A135463 -id:A135463 - cp's OEIS frontend

A134962 Numbers n with property that for each single digit d of n, we can also see the decimal expansion of d^2 as a substring of n. Also n may not contain any 0 digits.

1, 11, 111, 1111, 11111, 111111, 1111111, 3648169, 3649816, 3681649, 3698164, 8163649, 8164369, 8164936, 8169364, 9364816, 9368164, 9816364, 9816436, 11111111, 13648169, 13649816, 13681649, 13698164, 16364819, 16364981

Offset: 1

Zak Seidov and N. J. A. Sloane, Feb 03 2008

The number of terms less than 10^k: 1, 2, 3, 4, 5, 6, 19, 410, 8083, ... . - Robert G. Wilson v, Jan 06 2012

			In 3648169, for 3 we can see 9, for 6 we can see 36, for 4 we can see 16, for 8 we can see 64, for 1 we can see 1 and for 9 we can see 81.

Robert G. Wilson v, Table of n, a(n) for n = 1..11523 (first 300 terms from David Applegate)

Cf. A134439, A134692, A134698, A135463, A135464.

fQ[n_] := (id = IntegerDigits@ n; Union[id][[1]] != 0 && Sort[ StringPosition[ ToString[n], ToString[#]] & /@ Evaluate[ id^2]][[1]] != {}); k = 0; lst = {}; While[k < 2*10^7, If[fQ@k, AppendTo[lst, k]; Print@ k]; k++] (* Robert G. Wilson v, Jan 06 2012 *)

sq = {d:str(int(d)**2) for d in "123456789"}
def ok(n): return "0" not in (s:=str(n)) and all(sq[d] in s for d in set(s))
print([k for k in range(10**7) if ok(k)]) # Michael S. Branicky, May 05 2023

a(9) onwards computed by David Applegate, Feb 03 2008

A134692 Numbers k with the property that for each digit d from 1 to 9, the decimal expansion of d^3 is a substring of k.

Original entry on oeis.org

1251216427293438, 1251216427298343, 1251216434327298, 1251216434382729, 1251216482729343, 1251216483432729, 1251272921643438, 1251272921648343, 1251272934321648, 1251272934382164, 1251272982164343, 1251272983432164

Offset: 1

Zak Seidov, Feb 02 2008

See A134439 for further information.

David Applegate, Table of n, a(n) for n = 1..300

Cf. A134439, A134698.

Corrected by David Applegate, Feb 09 2008

A134439 Numbers n with property that for each digit d from 1 to 9, we can also see the decimal expansion of d^2 as a substring of n.

Original entry on oeis.org

2536497816, 2536498167, 2536781649, 2536816497, 2573649816, 2573681649, 2578163649, 2578164936, 2581636497, 2581649367, 2581649736, 2581673649, 3625781649, 3625816497, 3649257816, 3649258167, 3649725816, 3649781625

Offset: 1

Zak Seidov, Feb 02 2008

			In the first number, for 2 we can see 4, for 5 we can see 25, for 3 we can see 9, for 6 we can see 36, for 4 we can see 16, for 9 we can see 81, for 8 we can see 64, for 1 we can see 1 and for 7 we can see 49.

David Applegate, Table of n, a(n) for n = 1..300

Cf. A134692, A134698, A134962.

Corrected by David Applegate, Feb 09 2008

A050741 Numbers k such that the decimal expansion of k^2 contains no pair of consecutive equal digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 14, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 36, 37, 39, 41, 42, 43, 44, 45, 48, 49, 51, 52, 53, 54, 55, 56, 57, 59, 61, 63, 64, 66, 68, 69, 71, 72, 73, 74, 75, 77, 78, 79, 81, 82, 84, 86, 87, 89, 91, 92, 93, 95, 96, 97

Offset: 1

Patrick De Geest, Sep 15 1999

			10 is absent because 10^2=100 with repeating 0,
12 is absent because 12^2=144 with repeating 4,
21 is absent because 21^2=441 with repeating 4.

Zak Seidov and Michael De Vlieger, Table of n, a(n) for n = 1..10471, a(n) < 25,000 (First 1057 terms from Zak Seidov)

Cf. A043096, A050749, A135463, A134962.

Select[Range[0,97],FreeQ[Differences[IntegerDigits[#^2]],0]&] (* Jayanta Basu, May 31 2013 *)

Edited by N. J. A. Sloane, Mar 16 2008

A134948 Self-factorial numbers: numbers n with property that for each single digit d of n, we can also see the decimal expansion of d! as a substring of n.

Original entry on oeis.org

1, 2, 10, 11, 12, 21, 22, 24, 100, 101, 102, 110, 111, 112, 120, 121, 122, 124, 201, 210, 211, 212, 221, 222, 224, 241, 242, 244, 424, 1000, 1001, 1002, 1010, 1011, 1012, 1020, 1021, 1022, 1024, 1100, 1101, 1102, 1110, 1111, 1112, 1120, 1121, 1122, 1124, 1200

Offset: 1

Alexander R. Povolotsky, Feb 02 2008

As 9 does not occur in d! for all d in {0..9}, all self-factorials cannot contain 9 as a digit, cf. A007095. - Reinhard Zumkeller, Sep 26 2014

			24 is a self-factorial number because we can see both 2! = 2 and 4! = 24 in the decimal expansion 24.

D. Applegate and R. Zumkeller, Table of n, a(n) for n = 1..10000 (first 300 terms from David Applegate)

Cf. A134698, A134947, A134439, A134692.

Cf. A000142.

import Data.List (nub, sort, isInfixOf)
a134948 n = a134948_list !! (n-1)
a134948_list = filter h [0..] where
   h x = all (`isInfixOf` xs)
             (map (fss !!) $ map (read . return) $ sort $ nub xs)
         where xs = show x
   fss = map show $ take 10 a000142_list
-- Reinhard Zumkeller, Sep 26 2014

isA134948 := proc(n) local nbase10,dgs,d,dfac ; nbase10 := convert(n,base,10) ; dgs := convert(nbase10,set) ; for d in dgs do dfac := convert(d!,base,10) ; if verify(dfac,nbase10,'sublist') = false then RETURN(false) ; fi ; od: RETURN(true) ; end: for n from 1 to 10000 do if isA134948(n) then printf("%d ",n) ; fi ; od: # R. J. Mathar, Feb 05 2008

a(1) - a(18) computed by N. J. A. Sloane, Feb 02 2008
a(19) onwards from David Applegate, Feb 09 2008
More terms from R. J. Mathar, Feb 05 2008

A135015 Numbers n with property that for each single digit d of n, we can also see the decimal expansion of the d-th prime as a substring of n. Also n may not contain any zero digits.

Original entry on oeis.org

11235, 11253, 11325, 11352, 11523, 11532, 21135, 21153, 23115, 23511, 25113, 25311, 31125, 31152, 32115, 32511, 35112, 35211, 51123, 51132, 52113, 52311, 53112, 53211, 111235, 111253, 111325, 111352, 111523, 111532, 112135, 112153, 112235, 112253

Offset: 1

David Applegate and N. J. A. Sloane, Feb 10 2008

Robert Israel, Table of n, a(n) for n = 1..10000 (first 300 terms from David Applegate)

Cf. A134692, A134698, A134962.

filter:= proc (n) local L, LP;
  L := convert(n, base, 10);
  if has(L, 0) then return false end if;
if has(L, 1) and not has(L, 2) then return false end if;
if has(L, 2) and not has(L, 3) then return false end if;
if has(L, 3) and not has(L, 5) then return false end if;
if has(L, 4) and not has(L, 7) then return false end if;
LP := [seq([L[i], L[i+1]], i = 1 .. nops(L)-1)];
if has(L, 5) and not member([1, 1], LP) then return false end if;
if has(L, 6) and not member([3, 1], LP) then return false end if;
if has(L, 7) and not member([7, 1], LP) then return false end if;
if has(L, 8) and not member([9, 1], LP) then return false end if;
if has(L, 9) and not member([3, 2], LP) then return false end if;
true
end proc:
select(filter, [$1..1.5*10^5]);

A135016 Numbers n with property that for each single digit d of n, we can also see the decimal expansion of 2^d as a substring of n. Also n may not contain any zero digits.

Original entry on oeis.org

1642, 2164, 11642, 12164, 16264, 16412, 16421, 16422, 16424, 16426, 16442, 16462, 16642, 21164, 21641, 21642, 21644, 21646, 21664, 22164, 24164, 26164, 26416, 41642, 42164, 61642, 62164, 64162, 64216, 111642, 112164, 116264, 116412, 116421, 116422

Offset: 1

David Applegate and N. J. A. Sloane, Feb 10 2008

David Applegate, Table of n, a(n) for n = 1..300

Cf. A134692, A134698, A134962.

A135464 Numbers n with property that for each single digit d of the base 3 expansion of n, we can also see the base 3 expansion of d^2 as a substring. Also n may not contain any 0 digits.

Original entry on oeis.org

1, 11, 111, 112, 211, 1111, 1112, 1121, 1122, 1211, 2111, 2112, 2211, 11111, 11112, 11121, 11122, 11211, 11212, 11221, 11222, 12111, 12112, 12211, 21111, 21112, 21121, 21122, 21211, 22111, 22112, 22211, 111111, 111112, 111121

Offset: 1

David Applegate and N. J. A. Sloane, Feb 07 2008

The base 3 expansion of n may only contain 1's and 2's and if a 2 is present then 11 must also be present. These are the only restrictions.

David Applegate, Table of n, a(n) for n = 1..300

Base-3 analog of A134962. Cf. A135465.

awk 'function g(s,v,n){if (n==0){if (index(s,"2")==0||index(s,"11")>0)print v;}else{g(s "1",v*3+1,n-1); g(s "2",v*3+2,n-1);}}BEGIN{for (i=1; i<=8; i++)g("",0,i);}' # David Applegate
(C++) // See the Applegate link in A135463.

A135465 Decimal equivalents of A135465.

Original entry on oeis.org

1, 4, 13, 14, 22, 40, 41, 43, 44, 49, 67, 68, 76, 121, 122, 124, 125, 130, 131, 133, 134, 148, 149, 157, 202, 203, 205, 206, 211, 229, 230, 238, 364, 365, 367, 368, 373, 374, 376, 377, 391, 392, 394, 395, 400, 401, 403, 404, 445, 446, 448, 449

Offset: 1

David Applegate and N. J. A. Sloane, Feb 07 2008

David Applegate, Table of n, a(n) for n = 1..200 [Updated Feb 10 2008]

A135495 Numbers n with property that for each single digit d of n, we can also see the decimal expansion of 2d as a substring of n.

Original entry on oeis.org

0, 121648, 121684, 124168, 124816, 128164, 128416, 161248, 161284, 164128, 164812, 168124, 168412, 412168, 412816, 416128, 416812, 481216, 481612, 812164, 812416, 816124, 816412, 841216, 841612, 1121648, 1121684, 1124168

Offset: 1

Zak Seidov, Feb 08 2008

David Applegate, Table of n, a(n) for n = 1..301

Cf. A135463, A134962.

Select[Range[0, 10^6], Function[d, AllTrue[IntegerDigits[2 d], SequenceCount[d, #] >= 1 &]]@ IntegerDigits[#] &] (* Michael De Vlieger, Jan 11 2018 *)

Corrected by David Applegate, Feb 09 2008

cp's OEIS Frontend

A134962 Numbers n with property that for each single digit d of n, we can also see the decimal expansion of d^2 as a substring of n. Also n may not contain any 0 digits.

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A134692 Numbers k with the property that for each digit d from 1 to 9, the decimal expansion of d^3 is a substring of k.

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A134439 Numbers n with property that for each digit d from 1 to 9, we can also see the decimal expansion of d^2 as a substring of n.

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A050741 Numbers k such that the decimal expansion of k^2 contains no pair of consecutive equal digits.

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A134948 Self-factorial numbers: numbers n with property that for each single digit d of n, we can also see the decimal expansion of d! as a substring of n.

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A135015 Numbers n with property that for each single digit d of n, we can also see the decimal expansion of the d-th prime as a substring of n. Also n may not contain any zero digits.

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A135016 Numbers n with property that for each single digit d of n, we can also see the decimal expansion of 2^d as a substring of n. Also n may not contain any zero digits.

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A135464 Numbers n with property that for each single digit d of the base 3 expansion of n, we can also see the base 3 expansion of d^2 as a substring. Also n may not contain any 0 digits.

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A135465 Decimal equivalents of A135465.

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A135495 Numbers n with property that for each single digit d of n, we can also see the decimal expansion of 2d as a substring of n.