Displaying 1-10 of 36 results found.
0, 1, 4, 9, 100, 121, 144, 169, 400, 441, 484, 900, 961, 10000, 10201, 10404, 10609, 12100, 12321, 12544, 12769, 14400, 14641, 14884, 16900, 40000, 40401, 40804, 44100, 44521, 44944, 48400, 48841, 90000, 90601, 96100, 96721, 1000000, 1002001, 1004004, 1006009
a(n) = sum of digits of A061909(n).
+20
3
1, 2, 3, 1, 2, 3, 4, 2, 3, 4, 3, 4, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 4, 2, 3, 4, 3, 4, 5, 4, 5, 3, 4, 4, 5, 1, 2, 3, 4, 2, 3, 4, 5, 3, 4, 5, 4, 5, 2, 3, 4, 5, 3, 4, 5, 6, 4, 5, 6, 5, 3, 4, 5, 4, 5, 6, 5, 4, 5, 2, 3, 4, 3, 4, 5, 4, 5, 6, 3, 4, 5, 4, 5, 5, 6, 4, 5, 6, 5, 6, 3, 4, 4, 5, 4, 5, 5, 6, 1, 2, 3, 4, 2, 3
a(n) = A061909(n) with digits reversed.
+20
3
1, 2, 3, 1, 11, 21, 31, 2, 12, 22, 3, 13, 1, 101, 201, 301, 11, 111, 211, 311, 21, 121, 221, 31, 2, 102, 202, 12, 112, 212, 22, 122, 3, 103, 13, 113, 1, 1001, 2001, 3001, 101, 1101, 2101, 3101, 201, 1201, 2201, 301, 1301, 11, 1011, 2011
Complement of A061909 (skinny numbers).
+20
1
4, 5, 6, 7, 8, 9, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86
Partial sums of skinny numbers ( A061909).
+20
1
1, 3, 6, 16, 27, 39, 52, 72, 93, 115, 145, 176, 276, 377, 479, 582, 692, 803, 915, 1028, 1148, 1269, 1391, 1521, 1721, 1922, 2124, 2334, 2545, 2757, 2977, 3198, 3498, 3799, 4109, 4420, 5420, 6421, 7423, 8426, 9436, 10447, 11459, 12472, 13492, 14513, 15535
COMMENTS
The skinny partial sums of skinny numbers begin: a(1) = 1, a(2) = 3. The primes in the sequence begin: a(2) = 3, a(15) = 479, a(15) = 1721, a(38) = 6421, a(52) = 20899. The perfect powers in the sequence begin a(4) = 16 = 2^4, a(5) = 27 = 3^3, a(24) = 1521 = 39^2.
EXAMPLE
a(36) = 4420 = 1+2+3+10+11+12+13+20+21+22+30+31+100+101+102+103+110+111+112+113+120+121+122+130+200+201+202+210+211+212+220+221+300+301+310+311
MATHEMATICA
Accumulate[Select[Range[0, 1200], IntegerReverse[#^2]==IntegerReverse[#]^2&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 02 2017 *)
Skinny numbers ( A061909) containing no 3's.
+20
1
0, 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110, 111, 112, 120, 121, 122, 200, 201, 202, 210, 211, 212, 220, 221, 1000, 1001, 1002, 1010, 1011, 1012, 1020, 1021, 1022, 1100, 1101, 1102, 1110, 1111, 1112, 1120, 1121, 1122, 1200, 1201, 1202, 1210, 1211, 1212
Skinny numbers ( A061909) containing no 2's.
+20
1
0, 1, 3, 10, 11, 13, 30, 31, 100, 101, 103, 110, 111, 113, 130, 300, 301, 310, 311, 1000, 1001, 1003, 1010, 1011, 1013, 1030, 1031, 1100, 1101, 1103, 1110, 1111, 1113, 1130, 1300, 1301, 3000, 3001, 3010, 3011, 3100, 3101, 3110, 3111, 10000, 10001, 10003, 10010
Smallest skinny number ( A061909) with digit sum n.
+20
1
0, 1, 2, 3, 13, 113, 1113, 11113, 1011113, 101011113, 1101111211, 110101111211, 100110101111211, 10101010101101122, 1011111111100000013, 1010111111111000000022, 111000010111000111111111, 1010110111101110100000011111, 1111111110010101100001100000102
COMMENTS
The smallest m >= 0 with sum of digits of (m) = n and sum of digits of (m^2) = (sum of digits of (m))^2.
There are infinitely many natural numbers m >= 0 with sum of digits of (m) = n and sum of digits of (m^2) = (sum of digits of (m))^2.
MATHEMATICA
DS[n_] := Total[IntegerDigits[n]]; nn = 10; t = Table[0, {nn}]; n = 0; found = 0; While[n++; r = FromDigits[IntegerDigits[n, 4]]; found < nn, If[DS[r]^2 == DS[r^2] && DS[r] <= nn && t[[DS[r]]] == 0, t[[DS[r]]] = r; found++; Print[r]]]; Join[{0}, t] (* T. D. Noe, Apr 18 2013 *)
EXTENSIONS
a(10) corrected and a(11) added by T. D. Noe, Apr 18 2013
1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 30, 31, 100, 101, 102, 103, 110, 111, 112, 113
Number of Golomb rulers of length n.
+10
49
1, 1, 3, 3, 5, 7, 13, 15, 27, 25, 45, 59, 89, 103, 163, 187, 281, 313, 469, 533, 835, 873, 1319, 1551, 2093, 2347, 3477, 3881, 5363, 5871, 8267, 9443, 12887, 14069, 19229, 22113, 29359, 32229, 44127, 48659, 64789, 71167, 94625, 105699, 139119, 151145, 199657
COMMENTS
Wanted: a recurrence. Are any of A169940- A169954 related to any other entries in the OEIS? Leading entry in row n of triangle in A169940. Also the number of Sidon sets A with min(A) = 0 and max(A) = n. Odd for all n since {0,n} is the only symmetric Golomb ruler, and reversal preserves the Golomb property. Bounded from above by A032020 since the ruler {0 < r_1 < ... < r_t < n} gives rise to a composition of n: (r_1 - 0, r_2 - r_1, ... , n - r_t) with distinct parts. - Tomas Boothby, May 15 2012 Also the number of compositions of n such that every restriction to a subinterval has a different sum. This is a stronger condition than all distinct consecutive subsequences having a different sum (cf. A325676). - Gus Wiseman, May 16 2019
EXAMPLE
For n=2, there is one Golomb Ruler: {0,2}. For n=3, there are three: {0,3}, {0,1,3}, {0,2,3}. - Tomas Boothby, May 15 2012 The a(1) = 1 through a(8) = 15 compositions such that every restriction to a subinterval has a different sum:
(1) (2) (3) (4) (5) (6) (7) (8)
(12) (13) (14) (15) (16) (17)
(21) (31) (23) (24) (25) (26)
(32) (42) (34) (35)
(41) (51) (43) (53)
(132) (52) (62)
(231) (61) (71)
(124) (125)
(142) (143)
(214) (152)
(241) (215)
(412) (251)
(421) (341)
(512)
(521)
(End)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], UnsameQ@@ReplaceList[#, {___, s__, ___}:>Plus[s]]&]], {n, 15}] (* Gus Wiseman, May 16 2019 *)
PROG
(Sage)
R = QQ['x']
return sum(1 for c in cartesian_product([[0, 1]]*n) if max(R([1] + list(c) + [1])^2) == 2)
CROSSREFS
Cf. A000079, A103295, A103300, A108917, A143824, A325466, A325545, A325676, A325678, A325679, A325683, A325686.
Search completed in 0.023 seconds
| 