A256633 -id:A256633

A256631

Numbers n such that the decimal expansions of both n and n^2 have 0 as smallest digit and 5 as largest digit.

+10
24

50, 105, 150, 205, 350, 450, 500, 501, 502, 505, 550, 1005, 1015, 1050, 1055, 1105, 1150, 1205, 1450, 1500, 1501, 1550, 2005, 2050, 2055, 2105, 2305, 2350, 3350, 3500, 4500, 5000, 5001, 5002, 5005, 5010, 5011, 5012, 5015, 5020, 5021, 5032, 5045, 5050, 5055

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Felix Fröhlich, Table of n, a(n) for n = 1..10000

MATHEMATICA

fQ[n_] := Block[{c = DigitCount@ n}, And[Plus @@ Take[c, {6, 9}] == 0, c[[5]] > 0, c[[10]] > 0]]; Select[Range@ 5100, fQ@ # && fQ[#^2] &] (* Michael De Vlieger, Apr 12 2015 *)

Select[Range[5100], Min[IntegerDigits[#]]==0&&Max[IntegerDigits[#] ]== 5 && Min[IntegerDigits[#^2]]==0&&Max[IntegerDigits[#^2]]==5&] (* Harvey P. Dale, Jan 19 2020 *)

PROG

(PARI) is(n) = vecmin(digits(n))==0 && vecmin(digits(n^2))==0 && vecmax(digits(n))==5 && vecmax(digits(n^2))==5

CROSSREFS

Cf. A136817, A256630, A256633, A256634, A256708, A256709, A256889, A257086.

KEYWORD

nonn,base

AUTHOR

Felix Fröhlich, Apr 05 2015

STATUS

approved

A256630

Numbers n such that the decimal expansions of both n and n^2 have 0 as smallest digit and 4 as largest digit.

+10
23

142201, 1422010, 11141110, 11411110, 11412021, 14220100, 20323421, 21024111, 101203421, 110141011, 110142201, 111411100, 114111100, 114120210, 120013421, 141433102, 142201000, 203234210, 210241110, 1012034210, 1101410011, 1101410110, 1101422010, 1114111000

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..10000

MATHEMATICA

fQ[n_] := Block[{c = DigitCount@ n}, And[Plus @@ Take[c, {5, 9}] == 0, c[[4]] > 0, c[[10]] > 0]]; Select[Range@ 10000000, fQ@ # && fQ[#^2] &] (* Michael De Vlieger, Apr 12 2015 *)

PROG

(PARI) is(n) = vecmin(digits(n))==0 && vecmin(digits(n^2))==0 && vecmax(digits(n))==4 && vecmax(digits(n^2))==4

(Python)

from itertools import product

A256630_list = []

for l in range(11):

for a in ('1', '2', '3', '4'):

for b in product('01234', repeat = l):

for c in ('0', '1', '2'):

s = a+''.join(b)+c

if '0' in s and '4' in s:

n = int(s)

s2 = set(str(n**2))

if {'0', '4'} <= s2 <= {'0', '1', '2', '3', '4'}:

A256630_list.append(n)

print(A256630_list) # Chai Wah Wu, Apr 17 2015

CROSSREFS

Cf. A256631, A256633, A256634, A256708, A256709, A256889.

Subsequence of A136810.

KEYWORD

nonn,base

AUTHOR

Felix Fröhlich, Apr 05 2015

EXTENSIONS

More terms from Alois P. Heinz, Apr 16 2015

STATUS

approved

A256634

Numbers n such that the decimal expansions of both n and n^2 have 0 as smallest digit and 7 as largest digit.

+10
23

740, 760, 1071, 1740, 2074, 2705, 2710, 2740, 2750, 2760, 3705, 3710, 3760, 4071, 4705, 4740, 4760, 5071, 5705, 5760, 6740, 7074, 7240, 7260, 7400, 7550, 7560, 7600, 7601, 7760, 10076, 10174, 10274, 10275, 10371, 10375, 10376, 10571, 10710, 10724, 10726, 10740

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Felix Fröhlich, Table of n, a(n) for n = 1..10000

MATHEMATICA

fQ[n_] := Block[{c = DigitCount@ n}, And[Plus @@ Take[c, {8, 9}] == 0, c[[7]] > 0, c[[10]] > 0]]; Select[Range@ 10800, fQ@ # && fQ[#^2] &] (* Michael De Vlieger, Apr 12 2015 *)

PROG

(PARI) is(n) = vecmin(digits(n))==0 && vecmin(digits(n^2))==0 && vecmax(digits(n))==7 && vecmax(digits(n^2))==7

CROSSREFS

Cf. A136813, A256630, A256631, A256633, A256708, A256709.

KEYWORD

nonn,base

AUTHOR

Felix Fröhlich, Apr 05 2015

STATUS

approved

A256708

Numbers n such that the decimal expansions of both n and n^2 have 0 as smallest digit and 8 as largest digit.

+10
23

280, 508, 780, 805, 1028, 1078, 1280, 1308, 1680, 1780, 1805, 1840, 2078, 2608, 2680, 2780, 2800, 2801, 2802, 2805, 2840, 2850, 3280, 3580, 3780, 3805, 3808, 3850, 4048, 4078, 4280, 4780, 4804, 4805, 4880, 5008, 5018, 5028, 5048, 5078, 5080, 5084, 5180, 5280

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Felix Fröhlich, Table of n, a(n) for n = 1..10000

MATHEMATICA

fQ[n_] := Block[{c = DigitCount@ n}, And[c[[9]] == 0, c[[8]] > 0, c[[10]] > 0]]; Select[Range@ 5280, fQ@ # && fQ[#^2] &] (* Michael De Vlieger, Apr 12 2015 *)

maxdQ[n_]:=Module[{id1=IntegerDigits[n], id2=IntegerDigits[n^2]}, Max[ id1] == Max[ id2] == 8&&Min[id1]==Min[id2]==0]; Select[Range[6000], maxdQ] (* Harvey P. Dale, Oct 19 2021 *)

PROG

(PARI) is(n) = vecmin(digits(n))==0 && vecmin(digits(n^2))==0 && vecmax(digits(n))==8 && vecmax(digits(n^2))==8

CROSSREFS

Cf. A136814, A136820, A136825, A136829, A136832, A136840, A136845, A136849, A136904, A256630, A256631, A256633, A256634, A256709.

KEYWORD

nonn,base

AUTHOR

Felix Fröhlich, Apr 08 2015

STATUS

approved

A256709

Numbers n such that the decimal expansions of both n and n^2 have 0 as smallest digit and 9 as largest digit.

+10
23

509, 890, 903, 905, 930, 950, 960, 970, 980, 990, 1039, 1091, 1095, 1097, 1390, 1709, 1903, 1905, 1930, 1970, 1980, 1990, 2049, 2093, 2095, 2097, 2190, 2509, 2809, 2903, 2905, 2907, 2930, 2970, 2990, 3009, 3049, 3079, 3090, 3092, 3095, 3097, 3098, 3099, 3209

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Felix Fröhlich, Table of n, a(n) for n = 1..10000

MATHEMATICA

fQ[n_] := Block[{c = DigitCount@ n}, And[c[[9]] > 0, c[[10]] > 0]]; Select[Range@ 3209, fQ@ # && fQ[#^2] &] (* Michael De Vlieger, Apr 12 2015 *)

PROG

(PARI) is(n) = vecmin(digits(n))==0 && vecmin(digits(n^2))==0 && vecmax(digits(n))==9 && vecmax(digits(n^2))==9

CROSSREFS

Cf. A136836, A256630, A256631, A256633, A256634, A256708.

KEYWORD

nonn,base

AUTHOR

Felix Fröhlich, Apr 08 2015

STATUS

approved

A256889

Numbers k such that the decimal expansions of both k and k^2 have 1 as smallest digit and 5 as largest digit.

+10
20

115, 1115, 1235, 3515, 11115, 12335, 12415, 33515, 35415, 123335, 123512, 124235, 145415, 152132, 231115, 235211, 333515, 1114115, 1155211, 1233335, 1531115, 1534312, 2311115, 3333515, 11114115, 11141115, 11145511, 12333335, 12342335, 15334312, 15531115

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

k can only begin with 1, 2 or 3 and k mod 10 can only equal 1, 2 or 5. - Robert G. Wilson v, Apr 13 2015

Heuristics suggest that this sequence should be infinite and the sequence with 4 in place of 5 should be finite. The latter sequence contains no terms up to 10^30. - Charles R Greathouse IV, Mar 20 2022

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..2000 (n = 1..75 from Robert G. Wilson v).

MATHEMATICA

fQ[n_] := Block[{c = DigitCount@ n}, And[Plus @@ Take[c, {6, 10}] == 0, c[[1]] > 0, c[[5]] > 0]]; Select[Range@ 100000, fQ@ # && fQ[#^2] &] (* Michael De Vlieger, Apr 12 2015 *)

fQ[n_] := Block[{id1 = Union@ IntegerDigits[ n], id2 = Union@ IntegerDigits[ n^2]}, Min[id1] == Min[id2] == 1 && Max[id1] == Max[id2] == 5]; k = 1; lst = {}; While[k < 10^7, If[ fQ@ k, AppendTo[lst, k]]; k++; If[ fQ@ k, AppendTo[lst, k]]; k += 3; If[ fQ@ k, AppendTo[lst, k]]; k += 6]; lst (* Robert G. Wilson v, Apr 13 2015 *)

PROG

(PARI) is(n) = vecmin(digits(n))==1 && vecmin(digits(n^2))==1 && vecmax(digits(n))==5 && vecmax(digits(n^2))==5

CROSSREFS

Cf. A256630, A256631, A256633, A256634, A256708, A256709.

KEYWORD

base,nonn

AUTHOR

Felix Fröhlich, Apr 12 2015

STATUS

approved

A257197

Numbers k such that the decimal expansions of both k and k^2 have 1 as smallest digit and 6 as largest digit.

+10
16

116, 146, 1116, 1146, 1156, 1246, 1465, 1556, 1616, 3516, 3621, 4611, 4621, 4631, 11116, 11146, 11156, 11465, 11556, 11642, 15216, 16231, 21556, 22631, 25146, 25162, 25621, 33516, 34156, 35116, 35146, 35162, 36211, 36215, 36512, 46111, 46112, 46211, 46331

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Felix Fröhlich, Table of n, a(n) for n = 1..10000

MATHEMATICA

sd1Q[n_]:=Module[{idn=IntegerDigits[n]}, Min[idn]==1&&Max[idn]==6]; Select[ Range[50000], AllTrue[{#, #^2}, sd1Q]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 22 2020 *)

PROG

(PARI) is(n) = vecmin(digits(n))==1 && vecmin(digits(n^2))==1 && vecmax(digits(n))==6 && vecmax(digits(n^2))==6

CROSSREFS

Cf. A256630, A256631, A256633, A256634, A256708, A256709, A256889, A257210, A257211.

KEYWORD

base,nonn

AUTHOR

Felix Fröhlich, Apr 18 2015

EXTENSIONS

Corrected b-file from Felix Fröhlich, Mar 18 2022

STATUS

approved

A257210

Numbers n such that the decimal expansions of both n and n^2 have 1 as smallest digit and 7 as largest digit.

+10
16

271, 371, 1171, 1474, 1475, 1776, 2171, 2271, 2671, 2715, 2761, 3671, 3711, 4174, 4761, 4771, 6761, 7165, 7174, 7261, 7331, 11275, 11474, 11475, 11711, 11715, 11716, 11724, 11725, 11731, 12376, 12715, 12734, 12756, 12776, 13171, 13174, 13275, 13276, 14674

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

There are 2 3-digit terms, 19 4-digit terms, 122 5-digit terms, 646 6-digit terms, 3147 7-digit terms, 13300 8-digit terms, 54689 9-digit terms, and 216858 10-digit terms. - Charles R Greathouse IV, Apr 20 2015

LINKS

Felix Fröhlich, Table of n, a(n) for n = 1..10000

MATHEMATICA

fQ[n_] := Block[{d = DigitCount@ n}, Plus @@ Take[d, {8, 10}] == 0 && d[[1]] > 0 && d[[7]] > 0]; Select[Range@ 15000, fQ@ # && fQ[#^2] &] (* Michael De Vlieger, Apr 20 2015 *)

PROG

(PARI) is(n) = vecmin(digits(n))==1 && vecmin(digits(n^2))==1 && vecmax(digits(n))==7 && vecmax(digits(n^2))==7

(PARI) has(n)=my(d=Set(digits(n))); #d && d[1]==1 && d[#d]==7

is(n)=has(n) && has(n^2)

for(d=3, 7, for(i=6, 7^d-1, v=digits(i, 7); if(#v<=d, v=concat(vector(d-#v), v)); if(vecmax(v)==6 && vecmin(v)==0 && has((n=fromdigits(apply(k->k+1, v)))^2), print1(n", ")))) \\ Charles R Greathouse IV, Apr 20 2015

CROSSREFS

Cf. A256630, A256631, A256633, A256634, A256708, A256709, A256889, A257197, A257211.

KEYWORD

nonn,base

AUTHOR

Felix Fröhlich, Apr 18 2015

STATUS

approved

A257211

Numbers n such that the decimal expansions of both n and n^2 have 1 as smallest digit and 8 as largest digit.

+10
16

128, 178, 871, 1128, 1178, 1218, 1258, 1278, 1284, 1328, 1358, 1368, 1478, 1678, 1681, 1768, 1778, 1784, 1785, 1828, 1874, 1881, 2681, 2861, 2871, 3418, 3581, 3718, 3816, 3841, 4178, 4318, 4815, 4831, 4841, 4881, 5178, 5181, 5182, 5318, 5815, 5841, 5871, 5881

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Felix Fröhlich, Table of n, a(n) for n = 1..10000

MATHEMATICA

fQ[n_] := Block[{d = DigitCount@ n}, Plus @@ Take[d, {9, 10}] == 0 && d[[1]] > 0 && d[[8]] > 0]; Select[Range@ 6000, fQ@ # && fQ[#^2] &] (* Michael De Vlieger, Apr 20 2015 *)

sd1ld8Q[n_]:=With[{idn=IntegerDigits[n]}, Max[idn]==8&&Min[idn]==1]; Select[ Range[ 6000], AllTrue[{#, #^2}, sd1ld8Q]&] (* Harvey P. Dale, Oct 14 2022 *)

PROG

(PARI) is(n) = vecmin(digits(n))==1 && vecmin(digits(n^2))==1 && vecmax(digits(n))==8 && vecmax(digits(n^2))==8

CROSSREFS

Cf. A256630, A256631, A256633, A256634, A256708, A256709, A256889, A257197, A257210, A256601.

KEYWORD

nonn,base

AUTHOR

Felix Fröhlich, Apr 18 2015

STATUS

approved

A256601

Numbers n such that the decimal expansions of both n and n^2 have 1 as smallest digit and 9 as largest digit.

+10
13

139, 219, 519, 591, 719, 891, 911, 961, 971, 981, 1139, 1193, 1219, 1292, 1293, 1296, 1319, 1339, 1389, 1391, 1392, 1394, 1396, 1469, 1579, 1589, 1691, 1719, 1729, 1769, 1793, 1839, 1869, 1896, 1911, 1927, 1937, 1939, 1944, 1946, 1969, 1978, 1979, 1981, 1986

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Felix Fröhlich, Table of n, a(n) for n = 1..10000

MATHEMATICA

fQ[n_] := Block[{d = DigitCount@ n}, Last@ d == 0 && d[[1]] > 0 && d[[9]] > 0]; Select[Range@ 2000, fQ@ # && fQ[#^2] &] (* Michael De Vlieger, Apr 20 2015 *)

PROG

(PARI) is(n) = vecmin(digits(n))==1 && vecmin(digits(n^2))==1 && vecmax(digits(n))==9 && vecmax(digits(n^2))==9

CROSSREFS

Cf. A256630, A256631, A256633, A256634, A256708, A256709, A256889, A257197, A257210, A257211.

KEYWORD

nonn,base

AUTHOR

Felix Fröhlich, Apr 20 2015

STATUS

approved