A074841 -id:A074841 - cp's OEIS frontend

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

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

A232086 Third 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

2, 39, 48570, 70293094, 97959170, 383263523, 7141269931, 52167799575, 54592884236, 80834974860, 3224757993012, 8391216236921, 174753523862043, 2248771925089484, 355191775894066192, 758148263300700696, 3004862096444523247, 9336508574693449683, 71580261944407825851

Offset: 1

Robert Tanniru, Nov 17 2013

			97959170^(1/3) = 460.97959170151...

Robert Tanniru, PARI code

Cf. A074841, A232110, A074762

/* PARI functions provided in extra link. */
/* Sample Run Using a = [0,12], b=10, p=3 */
GetAllGIs(0,12,10,3,1)

a(11)-a(12) added by Robert Tanniru, Nov 20 2013

More terms from Bert Dobbelaere, Jun 23 2024

A232110 Fourth 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

3, 4, 27, 1913227, 9821998, 3588613885932, 7625632704605, 50859949338383, 21029300554772499, 97202454420912990, 440023525444970228, 783944985766933369, 1277151495727998611, 2283977463662240937, 72927208535053310211, 365439872472838714161, 740751647624914930138

Offset: 1

Robert Tanniru, Nov 18 2013

			1913227^(1/4) = 37.19132279207...

Sean A. Irvine, Table of n, a(n) for n = 1..19
Robert Tanniru, PARI code

Cf. A074841, A232086, A074762.

isok(n) = {if (ispower(n, 4), return (0)); fr = frac(n^(1/4)); while (frac(fr) < 1/10, fr *= 10); nd = length(digits(n)); fr *= 10^nd; floor(fr) == n;} \\ Michel Marcus, Nov 20 2013

/*Sample Run Using a = [0,14], b=10, p=4 using PARI code in link */
GetAllGIs(0,14,10,4,1)

More terms from Bert Dobbelaere, Jun 23 2024

A232087 Second-order base-10 grafting integers.

Original entry on oeis.org

0, 1, 8, 77, 98, 99, 100, 764, 765, 5711, 5736, 9797, 9998, 9999, 10000, 76394, 77327, 997997, 999998, 999999, 1000000, 2798254, 7639321, 8053139, 25225733, 42808341, 57359313, 60755907, 62996069, 99979997, 99999998, 99999999, 100000000, 127016654

Offset: 1

Robert Tanniru, Nov 17 2013

Second-order base-10 grafting integers are integers that, when expressed in base 10, will appear in their own square root before or directly after the decimal point (ignoring leading 0's and including trailing 0's).
All numbers of the form 10^2n, 10^2n - 1, and 10^2n - 2, n >= 1, are terms.
All numbers of the form (10^n-3)*(10^n+1), n > 0, are terms.

			sqrt(764) = 27.64054992...
sqrt(77327) = 278.0773273749...
sqrt(1000000) = 1000.000...

Robert Tanniru, A short note introducing Grafting Numbers and their connection to Catalan Numbers, J. Comb. Math. and Comb. Computing, 95 (2015), 309-312.

Robert Tanniru, Introduction to Grafting Numbers.
Robert Tanniru, PARI code.
Robert Tanniru, A short note introducing Grafting Numbers and their connection to Catalan Numbers, ResearchGate, 2015.

Cf. A074841 (subsequence).

/* Uses PARI functions provided in link
* Sample run uses a = [0,11], b=10, p=2, direct=FALSE */
GetAllGIs(0,11,10,2,0)

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

A105429 Numbers k such that square root of k contains the reverse of k as a string of digits to the immediate right of the decimal point (excluding leading zeros).

Original entry on oeis.org

8, 77, 98, 1439, 179848, 2723643, 3522548, 6428805, 805039212, 37574751287, 536098097002, 927422715988

Offset: 1

Gil Broussard, Apr 08 2005

a(11) > 2*10^11. If terms ending in 0 were allowed, then 3970 and 5490 would be terms, since sqrt(3970) = 63.00793... and sqrt(5490) = 74.0945... . - Giovanni Resta, Aug 08 2019

a(13) > 10^12. - Mauro Simonato, Sep 25 2021

			a(5)=179848 because the square root of 179848 (whose reverse is 848971) = 424.08489716...

Cf. A074841.

a(9)-a(10) from Giovanni Resta, Aug 08 2019

a(11)-a(12) from Mauro Simonato, Sep 25 2021

A272208 Numbers n such that n immediately follows the decimal point in the base-10 representation of sqrt(n).

Original entry on oeis.org

8, 77, 5711, 9797, 997997, 8053139, 60755907, 99979997, 9999799997, 71515443427, 93445113269, 999997999997, 26369408771424, 96872443448748, 99650905131203, 99999979999997, 751273714618266, 3368237924952647, 3493498117381256, 9999999799999997, 35399255736521405, 999999997999999997

Offset: 1

David W. Wilson, May 15 2016

Sequence includes all numbers of form (10^n-3)(10^n+1) for n >= 1, hence is infinite.

			5711 immediately follows the decimal in sqrt(5711) = 75.5711+, so 5711 is in the sequence.
77327 does not immediately follow the decimal in sqrt(77327) = 278.077327+, so 77327 is not in this sequence.

David W. Wilson, Table of n, a(n) for n = 1..27

Strict subsequence of A074841.

is(n)=my(t=10^#Str(n)); sqrtint(t^2*n)%t==n \\ Charles R Greathouse IV, May 16 2016

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

A099400 Square root of a(n) contains the n-th prime as a string of digits to the immediate right of the decimal point (excluding leading zeros).

Original entry on oeis.org

5, 11, 21, 3, 83, 124, 201, 27, 5, 28, 11, 179, 2, 89, 20, 91, 92, 58, 114, 50, 3, 23, 34, 24, 288, 2411, 581, 1377, 1031, 489, 1531, 366, 849, 632, 406, 536, 367, 2721, 13495, 537, 634, 492, 686, 331, 1866, 408, 52, 409, 485, 688, 297, 742, 1105, 12377, 856, 1174

Offset: 1

Gil Broussard, Nov 17 2004

			a(1)= 5 because sqrt( 5)=2.(2)236...
a(2)=11 because sqrt(11)=3.(3)316...
a(3)=21 because sqrt(21)=4.(5)825...
a(4)= 3 because sqrt( 3)=1.(7)320...
...
a(100) =1125 because sqrt(1125)=33.5410... and 541 is the 100th prime.

Cf. A074841, A074762.

A099401 Square root of a(n) contains the n-th Fibonacci number as a string of digits to the immediate right of the decimal point (excluding leading zeros).

Original entry on oeis.org

10, 10, 5, 11, 21, 8, 124, 52, 54, 43, 24, 970, 297, 457, 467, 1520, 2516, 7269, 12414, 3804, 11048, 25020, 135635, 56389, 710228, 44151, 21082, 762684, 696414, 1085414, 6472621, 2979828, 15220551, 72130, 9934617, 79533387

Offset: 1

Gil Broussard, Nov 17 2004

			a(1)= 10 because sqrt( 10)= 3.(1)622...
a(2)= 10 because sqrt( 10)= 3.(1)622...
a(3)= 5 because sqrt( 5)= 2.(2)360...
a(4)= 11 because sqrt( 11)= 3.(3)166...
a(5)= 21 because sqrt( 21)= 4.(5)825...
a(6)= 8 because sqrt( 8)= 2.(8)284...
a(7)= 124 because sqrt(124)=11.(13)55...
etc.

Cf. A074841, A074762.

Do[x = IntegerDigits[Fibonacci[n]]; i = 1; l = {}; While[l != x, d = RealDigits[N[Sqrt[i], 100]]; l = Take[Drop[First[d], Last[d]], Length[x]]; i++ ]; Print[i-1], {n, 1, 36}] (* Ryan Propper, Aug 11 2005 *)

Corrected and extended by Ryan Propper, Aug 11 2005

cp's OEIS Frontend

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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A232086 Third 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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A232110 Fourth 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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PARI

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A232087 Second-order base-10 grafting integers.

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A074119 Seventh 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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A105429 Numbers k such that square root of k contains the reverse of k as a string of digits to the immediate right of the decimal point (excluding leading zeros).

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A272208 Numbers n such that n immediately follows the decimal point in the base-10 representation of sqrt(n).

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PARI

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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Mathematica

Python

A099400 Square root of a(n) contains the n-th prime as a string of digits to the immediate right of the decimal point (excluding leading zeros).

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