Paul Lusch - cp's OEIS frontend

A292125 Positive numbers n such that the set of base-7 digits of n equals both the set of base-8 digits of n and the set of base-9 digits of n.

1, 2, 3, 4, 5, 6, 32942, 103302, 142489, 178248, 263180, 361369, 362370, 378454, 540929, 540940, 567333, 567337, 590697, 693302, 723659, 726195, 743217, 829792, 888934, 1070221, 1073830, 1074177, 1083411, 1083738, 1090352, 1106956, 1129738, 1139557, 1139881, 1201762, 1249562

Offset: 1

Paul Lusch, Sep 08 2017

			32942 in base 7 is 165020; in base 8 is 100256; and in base 9 is 50162. The set of digits for all three is {0, 1, 2, 5, 6}.

Robert Israel, Table of n, a(n) for n = 1..1000

Subsequence of A037438.

filter:= proc(n) local S;
  S:= convert(convert(n,base,9),set);
  nops(S intersect {7,8})=0 and evalb(convert(convert(n,base,8),set)=S) and evalb(convert(convert(n,base,7),set)=S)
end proc:
select(filter, [$1..6,seq(seq(9*a+b,b=0..6),a=1..7*9^5-1)]); # Robert Israel, Sep 11 2017

Select[Range[5/4*10^6], Function[k, SameQ @@ Map[Union@ IntegerDigits[k, #] &, {7, 8, 9}]]] (* Michael De Vlieger, Sep 10 2017 *)

isok(n) = my(s7 = Set(digits(n, 7)), s8 = Set(digits(n, 8)), s9 = Set(digits(n, 9))); (s7 == s8) && (s8 == s9); \\ Michel Marcus, Sep 09 2017

More terms from Michel Marcus, Sep 09 2017

A130604 Numbers whose base-10 and base-7 representations are permutations of the same multiset of digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 23, 46, 265, 316, 1030, 1234, 1366, 1431, 1454, 2060, 2116, 10144, 10342, 10542, 11425, 12415, 12450, 12564, 12651, 13045, 13245, 13534, 14610, 15226, 15643, 16255, 16546, 16633, 101046, 101264, 102615, 103260, 103316, 103460, 103461, 103462, 103463, 103464, 103465, 103466, 104126, 104632, 104650, 104651, 104652, 104653, 104654, 104655, 104656, 105266, 106235, 106253, 113256, 116336

Offset: 1

Paul Lusch, Aug 10 2007

The sequence is finite and full since any d-digit number is < 7^d in base 7 and > 10^(d-1) in base 10. But 1000000 = 10^6 > 7^7 = 823543, so any term must have 6 or fewer digits and all those are present. - Michael S. Branicky, Apr 22 2023

			14610 is represented as 14610 in base 10 and as 60411 in base 7. Each representation is a permutation of the multiset {0,1,1,4,6}.

Cf. A037440, A074233.

Select[Range[10,110000],Sort[IntegerDigits[#]]==Sort[IntegerDigits[#,7]]&] (* Harvey P. Dale, Sep 23 2017 *)

from sympy.ntheory import digits
def ok(n): return sorted(map(int, str(n))) == sorted(digits(n, 7)[1:])
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Apr 22 2023

a(1)-a(7) inserted and a(43)-a(61) from Michael S. Branicky, Apr 22 2023

A096257 The least k whose n-th root contains k as a string of digits to the immediate right of the decimal point (excluding leading zeros).

Original entry on oeis.org

8, 2, 3, 633, 19703, 89, 69, 56, 46, 39, 33, 29, 25, 22, 20, 18, 16, 14, 13, 12, 11, 10, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 138, 133, 128, 124, 120, 116, 113, 109, 106, 103, 100, 97, 95, 92, 90, 87, 85, 83, 81, 79, 77, 75, 74, 72, 70, 69, 67, 66, 65, 63, 62, 61, 59, 58, 57

Offset: 2

Paul Lusch and Robert G. Wilson v, Jul 31 2004

Cf. A074841, A074762.

f[k_, n_] := Block[{l = Floor[ Log[10, k] + 1], rd = RealDigits[ k^(1/n), 10, 24], id = IntegerDigits[k]}, rdd = Drop[ rd[[1]], rd[[2]]]; While[ rdd[[1]] == 0, rdd = Drop[rdd, 1]]; Take[rdd, l] == id]; g[n_] := Block[{k = 2}, While[IntegerQ[k^(1/n)] || f[k, n] == False, k++ ]; k]; Table[ g[n], {n, 2, 72}]

import re
from sympy import perfect_power
from decimal import *
getcontext().prec = 24
def lzs(s): return len(s) - 2 - len(s[2:].lstrip('0')) # # of leading zeros
def cond(sk, sroot, k, n): # is condition true, with precision verification
    if perfect_power(k, [n]): return False # decimal part should be all 0's
    assert lzs(sroot) + len(sk) < len(sroot) - 3, (n, "increase precision")
    return re.match("0.0*"+sk, sroot)
def a(n):
    k, power = 1, Decimal(1)/Decimal(n)
    rootk, sk = Decimal(k)**power, str(k)
    while not cond(sk, str(rootk - int(rootk)), k, n):
        k += 1
        rootk, sk = Decimal(k)**power, str(k)
    return k
print([a(n) for n in range(2, 73)]) # Michael S. Branicky, Aug 02 2021

A087672 n = k^2 - (reversal of k)^2 for two different values of k.

Original entry on oeis.org

891, 1485, 1584, 2376, 3168, 4455, 4752, 89991, 95931, 101871, 107811, 149985, 155925, 159984, 161865, 167805, 167904, 175824, 183744, 191664, 239976, 247896, 255816, 263736, 271656, 319968, 327888, 335808, 343728, 351648, 449955, 479655

Offset: 0

Paul Lusch, Sep 26 2003

nonn

Each n is a multiple of 99. The first few pairs for k: (30,54) (41,87) (40,53) (51,75) (62,97) (72,96) (71,84) (300,504) (310,534) (320,564) (330,594) (401,807) (411,837) (400,503) (421,867) (431,897) (410,523) (420,543) (430,563) (440,583).

			2376 = 51^2 - 15^2 = 75^2 - 57^2; 161865 = 421^2 - 124^2 = 867^2 - 768^2

A075761 Numbers n such that there are at least 3 integers k from the set {2,3,4,5,6,7,8,9} such that the digital sum of each base k representation of n equals k.

Original entry on oeis.org

13, 17, 19, 21, 25, 29, 31, 33, 37, 41, 57, 61, 65, 73, 85, 91, 121, 129, 133, 145, 169, 225, 253, 257, 261, 265, 281, 301, 325, 337, 361, 385, 505, 513, 757, 785, 841, 1057, 1081, 1093, 1381, 1625, 2080, 2241, 3241, 4117, 4377, 4401, 5125, 8281, 16416

Offset: 1

Paul Lusch, Oct 08 2002

13, 33, 37 and 57 work for 4 bases. 85 is the only integer found that works for 5 bases: 3, 4, 5, 7 and 8.

			91 in base 3 = 10101 and 1 + 0 + 1 + 0 + 1 = 3, which is the base. 91 in base 6 = 231 and 2 + 3 + 1 = 6, which is the base. 91 in base 7 = 160 and 1 + 6 + 0 = 7, which is the base.

A074762 Fifth root of n contains n as a string of digits to the immediate right of the decimal point (excluding leading zeros).

Original entry on oeis.org

633, 634, 635, 636, 637, 638, 639, 877, 878, 879, 880, 881, 882, 883, 884, 1185, 5061, 33459, 438240, 682290, 17263489, 188423892, 991790057, 7231603790, 75314706735, 62651040995719, 296757769625554, 4295141978111813, 14929328605861651, 516659008545595106

Offset: 1

Paul Lusch, Sep 06 2002

			Fifth root of 33459 = 8.033459...

Jon E. Schoenfield, Table of n, a(n) for n = 1..50
Jon E. Schoenfield, Magma code
Robert Tanniru, PARI Code

Cf. A074841, A232086, A232110.

f[n_] := Block[{l = Floor[ Log[10, n] + 1], rd = RealDigits[n^(1/5), 10, 24], id = IntegerDigits[n]}, rdd = Drop[ rd[[1]], rd[[2]]]; While[ rdd[[1]] == 0, rdd = Drop[ rdd, 1]]; Take[ rdd, l] == id]; Do[ If[ StringPosition[ ToString[ N[ n^(1/5), 24]], ToString[ n]] != {}, If[ f[n], Print[ n]]], {n, 2, 170000000}] (* Robert G. Wilson v, Jul 30 2004 *)

/* Uses PARI functions provided in link
* Note: does not predict 639 due to simplification error and
* 877-884 due to checking only first solutions to the Grafting Equation.
* Sample run uses a = [0,16], b=10, p=5, direct=True */
GetAllGIs(0,16,10,5,1) \\ Robert Tanniru, Nov 20 2013

Edited and extended by Robert G. Wilson v, Jul 31 2004
a(22)-a(25) by Robert Tanniru, Nov 20 2013
More terms from Jon E. Schoenfield, Aug 17 2014

A074841 Square root of n contains n as a string of digits to the immediate right of the decimal point (excluding leading zeros).

Original entry on oeis.org

8, 77, 5711, 9797, 77327, 997997, 8053139, 60755907, 62996069, 99979997, 9999799997, 71515443427, 76933604839, 93445113269, 999997999997

Offset: 1

Paul Lusch, Sep 09 2002

All numbers of the form (10^n-3)*(10^n+1), n > 0, are members. - Robert G. Wilson v, Aug 02 2004

			The square root of 77327 = 278.077327...

Robert Tanniru, PARI Code

Cf. A232086, A232110, A074762.

f[n_] := Block[{l = Floor[ Log[10, n] + 1], rd = RealDigits[ Sqrt[n], 10, 24], id = IntegerDigits[n]}, rdd = Drop[ rd[[1]], rd[[2]]]; While[ rdd[[1]] == 0, rdd = Drop[rdd, 1]]; Take[rdd, l] == id]; Do[ If[ StringPosition[ ToString[ N[ Sqrt[n], 24]], ToString[n]] != {}, If[ f[n], Print[n]]], {n, 2, 6 10^8}] (* Robert G. Wilson v, Aug 02 2004 *)

/* Uses PARI functions provided in link
* Sample run uses a = [0,12], b=10, p=2, direct=True */
GetAllGIs(0,12,10,2,1) \\ Robert Tanniru, Nov 20 2013

More terms from Robert G. Wilson v, Aug 02 2004

New term a(13) inserted by Robert Tanniru, Nov 20 2013

A073585 Square root of n has the same nonzero digit in each of the first 4 places to the right of the decimal point.

Original entry on oeis.org

5168, 6543, 8080, 9779, 11640, 13663, 15848, 18195, 20704, 23375, 26208, 29203, 32360, 35679, 39160, 42803, 46608, 50575, 51731, 54704, 55906, 58995, 60243, 63448, 64742, 68063, 69403, 72840, 74226, 77779, 79211, 81542, 82880, 84358, 86763, 88143, 89667

Offset: 1

Paul Lusch, Aug 28 2002

For each term given, the repeated digit is either 8 or 4.

			The square root of 5168 is 71.88880302...

Harvey P. Dale, Table of n, a(n) for n = 1..630

re[n_] := Union[First[RealDigits[Sqrt[n], 10, 4, -1]]]; t = {}; Do[If[Length[x = re[n]] == 1 && x != {0}, AppendTo[t, n]], {n, 90000}]; t (* Jayanta Basu, Jul 02 2013 *)
snd4Q[n_]:=Module[{rd=RealDigits[Sqrt[n],10,20],d4},d4=Take[ Drop[ rd[[1]], rd[[2]]], 4];d4[[1]]!=0&&Union[Differences[d4]]=={0}]; Select[ Range[100000],snd4Q] (* Harvey P. Dale, Jul 22 2016 *)

isA073585(n)=if(issquare(n),0,my(x=sqrt(n));x-=floor(x);x=floor(10000*x);x%1111==0)

A074233 Numbers whose base-4 and base-5 representations are permutations of the same multiset of digits.

Original entry on oeis.org

0, 1, 2, 3, 28, 137, 210, 692, 717, 933, 3138, 3213, 3416, 3467, 3538, 16068

Offset: 1

Paul Lusch, Sep 18 2002

This sequence is finite and full. Proof: the smallest base-5 number with m digits is 5^(m-1). The largest base-4 number with m digits is 4^m - 1. For m >= 8, 5^(m-1) > 4^m - 1. So any term is <= 4^8 which has been checked. - David A. Corneth, Jan 24 2022

			692 is represented as 22310 in base 4 and as 10232 in base 5. Each representation is a permutation of the multiset {0,1,2,2,3}.

Cf. A007090, A007091, A037423.

isok(m) = vecsort(digits(m, 4)) == vecsort(digits(m, 5)); \\ Michel Marcus, Jan 24 2022

First four terms inserted by David A. Corneth, Jan 24 2022

A074119 Seventh root of n contains n as a string of digits to the immediate right of the decimal point (excluding leading zeros).

Original entry on oeis.org

89, 90, 16874, 25077, 479505, 306577056, 3821075079, 18014062431, 23075041700, 7240367851167, 85944742335578, 359069276640550, 809162747122740, 41275883437937369, 3209114244021563000, 69831531751710887320, 236842309259501676015, 12355587970480207660102, 79903263070494746587634

Offset: 1

Paul Lusch, Sep 16 2002

			Seventh root of 16874 = 4.016874...

Sean A. Irvine, Table of n, a(n) for n = 1..29

Cf. A074762, A074841.

More terms from Bert Dobbelaere, Jun 23 2024

cp's OEIS Frontend

User: Paul Lusch

A292125 Positive numbers n such that the set of base-7 digits of n equals both the set of base-8 digits of n and the set of base-9 digits of n.

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A130604 Numbers whose base-10 and base-7 representations are permutations of the same multiset of digits.

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A096257 The least k whose n-th root contains k as a string of digits to the immediate right of the decimal point (excluding leading zeros).

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A087672 n = k^2 - (reversal of k)^2 for two different values of k.

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A075761 Numbers n such that there are at least 3 integers k from the set {2,3,4,5,6,7,8,9} such that the digital sum of each base k representation of n equals k.

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A074762 Fifth root of n contains n as a string of digits to the immediate right of the decimal point (excluding leading zeros).

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A074841 Square root of n contains n as a string of digits to the immediate right of the decimal point (excluding leading zeros).

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A073585 Square root of n has the same nonzero digit in each of the first 4 places to the right of the decimal point.

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A074233 Numbers whose base-4 and base-5 representations are permutations of the same multiset of digits.

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