ[n_]:= Length[NestWhileList[# + FromDigits[Reverse[IntegerDigits[#]]]&, n, #!= FromDigits[Reverse[IntegerDigits[#]]]&, 1, 10]] == 10; Select[Range[1000], lenQ] (* Vincenzo Librandi, Sep 24 2013 *)
Numbers such that ten 'Reverse and Add' steps are needed to reach a palindrome.
+10
3
829, 928, 9059, 9149, 9239, 9329, 9419, 9509, 9599, 9689, 9869, 9959, 10634, 10637, 10687, 10716, 10808, 10834, 10838, 10867, 10873, 10898, 10927, 10979, 11398, 11624, 11627, 11677, 11706, 11824, 11828, 11857, 11863, 11888, 11917, 11969
COMMENTS
The number of steps starts at 0, so palindromes (cf. A002113) are excluded.
MATHEMATICA
lenQ[n_]:= Length[NestWhileList[# + FromDigits[Reverse[IntegerDigits[#]]]&, n, #!= FromDigits[Reverse[IntegerDigits[#]]]&, 1, 11]] == 11; Select[Range[1000], lenQ] (* Vincenzo Librandi, Sep 24 2013 *)
Eleven 'Reverse and Add' steps are needed to reach a palindrome.
+10
3
167, 266, 365, 563, 662, 761, 860, 7069, 7159, 7249, 7339, 7429, 7519, 7609, 7699, 7789, 7879, 7969, 8068, 8158, 8248, 8428, 8518, 8608, 8698, 8788, 8878, 8968, 9039, 9067, 9129, 9157, 9219, 9247, 9309, 9337, 9399, 9427, 9489, 9517, 9579, 9607, 9697
COMMENTS
The number of steps starts at 0, so palindromes (cf. A002113) are excluded.
MATHEMATICA
lenQ[n_]:= Length[NestWhileList[# + FromDigits[Reverse[IntegerDigits[#]]]&, n, #!= FromDigits[Reverse[IntegerDigits[#]]]&, 1, 12]] == 12; Select[Range[1000], lenQ] (* Vincenzo Librandi, Sep 24 2013 *)
Search completed in 0.010 seconds
|