A134692 -id:A134692 - 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

A134698 Self-power numbers (or SPN's): numbers n with property that for each single digit d of n, we can also see the decimal expansion of the d-th power of some number as a substring of n and also n contains no 0's or 1's.

Original entry on oeis.org

32564, 232564, 256432, 322564, 325642, 325643, 325644, 325645, 325646, 325648, 325664, 332564, 432564, 532564, 632564, 643256, 832564, 2232564, 2256432, 2322564, 2325642, 2325643, 2325644, 2325645, 2325646, 2325648, 2325664, 2332564, 2432564, 2532564

Offset: 1

Eric Angelini, Jul 05 2005

Computed by M. F. Hasler, Feb 02 2008, Feb 03 2008

There are an infinite number of SPN's, since for example we can prefix n by any digit of n.

			n = 32564 is a member because we can see a cube (64) in n, a square (4 or 25), a fifth power (32), a sixth power (64) and a fourth power (256).

M. F. Hasler, Table of n, a(n) for n = 1..1158

Cf. A134949, A134947, A134439, A134692, A134948.

More terms from M. F. Hasler, Feb 02 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

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.

A135411 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.

Original entry on oeis.org

113579, 113597, 113759, 113795, 113957, 113975, 115379, 115397, 115739, 115793, 115937, 115973, 117359, 117395, 117539, 117593, 117935, 117953, 119357, 119375, 119537, 119573, 119735, 119753, 311579, 311597, 311759, 311795, 311957, 311975

Offset: 1

Zak Seidov, Feb 17 2008

There are exactly 120 six-digit such integers.
These include 24 primes: 113759, 113957, 115793, 117539, 311957, 351179, 375119, 511793, 531197, 539711, 573119, 579113, 579311, 591137, 593711, 711539, 739511, 751139, 751193, 759113, 793511, 911357, 937511, 971153;
and 36 semiprimes: 113579, 115397, 115937, 117953, 119537, 119573, 119753, 311579, 311759, 359117, 375911, 379511, 391157, 395711, 397115, 511379, 511397, 511739, 539117, 571139, 593117, 711593, 731195, 753119, 753911, 791135, 911573, 931157, 935117, 935711, 937115, 951137, 957113, 957311, 975113, 975311.

Cf. A134439, A134692, A134948, A135015, A135016, A135463, A135495.

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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A134698 Self-power numbers (or SPN's): numbers n with property that for each single digit d of n, we can also see the decimal expansion of the d-th power of some number as a substring of n and also n contains no 0's or 1's.

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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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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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A135411 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.

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