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User:Jason Kimberley/sqrt base

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Contents

1 Index of expansions of ${\sqrt {d}}$ in base $b$
2 Powers of ${\sqrt {d}}$
3 Integers written in base $b$
4 Magma

Index of expansions of ${\sqrt {d}}$ in base $b$

	$d$
$b$		2	3	4	5	6	7	8	9	10		60
	2	A004539	A004540	A004541	A004542	A004543	A004544	A004545	A004546	A002193		A070197
	3	A004547	A004548	A004549	A004550	A004551	A004552	A004553	A004554	A002194
	5	A004555	A004556	A004557	A004558	A004559	A004560	A004561	A004562	A002163
	6	A004609	A004610	A004563	A004564	A004565	A004566	A004567	A004568	A010464
	7	A004569	A004570	A004571	A004572	A004573	A004574	A004575	A004576	A010465
	8	A004539	A004578	A004579	A004580	A004581	A004582	A004583	A004584	A010466
	10	A004585	A004586	A004587	A004588	A004589	A004590	A004591	A004592	A010467
	11									A010468

	15									A010472
	17									A010473

	24									A010480
	26									A010481

	35									A010490
	37									A010491

	48									A010502
	51									A010504

	63									A010516
	65									A010517

	80									A010532
	82									A010533

	99									A010550

Powers of ${\sqrt {d}}$

Magma code by Librandi has rounding errors for large enough n.

$d$	$\left\lfloor {\sqrt {d^{n}}}\right\rfloor$		$\left\lceil {\sqrt {d^{n}}}\right\rceil$
10	A017934	A017935	A017936
11	A017937	A017938	A017939
12	A017940	A017941	A017942
13	A017943	A017944	A017945
14	A017946	A017947	A017948
15	A017949	A017950	A017951

17	A017955	A017956	A017957
18	A017958	A017959	A017960
19	A017961	A017962	A017963
20	A017964	A017965	A017966
21	A017967	A017968	A017969
22	A017970	A017971	A017972
23	A017973	A017974	A017975
24	A017976	A017977	A017978

Integers written in base $b$

$b$	$n$	Digits	$n^{2}$	$n^{3}$	$2^{n}$	$3^{n}$	$p_{n}$	$F_{n}$
1	A000042	A000012			A136308		A031974	A108047
2	A007088	A030190	A001737	A004632	A011557	A004656	A004676	A004685
3	A007089	A054635	A001738	A004633	A004642	A011557	A001363	A004686
4	A007090	A030373	A001739	A004634	A004643	A004658	A004678	A004687
5	A007091	A031219	A001740	A004635	A000866	A004659	A004679	A004688
6	A007092	A030548	A001741	A004636	A004645	A004660	A004680	A004689
7	A007093	A030998	A002440	A004637	A004646	A004661	A004681	A004690
8	A007094	A054634	A002441	A004638	A004647	A004662	A004682	A004691
9	A007095	A031076	A002442	A004639	A001357	A004663	A004683	A004692
10	A001477	A007376	A000290	A000578	A000079	A000244	A000040	A000045
11			A004629	A004640		A004665	A004684
12			A004630		A004651	A004666		A004693
13						A004667		A004694
14					A004653
15					A004654
16			A004631		A004655
26						A004668
27						A004669

Magma

 real_to_base :=
   func<
     r, b
   :
     n := Precision(r:Bits)div Ilog(2,b) - 1
   |
     Reverse(IntegerToSequence(Floor(r*b^n),b))
   >
 ;

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