A134962 -id:A134962 - cp's OEIS frontend

A135463 Numbers n with property that for each single digit d of n, we can also see the decimal expansions of d^2 and d^3 as substrings of n. Also n may not contain any 0 digits.

1, 11, 111, 1111, 11111, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, 111111111111, 1111111111111, 11111111111111, 111111111111111, 1111111111111111, 11111111111111111, 111111111111111111, 216343649812512729

Offset: 1

Proposed by Zak Seidov, Feb 04 2007; computed by David Applegate and N. J. A. Sloane, Feb 07 2008

David Applegate, Table of n, a(n) for n = 1..300
David Applegate, C++ program for sequences of this type

Cf. A134962.

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

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.

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

A135441 Numbers k such that for any single digit d of k the d-th semiprime sp(k) is substring of k.

Original entry on oeis.org

104, 410, 1004, 1014, 1040, 1041, 1044, 1104, 1410, 4010, 4100, 4101, 4104, 4110, 4410, 10004, 10014, 10040, 10041, 10044, 10104, 10114, 10140, 10141, 10144, 10145, 10400, 10401, 10404, 10410, 10411, 10414, 10440, 10441, 10444, 10514, 11004, 11014, 11040

Offset: 1

Zak Seidov, Feb 17 2008

sp(1..9)=4,6,9,10,14,15,21,22,25, with the convention that sp(0)=1.

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

Cf. A134439, A134962, A135463.

Corrected and extended by David Applegate, Feb 18 2008

A135496 Numbers k with the property that for each single digit d of k, the decimal expansion of 3d is a substring of k.

Original entry on oeis.org

0, 12182427369, 12182427396, 12182427639, 12182427693, 12182427936, 12182427963, 12182432769, 12182432796, 12182436279, 12182436927, 12182439276, 12182439627, 12182462739, 12182462793, 12182463279

Offset: 1

Zak Seidov, Feb 08 2008

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

Cf. A135463, A134962.

Corrected by David Applegate, Feb 10 2008

cp's OEIS Frontend

A135463 Numbers n with property that for each single digit d of n, we can also see the decimal expansions of d^2 and d^3 as substrings of n. Also n may not contain any 0 digits.

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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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Mathematica

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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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Maple

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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AWK

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.

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Programs

Mathematica

Extensions

A135441 Numbers k such that for any single digit d of k the d-th semiprime sp(k) is substring of k.

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Extensions

A135496 Numbers k with the property that for each single digit d of k, the decimal expansion of 3d is a substring of k.

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