A030175 -id:A030175 - cp's OEIS frontend

A043681 When cubed gives number composed just of the digits 0, 1, 2, 3.

0, 1, 10, 11, 100, 101, 110, 1000, 1001, 1010, 1100, 10000, 10001, 10010, 10100, 11000, 100000, 100001, 100010, 100100, 101000, 110000, 684917, 1000000, 1000001, 1000010, 1000100, 1001000, 1010000, 1100000, 6849170, 10000000, 10000001, 10000010, 10000100, 10001000

Offset: 1

Robert G. Wilson v, Jun 23 2001

684917*10^x (whose cube is 321302302131323213*10^3x) so far is the only entry not of the form 10^a + 10^b, a>b or simply 10^a.

Giovanni Resta, Table of n, a(n) for n = 1..187

Cf. A030175.

Do[ If[ Union[ IntegerDigits[n^3]] [[ -1]] < 4, Print[n] ], {n, 0, 10^8} ]

Offset corrected and terms a(33) and beyond added by Giovanni Resta, Mar 14 2020

A196516 Decimal expansion of the number x satisfying x*e^x=3.

Original entry on oeis.org

1, 0, 4, 9, 9, 0, 8, 8, 9, 4, 9, 6, 4, 0, 3, 9, 9, 5, 9, 9, 8, 8, 6, 9, 7, 0, 7, 0, 5, 5, 2, 8, 9, 7, 9, 0, 4, 5, 8, 9, 4, 6, 6, 9, 4, 3, 7, 0, 6, 3, 4, 1, 4, 5, 2, 9, 3, 2, 8, 7, 1, 5, 8, 3, 3, 1, 6, 6, 4, 9, 0, 5, 0, 4, 4, 4, 4, 4, 2, 9, 5, 7, 8, 8, 5, 6, 7, 8, 6, 6, 6, 8, 2, 2, 4, 3, 4, 6, 7, 4

Offset: 1

Clark Kimberling, Oct 03 2011

			1.049908894964039959988697070552897904589...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Plot[{E^x, 1/x, 2/x, 3/x, 4/x}, {x, 0, 2}]
t = x /. FindRoot[E^x == 1/x, {x, 0.5, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A030175 *)
t = x /. FindRoot[E^x == 2/x, {x, 0.5, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196515 *)
t = x /. FindRoot[E^x == 3/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196516 *)
t = x /. FindRoot[E^x == 4/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196517 *)
t = x /. FindRoot[E^x == 5/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196518 *)
t = x /. FindRoot[E^x == 6/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196519 *)
RealDigits[LambertW[3], 10, 50][[1]] (* _G. C. Greubel-, Nov 16 2017 *)

lambertw(3) \\ G. C. Greubel, Nov 16 2017

A196517 Decimal expansion of the number x satisfying x*e^x=4.

Original entry on oeis.org

1, 2, 0, 2, 1, 6, 7, 8, 7, 3, 1, 9, 7, 0, 4, 2, 9, 3, 9, 2, 1, 2, 0, 7, 4, 1, 6, 5, 4, 9, 5, 1, 5, 3, 4, 4, 7, 5, 0, 1, 5, 1, 2, 5, 2, 1, 8, 2, 9, 6, 2, 5, 9, 8, 1, 7, 3, 9, 2, 0, 3, 5, 9, 0, 7, 0, 0, 6, 3, 4, 1, 3, 2, 9, 8, 1, 7, 7, 2, 6, 7, 7, 2, 2, 7, 8, 2, 6, 1, 0, 4, 9, 7, 6, 5, 6, 8, 3, 7, 7

Offset: 1

Clark Kimberling, Oct 03 2011

			1.2021678731970429392120741654951534475015125218296259...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Plot[{E^x, 1/x, 2/x, 3/x, 4/x}, {x, 0, 2}]
t = x /. FindRoot[E^x == 1/x, {x, 0.5, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A030175 *)
t = x /. FindRoot[E^x == 2/x, {x, 0.5, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196515 *)
t = x /. FindRoot[E^x == 3/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196516 *)
t = x /. FindRoot[E^x == 4/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196517 *)
t = x /. FindRoot[E^x == 5/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196518 *)
t = x /. FindRoot[E^x == 6/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196519 *)
RealDigits[LambertW[4], 10, 50][[1]] (* G. C. Greubel, Nov 16 2017 *)

lambertw(4) \\ G. C. Greubel, Nov 16 2017

A196518 Decimal expansion of the number x satisfying x*e^x=5.

Original entry on oeis.org

1, 3, 2, 6, 7, 2, 4, 6, 6, 5, 2, 4, 2, 2, 0, 0, 2, 2, 3, 6, 3, 5, 0, 9, 9, 2, 9, 7, 7, 5, 8, 0, 7, 9, 6, 6, 0, 1, 2, 8, 7, 9, 3, 5, 5, 4, 6, 3, 8, 0, 4, 7, 4, 7, 9, 7, 8, 9, 2, 9, 0, 3, 9, 3, 0, 2, 5, 3, 4, 2, 6, 7, 9, 9, 2, 0, 5, 3, 6, 2, 2, 6, 7, 7, 4, 4, 6, 9, 9, 1, 6, 6, 0, 8, 4, 2, 6, 7, 8, 9

Offset: 1

Clark Kimberling, Oct 03 2011

			1.32672466524220022363509929775807966012...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Plot[{E^x, 1/x, 2/x, 3/x, 4/x}, {x, 0, 2}]
t = x /. FindRoot[E^x == 1/x, {x, 0.5, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A030175 *)
t = x /. FindRoot[E^x == 2/x, {x, 0.5, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196515 *)
t = x /. FindRoot[E^x == 3/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196516 *)
t = x /. FindRoot[E^x == 4/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196517 *)
t = x /. FindRoot[E^x == 5/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196518 *)
t = x /. FindRoot[E^x == 6/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196519 *)
RealDigits[LambertW[5], 10, 50][[1]] (* G. C. Greubel, Nov 16 2017 *)

lambertw(5) \\ G. C. Greubel, Nov 16 2017

A196519 Decimal expansion of the number x satisfying x*e^x=6.

Original entry on oeis.org

1, 4, 3, 2, 4, 0, 4, 7, 7, 5, 8, 9, 8, 3, 0, 0, 3, 1, 1, 2, 3, 4, 0, 7, 8, 0, 0, 7, 2, 1, 2, 0, 5, 8, 6, 9, 4, 7, 8, 6, 4, 3, 4, 6, 0, 8, 8, 0, 4, 3, 0, 2, 0, 2, 5, 6, 5, 5, 9, 4, 8, 4, 9, 6, 3, 4, 3, 3, 9, 9, 5, 9, 3, 2, 5, 9, 8, 3, 1, 1, 1, 6, 8, 5, 7, 6, 3, 8, 4, 2, 2, 2, 9, 9, 4, 4, 5, 6, 5, 1

Offset: 1

Clark Kimberling, Oct 03 2011

			1.43240477589830031123407800721205869478643460...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

Plot[{E^x, 1/x, 2/x, 3/x, 4/x}, {x, 0, 2}]
t = x /. FindRoot[E^x == 1/x, {x, 0.5, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A030175 *)
t = x /. FindRoot[E^x == 2/x, {x, 0.5, 1}, WorkingPrecision -> 100]
RealDigits[t]  (* A196515 *)
t = x /. FindRoot[E^x == 3/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196516 *)
t = x /. FindRoot[E^x == 4/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196517 *)
t = x /. FindRoot[E^x == 5/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196518 *)
t = x /. FindRoot[E^x == 6/x, {x, 0.5, 2}, WorkingPrecision -> 100]
RealDigits[t]  (* A196519 *)
RealDigits[LambertW[6], 10, 50][[1]] (* G. C. Greubel, Nov 16 2017 *)

lambertw(6) \\ G. C. Greubel, Nov 16 2017

Terms a(95) onward corrected by G. C. Greubel, Nov 16 2017

A030174 Squares composed of digits {1,2,3}.

Original entry on oeis.org

1, 121, 12321, 1322122321, 132233322321, 213223221121, 13223121322321, 1212111311221321, 131332121232312121, 31121322221111133211321, 3322133322121313132333322321, 12232213113321121113332133323121, 3231221313313311221231322223122312333121, 311323333121312322332133323111223321313321

Offset: 1

Patrick De Geest

Patrick De Geest, Squares containing at most three distinct digits, Index entries for related sequences.
Hisanori Mishima, Sporadic tridigital solutions.
Eric Weisstein's World of Mathematics, Square Number.
Author?, Source(txt)

Cf. A030175.

a(n) = A030175(n)^2. - Elmo R. Oliveira, Jul 17 2025

More terms from Patrick De Geest, Mar 15 2000.
More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 14 2005
a(13)-a(14) from Mishima's website added by Elmo R. Oliveira, Jul 17 2025

A031997 Odd numbers which when cubed give number composed just of the digits 0, 1, 2, 3.

Original entry on oeis.org

1, 11, 101, 1001, 10001, 100001, 684917, 1000001, 10000001, 100000001, 1000000001, 10000000001, 100000000001, 1000000000001, 10000000000001, 100000000000001, 1000000000000001, 10000000000000001, 100000000000000001, 1000000000000000001

Offset: 1

Robert G. Wilson v, Jun 23 2001

Note that 684917 (whose cube is 321302302131323213) so far is the only entry not of the form 10^x + 1.

Cf. A030175, A043681, A278936, A278937.

Do[ If[ Union[ IntegerDigits[ n^3 ] ] [ [ -1 ] ] < 4, Print[ n ] ], {n, 1, 10^9, 2} ] (* corrected by Friedjof Tellkamp, Apr 24 2025 *)
(* faster code *)
DigitsLEQ3[n_] := And @@ (LessEqual[#, 3] & /@ IntegerDigits[n])
Arr = {1, 7}; For[i = 1, i < 10, i++, Arr = Flatten[Table[Select[Arr + 10^i j, DigitsLEQ3[Mod[#^3, 10^(i+1)]] &], {j, 0, 9}]]];
Select[Arr, DigitsLEQ3[#^3] &] (* Friedjof Tellkamp, Apr 25 2025 *)

A031997_list = [n for n in range(1,10**6,2) if max(str(n**3)) <= '3'] # Chai Wah Wu, Feb 23 2016

Term 0 removed and a(12)-a(17) added by Chai Wah Wu, Feb 25 2016

a(18)-a(20) from Giovanni Resta, Mar 14 2020

A136812 Numbers k such that k and k^2 use only the digits 0, 1, 2, 3 and 6.

Original entry on oeis.org

0, 1, 6, 10, 11, 60, 100, 101, 106, 110, 111, 361, 600, 601, 1000, 1001, 1006, 1010, 1011, 1060, 1100, 1101, 1106, 1110, 1631, 3606, 3610, 6000, 6001, 6010, 6011, 10000, 10001, 10006, 10010, 10011, 10060, 10100, 10101, 10106, 10110, 10111, 10301, 10306, 10600, 11000, 11001, 11006, 11010, 11060, 11100, 11101, 16310, 32111, 36060, 36100, 36361

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.

			1031316261^2 = 1063613230203020121.

Jonathan Wellons, Table of n, a(n) for n = 1..1315
J. Wellons, Tables of Shared Digits [archived]
Index entries for sequences related to squares.

Cf. A136808, ..., A137147.

Cf. A006716 = A027675^2, A030174 = A030175^2, A030176 = A030177^2, A058411^2 = A058412, ..., A058473^2 = A058474.

A294660 Least nonnegative integer not occurring earlier whose square has no digit in common with the square of the previous term, a(0) = 0.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 8, 10, 15, 12, 16, 20, 11, 22, 13, 18, 14, 28, 19, 17, 21, 23, 26, 29, 24, 30, 25, 33, 58, 27, 34, 47, 38, 45, 31, 48, 41, 50, 37, 52, 44, 65, 40, 57, 76, 32, 63, 35, 60, 39, 62, 36, 88, 46, 67, 51, 183, 75, 43, 55, 42, 53, 56, 70, 61, 64, 85, 59, 77, 69, 73, 78, 89

Offset: 0

M. F. Hasler, Nov 08 2017

This is not a permutation of the nonnegative integers, since numbers whose square has all digits '1' through '9' (cf. A294661, e.g., 11826 with 11826^2 = 139854276) can never appear - and these numbers have asymptotic density 1.

Will all integers whose square does not have all of the digits 1-9, eventually appear? Or might the sequence be finite? Since a(n)^2 has no digits in common with a(n-1)^2, it is sufficient for a(n+1) to exist, to find a number whose square has a subset of the digits of a(n-1)^2. Is this always possible? This problem sometimes has only "sporadic k-digital solutions", see, e.g., A058430, A030175, ... and the link to De Geest's page.

			Since a(7)^2 = 7^2 = 49, the subsequent term cannot be 8, since 8^2 = 64 has the digit 4 in common with 49. Therefore, a(8) = 9, with 9^2 = 81 having no common digit with 49.
a(1201) = 1037. So the square of the next term must not have any of the digits in {0, 1, 3, 5, 6, 7, 9}, only 2, 4, 8 are allowed. The least such number that has not been used before is a(1202) = 210912978, with a(1202)^2 = 210912978^2 = 44484284288828484. - _Alois P. Heinz_, Nov 09 2017

M. F. Hasler, Table of n, a(n) for n = 0..4829
P. De Geest, Squares containing at most three distinct digits, Index entries for related sequences

Cf. A030287 (strictly increasing), A067581 (do not take squares).

{u=a=0; for(n=0, 99, print1(a", "); u+=1<

{u=[a=0]; for(n=0, 99, print1(a", "); D=Set(if(a, digits(a^2))); for(k=u[1]+1, oo, setsearch(u, k)&&next; #setintersect(D, Set(digits(k^2)))&&next; u=setunion(u,[a=k]); break); while(#u>1&&u[2]==u[1]+1,u=u[^1])); a}

cp's OEIS Frontend

A043681 When cubed gives number composed just of the digits 0, 1, 2, 3.

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A196516 Decimal expansion of the number x satisfying x*e^x=3.

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A196517 Decimal expansion of the number x satisfying x*e^x=4.

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A196518 Decimal expansion of the number x satisfying x*e^x=5.

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A196519 Decimal expansion of the number x satisfying x*e^x=6.

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A030174 Squares composed of digits {1,2,3}.

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A031997 Odd numbers which when cubed give number composed just of the digits 0, 1, 2, 3.

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A136812 Numbers k such that k and k^2 use only the digits 0, 1, 2, 3 and 6.

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A294660 Least nonnegative integer not occurring earlier whose square has no digit in common with the square of the previous term, a(0) = 0.

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