A049329 -id:A049329 - cp's OEIS frontend

A176905 Primes p such that p^p contains the string 'p' as a substring.

5, 11, 17, 19, 31, 37, 41, 43, 53, 59, 61, 71, 73, 79, 83, 97, 101, 103, 127, 131, 151, 173, 191, 193, 227, 233, 251, 263, 269, 271, 293, 313, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 401, 419, 421, 431, 433, 439, 443, 461, 467, 487, 491, 499, 503, 521

Offset: 1

Harvey P. Dale & Vladimir Joseph Stephan Orlovsky, Apr 28 2010, May 01 2010

17^17=827240261886336764"17"7, 19^19="19"784"19"655660313589123979,..

N. J. A. Sloane, Table of n, a(n) for n = 1..8114 [Computed using the Mma program]

Cf. A169753, A169754, A049329.

okQ[n_]:=Module[{idn=IntegerDigits[n], len, idp=IntegerDigits[n^n]}, len=Length[idn];Length[Flatten[Position[Partition[idp, len, 1], idn]]]>=1]; Select[Prime[Range[250]], okQ]

A229629 Numbers k such that k is in the middle of decimal expansion of k^k.

Original entry on oeis.org

1, 6, 888, 1808, 2138, 65246, 268105

Offset: 1

Farideh Firoozbakht, Oct 04 2013

a(6) is greater than 50000.

a(8) > 600000. - Giovanni Resta, Oct 08 2013

			6 is in the sequence because 6^6 = 46656, which includes a 6 in the middle.
11 is not in the sequence, because even though the substring 11 appears twice in 11^11 = 285311670611, neither occurrence is precisely in the middle.

Cf. A033147, A049329, A131495.

Do[a = IntegerDigits[n^n]; b = Length[a]; c = IntegerLength[n]; If[EvenQ[b - c] && FromDigits[Take[a, {(b - c)/2 + 1, (b + c)/2}]] == n, Print[n]], {n, 50000}]

is(n)=my(d=digits(n),D=digits(n^n)); if((#d+#D)%2, return(0)); for(i=1,#d, if(d[i]!=D[#D/2-#d/2+i], return(0))); 1 \\ Charles R Greathouse IV, Jul 30 2016

from itertools import islice
def A229629(): # generator of terms
    n = 1
    while True:
        s, sn = str(n**n), str(n)
        l, ln = len(s), len(sn)
        if (ln-l) % 2 == 0 and s[l//2-ln//2:l//2+(ln+1)//2] == sn: yield n
        n += 1
A229629_list = list(islice(A229629(),5)) # Chai Wah Wu, Nov 21 2021

a(6)-a(7) from Giovanni Resta, Oct 08 2013

A236314 Number of non-overlapping occurrences of n in the decimal representation of n^n.

Original entry on oeis.org

1, 0, 0, 0, 1, 3, 0, 0, 1, 1, 2, 0, 0, 0, 0, 2, 1, 0, 2, 0, 2, 0, 0, 1, 2, 0, 0, 1, 0, 0, 2, 2, 2, 0, 3, 1, 1, 0, 1, 0, 1, 1, 1, 0, 3, 1, 0, 1, 1, 1, 1, 2, 2, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 1, 2, 5, 2, 1, 2, 0, 3, 3, 2, 1, 0, 1, 0

Offset: 1

Christian Perfect, Jan 22 2014

			6^6 is 46656 with 3 6's, hence a(6) = 3.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

A049329 lists n where a(n) is nonzero.

The same sequence but allowing for overlapping occurrences is at A236322.

a[n_] := Length@ StringPosition[ToString[n^n], ToString[n], Overlaps -> False]; (* Giovanni Resta, Jan 22 2014 *)

from itertools import count
a=(str(n**n).count(str(n)) for n in count(1))

A236322 Number of (potentially overlapping) occurrences of n in the decimal representation of n^n.

Original entry on oeis.org

1, 0, 0, 0, 1, 3, 0, 0, 1, 1, 2, 0, 0, 0, 0, 2, 1, 0, 2, 0, 2, 0, 0, 1, 2, 0, 0, 1, 0, 0, 2, 2, 2, 0, 3, 1, 1, 0, 1, 0, 1, 1, 1, 0, 3, 1, 0, 1, 1, 1, 1, 2, 2, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 1, 0, 2, 0, 0, 0, 0, 1, 0, 1, 1, 2, 5, 2, 1, 2, 0, 3, 3, 2, 1, 0, 1, 0, 0, 0, 0, 5, 1, 3, 4, 2, 2, 1, 1, 10

Offset: 1

Christian Perfect, Jan 22 2014

Differs from A236314 at n=99.

Scott R. Shannon, Table of n, a(n) for n = 1..10000 (terms 1..999 from Christian Perfect).

A049329 lists n where a(n) is nonzero.

Non-overlapping occurrences are counted by A236314.

a[n_] := Length[StringPosition @@ ToString /@ {n^n, n}]; Array[a, 99] (* Giovanni Resta, Jan 22 2014 *)

a(n) = my(m=Mod(n,10^#Str(n)));(m==n=n^n)+sum(i=0,1+log(n)/log(10),m==n\=10) \\ - M. F. Hasler, Jan 23 2014

from itertools import count
def occurrences(string, sub):
    count = start = 0
    while True:
        start = string.find(sub, start) + 1
        if start > 0:
            count+=1
        else:
            return count
def a(n):
    return occurrences(str(n**n), str(n))

A050763 Numbers k such that the decimal expansion of k^k contains no pair of consecutive equal digits (probably finite).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 9, 15, 18

Offset: 1

Patrick De Geest, Sep 15 1999

Cf. A043096, A000312, A050764, A049329.

Changed offset to 1. - N. J. A. Sloane, Aug 24 2019

A050764 Numbers of form k^k (for values of k see A050763) containing no pair of consecutive equal digits (probably finite).

Original entry on oeis.org

1, 4, 27, 256, 3125, 823543, 387420489, 437893890380859375, 39346408075296537575424

Offset: 1

Patrick De Geest, Sep 15 1999

Cf. A043096, A000312, A050763, A049329.

Select[Table[k^k,{k,20}],FreeQ[Differences[IntegerDigits[#]],0]&] (* Harvey P. Dale, Aug 23 2019 *)

Since this is a list, it has offset 1 and does not contain any repeated terms. - N. J. A. Sloane, Aug 24 2019

A169754 Numbers of the form k^k which contain k as a substring in base 10.

Original entry on oeis.org

1, 3125, 46656, 387420489, 10000000000, 285311670611, 18446744073709551616, 827240261886336764177, 1978419655660313589123979, 5842587018385982521381124421

Offset: 1

N. J. A. Sloane, May 03 2010

k^k for k in A049329.

Cf. A176905, A049329.

A343594 Numbers k that, when written in all bases from base 2 to base 10, are a substring of k^k when written in the same base.

Original entry on oeis.org

1, 5, 17, 25, 31, 41, 63, 92, 151, 170, 202, 221, 263, 266, 278, 322, 327, 347, 364, 401, 404, 412, 421, 423, 437, 467, 470, 482, 490, 498, 501, 515, 519, 543, 558, 578, 590, 612, 623, 636, 646, 647, 671, 683, 685, 705, 707, 717, 718, 726, 764, 785, 795, 859, 867, 872, 875, 881, 890, 892, 897

Offset: 1

Scott R. Shannon, Apr 21 2021

			5 is a term. See below table:
.
   base  |  5 in base  |  5^5 in base
---------+-------------+-------------
    10          5                3125
     9          5                4252
     8          5                6065
     7          5               12053
     6          5               22245
     5         10              100000
     4         11              300311
     3         12            11021202
     2        101        110000110101
.
5^5 in all bases contains 5 in that base as a substring.

Cf. A049329, A045537, A051248.

str(v) = my(s=""); for (k=1, #v, s = concat(s, Str(v[k]))); s;
isok(k) = {for (b=2, 10, my(kb = digits(k, b), kkb = digits(k^k, b)); if (#strsplit(str(kkb), str(kb)) <=1 , return (0));); return (1);} \\ Michel Marcus, Apr 26 2021

from sympy.ntheory import digits
def nstr(n, b): return "".join(map(str, digits(n, b=b)[1:]))
def ok(k): return all(nstr(k, b) in nstr(k**k, b) for b in range(10, 1, -1))
print(list(filter(ok, range(900)))) # Michael S. Branicky, Apr 25 2021

cp's OEIS Frontend

A176905 Primes p such that p^p contains the string 'p' as a substring.

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A229629 Numbers k such that k is in the middle of decimal expansion of k^k.

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A236314 Number of non-overlapping occurrences of n in the decimal representation of n^n.

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A236322 Number of (potentially overlapping) occurrences of n in the decimal representation of n^n.

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A050763 Numbers k such that the decimal expansion of k^k contains no pair of consecutive equal digits (probably finite).

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A050764 Numbers of form k^k (for values of k see A050763) containing no pair of consecutive equal digits (probably finite).

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Mathematica

Extensions

A169754 Numbers of the form k^k which contain k as a substring in base 10.

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A343594 Numbers k that, when written in all bases from base 2 to base 10, are a substring of k^k when written in the same base.

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Keywords

Examples

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PARI

Python