login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
Search: keyword:new
Displaying 1-10 of 324 results found. page 1 2 3 4 5 6 7 8 9 10 ... 33
     Sort: relevance | references | number | modified | created      Format: long | short | data
A367090 Numbers that cannot be written as a sum of distinct powers of 3 and distinct powers of 4. +0
0
62, 63, 143, 144, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
These numbers come mostly in chunks/runs of length 2 or 36 or 23 (first occurring as length of the 7th run starting with 706) or later 14 (length of a run starting at 6768), 1081 (length of a run starting with 15303), 20 (length of a run starting with 21186), ...
The first isolated terms are a(2701) = 25595 and a(2702) = 25774.
This set exhibits an interesting self-similar, pseudo-symmetric structure. This is due to the following
Proposition: Let d(n) = (3^r(n)-1)/2 + (4^s(n)-1)/3, where r and s are exponents such that 4^(s(n)-1) <= 3^(r(n)-1) < 3^r(n) < 4^s(n), r(0) = s(0) = 1 being the only case with equality. Then any x <= d(n) is in this sequence iff d(n) - x is in the sequence.
The study of this set is certainly useful in view of a proof of Erdös' conjecture mentioned in A327621, namely, the positive density of A005836 + A000695 (set-wise sum). This is obviously equivalent to an asymptotic density strictly smaller than 1 of the present sequence which is the complement.
LINKS
Table of n, a(n) for n=1..58.
FORMULA
Complement of A005836 + A000695 = { x+y ; x in A005836 and y in A000695 }.
EXAMPLE
0 is the empty sum, 1 = 3^0 = 4^0, 2 = 3^0 + 4^0 (*), 3 = 3^1, 4 = 4^1, 5 = 4^0 + 4^1, 6 = 3^0 + 4^0 + 4^1 (*), ..., so these numbers are not in this sequence.
After 61 = 3^0 + 3^1 + 3^2 + 3^3 + 4^0 + 4^1 + 4^2, the next larger number that is such a sum of powers is 64 = 4^3, which creates the first gap of [62, 63], first terms of this sequence.
PROG
(PARI) S34_upto(N)={my( p(N, b)=[b^k|k<-[0..logint(N, b)]] , S=[0] , i); for(b=3, 4,
foreach(p(N, b), x, S = setunion(S, [x+y| y<-S[1 .. if( S[#S]+x < N, -1,
i=setsearch(S, N-x, 1), i-1, setsearch(S, N-x)) ]]))); S}
A367090_upto(N) = setminus([1..N], S34_upto(N)) \\ not very efficient...
CROSSREFS
Cf. A000244 (powers of 3), A000302 (powers of 4).
Cf. A005836 and A000695 (sums of distinct powers of 3 and of 4).
Cf. A327621.
KEYWORD
nonn,new
AUTHOR
M. F. Hasler, Nov 08 2023
STATUS
approved
A367198 T(n, k) = Sum_{m = 0..n-1} Stirling1(m+1, k)*binomial(n, m)*(-1)^(n + k), where "Stirling1" are the signed Stirling numbers of the first kind. +0
0
1, 1, 2, 4, 6, 3, 15, 30, 18, 4, 76, 165, 125, 40, 5, 455, 1075, 930, 380, 75, 6, 3186, 8015, 7679, 3675, 945, 126, 7, 25487, 67536, 70042, 37688, 11550, 2044, 196, 8, 229384, 634935, 702372, 414078, 144417, 30870, 3990, 288, 9, 2293839, 6591943, 7696245, 4886390, 1885065, 463092, 73080, 7200, 405, 10 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
To use the unsigned Stirling numbers rewrite the formula as: T(n, k) = Sum_{m = 0..n-1} abs(Stirling1(m+1, k))*binomial(n, m)*(-1)^(1+m+n). Replacing in this formula Stirling1 (A008275) by Stirling2 (A048993) one obtains a shifted version of A321331.
LINKS
Table of n, a(n) for n=1..55.
FORMULA
T(n+1, n) = n^2*(n+1)/2 = A002411(n).
T(n, n-2) = 6*T(n-1, n-3) - 15*T(n-2, n-4) + 20*T(n-3, n-5) - 15*T(n-4, n-6) + 6*T(n-5, n-7) - T(n-6, n-8), for n > 8.
T(n, n-k) = (-1)^k*Sum_{m=0..n-1} Stirling1(m+1, n-k)*binomial(n, m).
EXAMPLE
Triangle begins:
1;
1, 2;
4, 6, 3;
15, 30, 18, 4;
76, 165, 125, 40, 5;
455, 1075, 930, 380, 75, 6;
MAPLE
T := (n, k) -> local m; add(Stirling1(m+1, k)*binomial(n, m)*(-1)^(n + k), m = 0..n-1): seq(seq(T(n, k), k = 1..n), n = 1..9); # Peter Luschny, Nov 10 2023
PROG
(PARI) T(n, k) = sum(m=0, n-1, stirling(m+1, k)*binomial(n, m)*(-1)^(n+k))
CROSSREFS
Cf. A002411, A002467 (first column), A000027 (main diagonal), A008275.
Cf. A180191(n+1) (row sums), A321331 (variant with Stirling2).
KEYWORD
nonn,tabl,new
AUTHOR
Thomas Scheuerle, Nov 10 2023
STATUS
approved
A366959 Numbers whose difference between the largest and smallest digits is equal to 2. +0
0
13, 20, 24, 31, 35, 42, 46, 53, 57, 64, 68, 75, 79, 86, 97, 102, 113, 120, 123, 131, 132, 133, 200, 201, 202, 210, 213, 220, 224, 231, 234, 242, 243, 244, 311, 312, 313, 321, 324, 331, 335, 342, 345, 353, 354, 355, 422, 423, 424, 432, 435, 442, 446, 453, 456, 464, 465, 466 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The number of n-digit terms of this sequence is (46*3^n - 93*2^n + 48)/6.
LINKS
Table of n, a(n) for n=1..58.
MATHEMATICA
Select[Range[500], Max[d=IntegerDigits[#]]-Min[d]==2 &]
PROG
(Python)
def ok(n): return max(d:=list(map(int, str(n))))-min(d) == 2
print([k for k in range(500) if ok(k)]) # Michael S. Branicky, Oct 30 2023
(PARI) isok(n) = my(d=digits(n)); vecmax(d) - vecmin(d) == 2; \\ Michel Marcus, Nov 05 2023
CROSSREFS
Cf. A037904.
Cf. A010785 (difference = 0), A366958 (difference = 1), A366960 (difference = 3), A366961 (difference = 4), A366962 (difference = 5), A366963 (difference = 6), A366964 (difference = 7), A366965 (difference = 8), A366966 (difference = 9).
KEYWORD
nonn,base,easy,new
AUTHOR
Stefano Spezia, Oct 30 2023
STATUS
approved
A366960 Numbers whose difference between the largest and smallest digits is equal to 3. +0
0
14, 25, 30, 36, 41, 47, 52, 58, 63, 69, 74, 85, 96, 103, 114, 124, 130, 134, 141, 142, 143, 144, 203, 214, 225, 230, 235, 241, 245, 252, 253, 254, 255, 300, 301, 302, 303, 310, 314, 320, 325, 330, 336, 341, 346, 352, 356, 363, 364, 365, 366, 411, 412, 413, 414 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The number of n-digit terms of this sequence is 27*4^(n-1) - 41*3^(n-1) + 7*2^n.
LINKS
Table of n, a(n) for n=1..55.
MATHEMATICA
Select[Range[415], Max[d=IntegerDigits[#]]-Min[d]==3 &]
PROG
(Python)
def ok(n): return max(d:=list(map(int, str(n))))-min(d) == 3
print([k for k in range(420) if ok(k)]) # Michael S. Branicky, Oct 30 2023
(PARI) isok(n) = my(d=digits(n)); vecmax(d) - vecmin(d) == 3; \\ Michel Marcus, Nov 05 2023
CROSSREFS
Cf. A037904.
Cf. A010785 (difference = 0), A366958 (difference = 1), A366959 (difference = 2), A366961 (difference = 4), A366962 (difference = 5), A366963 (difference = 6), A366964 (difference = 7), A366965 (difference = 8), A366966 (difference = 9).
KEYWORD
nonn,base,easy,new
AUTHOR
Stefano Spezia, Oct 30 2023
STATUS
approved
A366962 Numbers whose difference between the largest and smallest digits is equal to 5. +0
0
16, 27, 38, 49, 50, 61, 72, 83, 94, 105, 116, 126, 136, 146, 150, 156, 161, 162, 163, 164, 165, 166, 205, 216, 227, 237, 247, 250, 257, 261, 267, 272, 273, 274, 275, 276, 277, 305, 316, 327, 338, 348, 350, 358, 361, 368, 372, 378, 383, 384, 385, 386, 387, 388 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The number of n-digit terms of this sequence is 29*6^(n-1) - 49*5^(n-1) + 5*4^n.
LINKS
Table of n, a(n) for n=1..54.
MATHEMATICA
Select[Range[400], Max[d=IntegerDigits[#]]-Min[d]==5 &]
PROG
(Python)
def ok(n): return max(d:=list(map(int, str(n))))-min(d) == 5
print([k for k in range(400) if ok(k)]) # Michael S. Branicky, Oct 30 2023
(PARI) isok(n) = my(d=digits(n)); vecmax(d) - vecmin(d) == 5; \\ Michel Marcus, Nov 05 2023
CROSSREFS
Cf. A037904.
Cf. A010785 (difference = 0), A366958 (difference = 1), A366959 (difference = 2), A366960 (difference = 3), A366961 (difference = 4), A366963 (difference = 6), A366964 (difference = 7), A366965 (difference = 8), A366966 (difference = 9).
KEYWORD
nonn,base,easy,new
AUTHOR
Stefano Spezia, Oct 30 2023
STATUS
approved
A366961 Numbers whose difference between the largest and smallest digits is equal to 4. +0
0
15, 26, 37, 40, 48, 51, 59, 62, 73, 84, 95, 104, 115, 125, 135, 140, 145, 151, 152, 153, 154, 155, 204, 215, 226, 236, 240, 246, 251, 256, 262, 263, 264, 265, 266, 304, 315, 326, 337, 340, 347, 351, 357, 362, 367, 373, 374, 375, 376, 377, 400, 401, 402, 403, 404, 410 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The number of n-digit terms is 29*5^(n-1) - 47*4^(n-1) + 2*3^(n+1).
LINKS
Table of n, a(n) for n=1..56.
MATHEMATICA
Select[Range[410], Max[d=IntegerDigits[#]]-Min[d]==4 &]
PROG
(Python)
def ok(n): return max(d:=list(map(int, str(n))))-min(d) == 4
print([k for k in range(411) if ok(k)]) # Michael S. Branicky, Oct 30 2023
(PARI) isok(n) = my(d=digits(n)); vecmax(d) - vecmin(d) == 4; \\ Michel Marcus, Nov 05 2023
CROSSREFS
Cf. A037904.
Cf. A010785 (difference = 0), A366958 (difference = 1), A366959 (difference = 2), A366960 (difference = 3), A366962 (difference = 5), A366963 (difference = 6), A366964 (difference = 7), A366965 (difference = 8), A366966 (difference = 9).
KEYWORD
nonn,base,easy,new
AUTHOR
Stefano Spezia, Oct 30 2023
STATUS
approved
A366963 Numbers whose difference between the largest and smallest digits is equal to 6. +0
0
17, 28, 39, 60, 71, 82, 93, 106, 117, 127, 137, 147, 157, 160, 167, 171, 172, 173, 174, 175, 176, 177, 206, 217, 228, 238, 248, 258, 260, 268, 271, 278, 282, 283, 284, 285, 286, 287, 288, 306, 317, 328, 339, 349, 359, 360, 369, 371, 379, 382, 389, 393, 394, 395, 396, 397, 398, 399 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The number of n-digit terms of this sequence is 27*7^(n-1) - 47*6^(n-1) + 4*5^n.
LINKS
Table of n, a(n) for n=1..58.
MATHEMATICA
Select[Range[400], Max[d=IntegerDigits[#]]-Min[d]==6 &]
PROG
(Python)
def ok(n): return max(d:=list(map(int, str(n))))-min(d) == 6
print([k for k in range(400) if ok(k)]) # Michael S. Branicky, Oct 30 2023
(PARI) isok(n) = my(d=digits(n)); vecmax(d) - vecmin(d) == 6; \\ Michel Marcus, Nov 05 2023
CROSSREFS
Cf. A037904.
Cf. A010785 (difference = 0), A366958 (difference = 1), A366959 (difference = 2), A366960 (difference = 3), A366961 (difference = 4), A366962 (difference = 5), A366964 (difference = 7), A366965 (difference = 8), A366966 (difference = 9).
KEYWORD
nonn,base,easy,new
AUTHOR
Stefano Spezia, Oct 30 2023
STATUS
approved
A366964 Numbers whose difference between the largest and smallest digits is equal to 7. +0
0
18, 29, 70, 81, 92, 107, 118, 128, 138, 148, 158, 168, 170, 178, 181, 182, 183, 184, 185, 186, 187, 188, 207, 218, 229, 239, 249, 259, 269, 270, 279, 281, 289, 292, 293, 294, 295, 296, 297, 298, 299, 307, 318, 329, 370, 381, 392, 407, 418, 429, 470, 481, 492 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The number of n-digit terms is 23*8^(n-1) - 41*7^(n-1) + 2^n*3^(n+1).
LINKS
Table of n, a(n) for n=1..53.
MATHEMATICA
Select[Range[500], Max[d=IntegerDigits[#]]-Min[d]==7 &]
PROG
(Python)
def ok(n): return max(d:=list(map(int, str(n))))-min(d) == 7
print([k for k in range(500) if ok(k)]) # Michael S. Branicky, Oct 30 2023
(PARI) isok(n) = my(d=digits(n)); vecmax(d) - vecmin(d) == 7; \\ Michel Marcus, Nov 05 2023
CROSSREFS
Cf. A037904.
Cf. A010785 (difference = 0), A366958 (difference = 1), A366959 (difference = 2), A366960 (difference = 3), A366961 (difference = 4), A366962 (difference = 5), A366963 (difference = 6), A366965 (difference = 8), A366966 (difference = 9).
KEYWORD
nonn,base,easy,new
AUTHOR
Stefano Spezia, Oct 30 2023
STATUS
approved
A366965 Numbers whose difference between the largest and smallest digits is equal to 8. +0
0
19, 80, 91, 108, 119, 129, 139, 149, 159, 169, 179, 180, 189, 191, 192, 193, 194, 195, 196, 197, 198, 199, 208, 219, 280, 291, 308, 319, 380, 391, 408, 419, 480, 491, 508, 519, 580, 591, 608, 619, 680, 691, 708, 719, 780, 791, 800, 801, 802, 803, 804, 805, 806, 807, 808, 810 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The number of n-digit terms of this sequence is 17*9^(n-1) - 31*8^(n-1) + 2*7^n.
LINKS
Table of n, a(n) for n=1..56.
MATHEMATICA
Select[Range[810], Max[d=IntegerDigits[#]]-Min[d]==8 &]
PROG
(Python)
def ok(n): return max(d:=list(map(int, str(n))))-min(d) == 8
print([k for k in range(900) if ok(k)]) # Michael S. Branicky, Oct 30 2023
(PARI) isok(n) = my(d=digits(n)); vecmax(d) - vecmin(d) == 8; \\ Michel Marcus, Nov 05 2023
CROSSREFS
Cf. A037904.
Cf. A010785 (difference = 0), A366958 (difference = 1), A366959 (difference = 2), A366960 (difference = 3), A366961 (difference = 4), A366962 (difference = 5), A366963 (difference = 6), A366964 (difference = 7), A366966 (difference = 9).
KEYWORD
nonn,base,easy,new
AUTHOR
Stefano Spezia, Oct 30 2023
STATUS
approved
A366966 Numbers whose difference between the largest and smallest digits is equal to 9. +0
0
90, 109, 190, 209, 290, 309, 390, 409, 490, 509, 590, 609, 690, 709, 790, 809, 890, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 920, 930, 940, 950, 960, 970, 980, 990, 1009, 1019, 1029, 1039, 1049, 1059, 1069, 1079, 1089, 1090, 1091, 1092, 1093, 1094, 1095 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The number of n-digit terms of this sequence is 9*10^(n-1) - 17*9^(n-1) + 8^n.
LINKS
Table of n, a(n) for n=1..51.
MATHEMATICA
Select[Range[1095], Max[d=IntegerDigits[#]]-Min[d]==9 &]
PROG
(Python)
def ok(n): return max(d:=list(map(int, str(n))))-min(d) == 9
print([k for k in range(1100) if ok(k)]) # Michael S. Branicky, Oct 30 2023
(PARI) isok(n) = my(d=digits(n)); vecmax(d) - vecmin(d) == 9; \\ Michel Marcus, Nov 05 2023
CROSSREFS
Cf. A037904.
Cf. A010785 (difference = 0), A366958 (difference = 1), A366959 (difference = 2), A366960 (difference = 3), A366961 (difference = 4), A366962 (difference = 5), A366963 (difference = 6), A366964 (difference = 7), A366965 (difference = 8).
KEYWORD
nonn,base,easy,new
AUTHOR
Stefano Spezia, Oct 30 2023
STATUS
approved
page 1 2 3 4 5 6 7 8 9 10 ... 33

Search completed in 0.095 seconds

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 10 19:25 EST 2023. Contains 366923 sequences. (Running on oeis4.)