James Carruthers - cp's OEIS frontend

A367857 a(n) is the smallest base b such that (b+1)^n in base b is a palindrome.

2, 2, 2, 7, 11, 21, 36, 71, 127, 253, 463, 925, 1717, 3433, 6436, 12871, 24311, 48621, 92379, 184757, 352717, 705433, 1352079, 2704157, 5200301, 10400601, 20058301, 40116601, 77558761, 155117521, 300540196, 601080391, 1166803111

Offset: 1

James Carruthers, Dec 03 2023

Empirically the same as A001405(n)+1 apart from a(2) where 11^10=1001 in base 2 (3^2=9 in base 10) and a(3) where 11^11=11011 in base 2 (3^3=27 in base 10).

			For n=5 the minimum base required is 11, giving 11^5=15aa51 (12^5=248832 in base 10).

Cf. A001405.

a[n_] := Module[{b = 2, d},  While[(d = IntegerDigits[(b + 1)^n, b]) != Reverse[d],   b++  ];  b] ;
Table[a[n],{n,33}] (* James C. McMahon, Dec 13 2023 after PARI *)

a(n) = my(b=2,d); while ((d=digits((b+1)^n, b)) != Vecrev(d), b++); b; \\ Michel Marcus, Dec 05 2023

from itertools import count
from sympy.ntheory.factor_ import digits
def ispal(d): return d == d[::-1]
def a(n): return next(b for b in count(2) if ispal(digits((b+1)**n, b)[1:]))
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Dec 04 2023

Conjecture: a(n) = binomial(n, floor(n/2))+1 for n>3.

a(8)-a(33) from Michael S. Branicky, Dec 04 2023

A327788 a(n) is the smallest nonnegative integer m such that the integer part of tan(m) is equal to n.

Original entry on oeis.org

0, 1, 20, 17, 105, 237, 303, 14, 58, 80, 124, 146, 11151, 168, 46318, 190, 46695, 212, 23997, 58432, 234, 13014, 38574, 61649, 82949, 256, 16586, 33271, 48891, 63091, 76581, 89361, 278, 8088, 18738, 28678, 37908, 46783, 54948, 63113, 70568, 77668, 84768, 91158, 97193, 300, 4915, 10240, 15565, 20535, 25150

Offset: 0

James Carruthers, Sep 25 2019

			tan(0) = 0, so a(0) = 0.
tan(1) = 1.557407724654902230506974807... so a(1) = 1.
For m = 2, 3, 4, ... , 13, 15, 16, 18, 19, tan(m) < 2, tan (14) = 7.24460661609480..., tan(17) = 3.493915645474... and tan(20) = 2.2371609442247422652871732477... so a(2) = 20.

Rémy Sigrist, Table of n, a(n) for n = 0..5000

Cf. A000503 (floor(tan(n))).

a:=[]; for n in [0..50] do m:=0; while Floor(Tan(m)) ne n do m:=m+1; end while; Append(~a,m); end for; a; // Marius A. Burtea, Oct 05 2019

Array[Block[{m = 0}, While[IntegerPart@ Tan@ m != #, m++]; m] &, 40, 0] (* Michael De Vlieger, Oct 05 2019 *)

a(n) = my(k=0); while (floor(tan(k)) != n, k++); k; \\ Michel Marcus, Oct 05 2019

import numpy as np
import math as m
n = 1
i = 0
inp = np.zeros(1)
out = inp
while n < 10001:
    k=m.trunc(m.tan(i))
    if k==n:
        inp = np.append(inp,int(n))
        out = np.append(out,int(i))
        print(n,i)
        n += 1
        i = 0
        continue
    else:
        i+=1

A309320 a(n) is the smallest positive integer m such that the digits of n in base 10 are also the first digits of sin(m) in base 10 after the decimal point.

Original entry on oeis.org

0, 3, 6, 53, 9, 10, 7, 4, 1, 2, 91, 69, 47, 25, 3, 41, 63, 85, 107, 129, 151, 160, 138, 116, 94, 72, 50, 6, 16, 38, 60, 82, 104, 148, 170, 163, 141, 97, 75, 53, 31, 9, 13, 57, 79, 101, 145, 167, 166, 122, 100, 78, 34, 12, 10, 32, 76, 98, 120, 164, 147, 125, 81

Offset: 0

James Carruthers, Sep 21 2019

			a(1) = 3 because we have sin(1.) = 0.8414709848..., sin(2.) = 0.9092974268..., sin(3.) = 0.1411200081.. - _N. J. A. Sloane_, Sep 28 2019

James Carruthers, Table of n, a(n) for n = 0..10000

a[0] =0; a[n_] := Module[{m = 1}, While[(d = IntegerDigits[n]) != RealDigits[ Sin[m], 10, Length[d], -1][[1]], m++]; m]; Array[a, 100, 0] (* Amiram Eldar, Sep 28 2019 *)

import numpy as np
import math as m
n = 1
i = 0
inp = np.zeros(1)
out = inp
while n < 10001:
    k=m.fabs( m.trunc( m.sin(i) * m.pow( 10,m.floor( m.log10(n)+1 ) ) ) )
    if k==n:
        inp = np.append(inp,int(n))
        out = np.append(out,int(i))
        print(n,i)
        n += 1
        i = 0
        continue
    else:
        i+=1

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User: James Carruthers

A367857 a(n) is the smallest base b such that (b+1)^n in base b is a palindrome.

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A327788 a(n) is the smallest nonnegative integer m such that the integer part of tan(m) is equal to n.

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A309320 a(n) is the smallest positive integer m such that the digits of n in base 10 are also the first digits of sin(m) in base 10 after the decimal point.

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Python