| Displaying 1-7 of 7 results found.
page
1
Triangle read by rows in which k-th entry in row n is representation of n in base k, for 1 <= k <= n.
+10
6
1, 11, 10, 111, 11, 10, 1111, 100, 11, 10, 11111, 101, 12, 11, 10, 111111, 110, 20, 12, 11, 10, 1111111, 111, 21, 13, 12, 11, 10, 11111111, 1000, 22, 20, 13, 12, 11, 10, 111111111, 1001, 100, 21, 14, 13, 12, 11, 10, 1111111111, 1010, 101, 22, 20, 14, 13
COMMENTS
Representation of n in base 1 is defined to be a concatenation of n 1's.
It is difficult to write twenty-one in base 11 using decimal digits.
Representation in bases greater than 10 are written in base 10. This is really nasty! - N. J. A. Sloane, Dec 06 2002
EXAMPLE
Rows start (1), (11, 10), (111, 11, 10), (1111, 100, 11, 10), etc.
MATHEMATICA
f[n_] := Flatten[ Append[ {FromDigits[ Table[1, {n}]] }, Table[ FromDigits[ IntegerDigits[n, i]], {i, 2, n}]]]; Flatten[ Table[ f[n], {n, 1, 10}]] (* Robert G. Wilson v *)
CROSSREFS
Rows are effectively the reverse of A001731, A001732, A001733, A001734, A001735, A001736, A008707, A008708, A008709, A008710, A008711, A008712, A008713, A008714, A008715, A008716, A008717, etc. Columns are truncated versions of A000042, A007088, A007089, A007090, A007091, A007092, A007093, A007094, A007095, A000027 and perhaps A055649, etc. Without the 1st column becomes A004053.
a(n) is n written in base n mod 10, or 0 if n mod 10 = 0.
+10
5
1, 10, 10, 10, 10, 10, 10, 10, 10, 0, 11111111111, 1100, 111, 32, 30, 24, 23, 22, 21, 0, 111111111111111111111, 10110, 212, 120, 100, 42, 36, 34, 32, 0, 1111111111111111111111111111111, 100000, 1020, 202, 120, 100, 52, 46, 43, 0
EXAMPLE
In base 1, n =11...11=n written n times; in base 0 baseform is taken as 0.
MATHEMATICA
Table[BaseForm[w, Mod[w, 10]], {w, 1, 128}]
Primes of form 4k+3 written in base 3.
+10
5
10, 21, 102, 201, 212, 1011, 1121, 1202, 2012, 2111, 2122, 2221, 10002, 10211, 10222, 11201, 11212, 12011, 12121, 20001, 20012, 20122, 21002, 21101, 21211, 22021, 22102, 22212, 100022, 100202, 101001, 101111, 102101, 102112, 110021
MATHEMATICA
Do[s=Prime[n]; If[Mod[s, 4]==3, Print[BaseForm[s, 3]]], {n, 1, 256}]
n-th prime prime(n) written in base (prime(n) (mod prime(n-1))).
+10
5
111, 101, 111, 23, 1101, 101, 10011, 113, 45, 11111, 101, 221, 101011, 233, 125, 135, 111101, 151, 1013, 1001001, 211, 1103, 225, 141, 1211, 1100111, 1223, 1101101, 1301, 91, 2003, 345, 10001011, 149, 10010111, 421, 431, 2213, 445, 455, 10110101
EXAMPLE
Eventually non-decimal digit symbols appear, as in case of 307=17d, in base 14 = 307 mod 293.
MAPLE
a:= proc(n) local b, p, l;
p:= ithprime(n); b:= irem(p, prevprime(p));
if b=1 then l:= 1$p
else l:= ""; while p>0 do l:= irem(p, b, 'p'), l od
fi; parse(cat(l))
end:
MATHEMATICA
Table[BaseForm[Prime[w], Mod[Prime[w], Prime[w-1]]], {w, 2, 128}]
Join[{111}, FromDigits[IntegerDigits[#[[2]], Mod[#[[2]], #[[1]]]]]&/@ Partition[ Prime[Range[2, 50]], 2, 1]] (* Harvey P. Dale, Jul 03 2021 *)
PROG
(PARI) a(n) = {my(p=prime(n), q=prime(n-1)); if ((p % q) != 1, d=digits(p, p % q); if (#select(x->(x>9), d), 0, fromdigits(d, 10)), fromdigits(vector(p, k, 1), 10)); } \\ Michel Marcus, Sep 05 2019
n-th prime prime(n) written in base (prime(n) (mod 4)).
+10
4
10, 10, 11111, 21, 102, 1111111111111, 11111111111111111, 201, 212, 11111111111111111111111111111, 1011, 1111111111111111111111111111111111111, 11111111111111111111111111111111111111111, 1121, 1202, 11111111111111111111111111111111111111111111111111111, 2012
EXAMPLE
4k+1 primes are written in base 1, while 4k+3 primes are in base 3.
MATHEMATICA
Table[FromDigits@ If[#2 == 1, ConstantArray[1, #1], IntegerDigits[#1, #2]] & @@ {#, Mod[#, 4]} &@ Prime@ w, {w, 17}] (* Michael De Vlieger, Sep 04 2019 *)
PROG
(PARI) a(n) = {my(p=prime(n)); if ((p % 4) != 1, fromdigits(digits(p, p % 4), 10), fromdigits(vector(p, k, 1), 10)); } \\ Michel Marcus, Sep 04 2019
Primes of the form 6k+5 written in base 5.
+10
3
10, 21, 32, 43, 104, 131, 142, 203, 214, 241, 313, 324, 401, 412, 423, 1011, 1022, 1044, 1132, 1143, 1204, 1231, 1242, 1402, 1413, 1424, 2001, 2012, 2023, 2034, 2111, 2133, 2221, 2232, 2342, 2403, 2414, 3013, 3024, 3101, 3134, 3211, 3233, 3244, 3321
EXAMPLE
41 = 25 + 3*5 + 1 = 131_5.
MATHEMATICA
Do[s=Prime[n]; If[Mod[s, 6]==5, Print[BaseForm[s, 5]]], {n, 1, 256}]
FromDigits[IntegerDigits[#, 5]] & /@ Select[Table[6 n + 5, {n, 0, 100}], PrimeQ] (* Harvey P. Dale, Oct 05 2023 *)
PROG
(PARI) lista(nn) = for (n=0, nn, if (isprime(p=6*n+5), print1(fromdigits(digits(p, 5)), ", "))); \\ Michel Marcus, Jul 09 2018
Square array read by antidiagonals of n written in base k (n,k>0).
+10
1
1, 1, 11, 1, 10, 111, 1, 2, 11, 1111, 1, 2, 10, 100, 11111, 1, 2, 3, 11, 101, 111111, 1, 2, 3, 10, 12, 110, 1111111, 1, 2, 3, 4, 11, 20, 111, 11111111, 1, 2, 3, 4, 10, 12, 21, 1000, 111111111, 1, 2, 3, 4, 5, 11, 13, 22, 1001, 1111111111, 1, 2, 3, 4, 5, 10, 12, 20, 100
COMMENTS
It is difficult to write ten in base 11 using decimal digits.
EXAMPLE
Rows start (1, 1, 1, 1, 1,...), (11, 10, 2, 2, 2,...), (111, 11, 10, 3, 3,...), (1111, 100, 11, 10, 4,...), etc.
CROSSREFS
Cf. A063432. Rows start effectively as the reverse of A001731, A001732, A001733, A001734, A001735, A001736, A008707, A008708, A008709, A008710, A008711, A008712, A008713, A008714, A008715, A008716, A008717 etc. Columns are A000042, A007088, A007089, A007090, A007091, A007092, A007093, A007094, A007095, A000027 and perhaps A055649 etc.
Search completed in 0.010 seconds
|