A032749 -id:A032749 - cp's OEIS frontend

A014569 Super-3 Numbers (3n^3 contains substring '333' in its decimal expansion).

261, 462, 471, 481, 558, 753, 1036, 1046, 1471, 1645, 1752, 1848, 1923, 1926, 1968, 2031, 2231, 2232, 2363, 2395, 2471, 2591, 2610, 3058, 3087, 3148, 3163, 3172, 3181, 3471, 3494, 3542, 3851, 3884, 4143, 4269, 4314, 4471, 4527, 4554, 4620, 4710, 4732

Offset: 1

Eric W. Weisstein

For any term a(n), all numbers a(n)*10^k, k >= 0, are also in the sequence. More interestingly, all numbers N == 471 (mod 1000) are in the sequence, since 471^3*3 == 333 (mod 1000). - M. F. Hasler, Jul 16 2024

Conjecture: a(n) ~ n. - Charles R Greathouse IV, Dec 04 2024

			1752 is in the sequence since 3 * 1752^3 = 161'333'13024.

C. A. Pickover, Keys to Infinity. New York: Wiley, p. 7, 1995.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Giovanni Resta, super-d numbers, personal web site "Numbers Aplenty", 2013.
Eric Weisstein's World of Mathematics, Super-d Number.

Cf. A032743-A032749 (similar for d=2, ..., 9).

Select[Range[5000],MemberQ[Partition[IntegerDigits[3#^3],3,1],{3,3,3}]&] (* Harvey P. Dale, Feb 01 2013 *)

select( {is_A014569(n, d=3, m=10^d, r=m\9*d)=n=d*n^d; until(r>n\=10, n%m==r && return(1))}, [0..4999]) \\ Using the (optional) 2nd arg d=2..9 allows to compute the sequences A032743-A032749. - M. F. Hasler, Jul 16 2024

is_A014569=lambda n, d=3: str(d)*d in str(d*n**d) # M. F. Hasler, Jul 16 2024

n < a(n) < 200n for n > 2. - Charles R Greathouse IV, Dec 04 2024

Corrected and extended by Patrick De Geest, May 15 1998

Offset changed to 1 by M. F. Hasler, Jul 16 2024

A032743 Super-2 Numbers (2 * n^2 contains substring '22' in its decimal expansion).

Original entry on oeis.org

19, 31, 69, 81, 105, 106, 107, 119, 127, 131, 169, 181, 190, 219, 231, 247, 269, 281, 310, 318, 319, 331, 332, 333, 334, 335, 336, 337, 338, 339, 348, 369, 381, 419, 431, 454, 469, 481, 511, 519, 531, 558, 569, 581, 601, 619, 631, 669, 679, 681, 690, 715

Offset: 1

Patrick De Geest, May 15 1998

For any term a(n), all numbers a(n)*10^k, k >= 0, are also in the sequence. Moreover, the first four terms satisfy 2*a(n)^2 == 22 (mod 100), therefore any number ending in 19, 31, 69 or 81 (possibly followed by trailing '0's) is in the sequence. - M. F. Hasler, Jul 16 2024

Conjecture: a(n) ~ n. - Charles R Greathouse IV, Dec 04 2024

C. A. Pickover, "Keys to Infinity", New York: Wiley, p. 7, 1995.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Giovanni Resta, super-d numbers, personal web site "Numbers Aplenty", 2013
Eric Weisstein's World of Mathematics, Super-d Number.

Cf. A014569 (similar for d=3), A032744 - A032749 (similar for d=4, ..., 9).

Select[Range[1000],MemberQ[Partition[IntegerDigits[2#^2],2,1],{2,2}]&] (* Harvey P. Dale, May 09 2012 *)
Select[Range[750],SequenceCount[IntegerDigits[2#^2],{2,2}]>0&] (* Harvey P. Dale, May 13 2022 *)

select( {is_A032743(n, d=2, m=10^d, r=m\9*d)=n=d*n^d; until(r>n\=10, n%m==r && return(1))}, [0..999]) \\ M. F. Hasler, Jul 16 2024

is_A032743=lambda n, d=2: str(d)*d in str(d*n**d) # M. F. Hasler, Jul 16 2024

A032744 Super-4 Numbers (4 * n^4 contains substring '4444' in its decimal expansion).

Original entry on oeis.org

1168, 4972, 7423, 7752, 8431, 10267, 11317, 11487, 11549, 11680, 16588, 16664, 16837, 18257, 18597, 19784, 19933, 22217, 22504, 22819, 22829, 24078, 24331, 24514, 25296, 25698, 26685, 26738, 27812, 27973, 28988, 32466, 32467, 32735, 34078, 34636, 35248, 36219, 36602

Offset: 1

Patrick De Geest, May 15 1998

a(17) = 19933 and a(20) = 22819 are such that a(n)^4 == 111121 (mod 10^6), therefore any number ending in (0|5)19933 or in (0|5)22819, where (a|b) means a or b, is in the sequence. Of course, for each term a(n), all numbers a(n)*10^k, k >= 0, are also in the sequence. - M. F. Hasler, Jul 16 2024

Conjecture: a(n) ~ n. - Charles R Greathouse IV, Dec 04 2024

C. A. Pickover, "Keys to Infinity", New York: Wiley, p. 7, 1995.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Giovanni Resta, super-d numbers, personal web site "Numbers Aplenty", 2013
Eric Weisstein's World of Mathematics, Super-d Number.

Cf. A014569 (d=3), A032743-A032749 (d=2, ..., 9).

Select[Range[33000],MemberQ[Partition[IntegerDigits[4*#^4],4,1],{4,4,4,4}]&] (* Harvey P. Dale, Mar 22 2012 *)

select( is_A032744(n)=is_A032743(n,4), [1..33333]) \\ M. F. Hasler, Jul 16 2024

is_A032744=lambda n: '4444' in str(4*n**4) # M. F. Hasler, Jul 16 2024

Offset changed to 1 by M. F. Hasler, Jul 16 2024

A032745 Super-5 Numbers (5 * n^5 contains substring '55555' in its decimal expansion).

Original entry on oeis.org

4602, 5517, 7539, 12955, 14555, 20137, 20379, 26629, 32767, 35689, 35825, 37706, 46020, 46715, 51988, 55170, 66344, 73338, 73974, 75390, 76157, 86025, 91497, 105852, 114488, 129550, 132234, 145550, 146399, 158651, 160897, 171673, 174782, 176988, 184471, 188421, 191261, 192607

Offset: 1

Patrick De Geest, May 15 1998

The terms a({15, 25, 34}) = {51988, 114488, 176988} are such that 5*a(n)^5 == 55555840 (mod 10^8). Therefore any number congruent to one of these, modulo 5*10^5, is also in the sequence. Of course, for any a(n) in the sequence, any a(n)*10^k, k >= 0, is also in the sequence. - M. F. Hasler, Jul 16 2024

Conjecture: a(n) ~ n. - Charles R Greathouse IV, Dec 04 2024

C. A. Pickover, "Keys to Infinity", New York: Wiley, p. 7, 1995.

Robert Israel, Table of n, a(n) for n = 1..10000
Giovanni Resta, super-d numbers, personal web site "Numbers Aplenty", 2013
Eric Weisstein's World of Mathematics, Super-d Number

Cf. A014569, A032743-A032749.

filter:= proc(n) local S;
  StringTools:-Search("55555",sprintf("%d",5*n^5))<> 0
end proc:
select(filter, [$1..200000]); # Robert Israel, Jul 14 2025

Select[Range[200000],SequenceCount[IntegerDigits[5#^5],{5,5,5,5,5}]>0&] (* Harvey P. Dale, Jul 16 2016 *)

select( {is_A032745(n)=is_A032743(n, 5)}, [1..2^18]) \\ M. F. Hasler, Jul 16 2024

Offset changed to 1 by Andrew Howroyd, Jul 16 2024

A032746 Super-6 Numbers (6 * n^6 contains substring '666666' in its decimal expansion).

Original entry on oeis.org

27257, 272570, 302693, 323576, 364509, 502785, 513675, 537771, 676657, 678146, 731378, 831122, 836553, 913797, 920456, 921269, 1045361, 1144983, 1169054, 1283069, 1288697, 1292673, 1343642, 1346117, 1472078, 1523993, 1640026

Offset: 1

Patrick De Geest, May 15 1998

The terms a({5, 9, 11, 12}) = {364509, 676657, 731378, 831122} are such that 6*a(n)^6 == 66666646, 66666694, or 66666624 (mod 10^8). Therefore, any number congruent to one of these (mod 5*10^7) is also in the sequence. Of course, for any term a(n), all numbers a(n)*10^k, k >= 0, are also in the sequence. - M. F. Hasler, Jul 16 2024

Conjecture: a(n) ~ n. - Charles R Greathouse IV, Dec 04 2024

C. A. Pickover, "Keys to Infinity", New York: Wiley, p. 7, 1995.

Giovanni Resta, super-d numbers, personal web site "Numbers Aplenty", 2013
Eric Weisstein's World of Mathematics, Super-d Number.

Cf. A014569 (d=3), A032743 - A032749 (d=2, ..., 9).

With[{c=6},Select[Range[165*10^4],SequenceCount[IntegerDigits[c #^c],PadRight[ {},c,c]]>0&]] (* Harvey P. Dale, Jan 18 2023 *)

select( {is_A032746(n)=is_A014569(n,6)}, [1..10^5])
for(n=1, oo, is_A032746(n)&& print1(n", ")) \\ Quite slow... - M. F. Hasler, Jul 16 2024

Offset changed to 1 by M. F. Hasler, Jul 16 2024

A032748 Super-8 Numbers (8 * n^8 contains substring '88888888' in its decimal expansion).

Original entry on oeis.org

185423, 641519, 1551728, 1854230, 6415190, 12043464, 12147605, 15517280, 16561735, 18542300, 26908132, 29242698, 33491333, 34982204, 35866945, 37584428, 44263715, 45980752, 54555936, 56148739, 60883944, 64151900

Offset: 1

Patrick De Geest, May 15 1998

All numbers congruent to a(2) = 641519 (mod 5*10^10) are also in the sequence. - M. F. Hasler, Jul 17 2024

Conjecture: a(n) ~ n. - Charles R Greathouse IV, Dec 04 2024

C. A. Pickover, "Keys to Infinity", New York: Wiley, p. 7, 1995.

Giovanni Resta, super-d numbers, personal web site "Numbers Aplenty", 2013
Eric Weisstein's World of Mathematics, Super-d Number.

Cf. A014569 (similar for d=3), A032743 - A032749 (similar for d=2, ..., 9).

d := 8 ;
for n from 1 do
        convert(d*n^d,base,10) ;
        if verify([8,8,8,8,8,8,8,8],%,'sublist') then
                print(n) ;
        end if;
end do: # R. J. Mathar, Jan 11 2013

Select[Range[65*10^6],SequenceCount[IntegerDigits[8*#^8],{8,8,8,8,8,8,8,8}]>0&] (* Harvey P. Dale, Dec 24 2016 *)

\\ See A014569 (not very efficient for d=8).

# See A014569 (not very efficient for d=8).

Offset changed to 1 by M. F. Hasler, Jul 17 2024

A032747 Super-7 Numbers (7 * n^7 contains substring '7777777' in its decimal expansion).

Original entry on oeis.org

140997, 490996, 1184321, 1259609, 1409970, 1783166, 1886654, 1977538, 2457756, 2714763, 2750425, 2980991, 3043607, 3283057, 3689639, 4191601, 4258476, 4642725, 4909960, 4973029, 5242829, 5349973, 5444788, 5523544, 5682065

Offset: 1

Patrick De Geest, May 15 1998

The term a(6) = 1783166 is such that 7*a(6)^7 == 777777792 (mod 10^9). Therefore, all numbers congruent to a(6) (mod 5*10^8) are also in the sequence. Of course, for any term a(n), all numbers a(n)*10^k, k >= 0, are also in the sequence. - M. F. Hasler, Jul 16 2024

Conjecture: a(n) ~ n. - Charles R Greathouse IV, Dec 04 2024

C. A. Pickover, "Keys to Infinity", New York: Wiley, p. 7, 1995.

Giovanni Resta, super-d numbers, personal web site "Numbers Aplenty", 2013.
Eric Weisstein's World of Mathematics, Super-d Number.

Cf. A014569 (d=3), A032743 (d=2) - A032749 (d=9).

Select[Range[6*10^6],MemberQ[Partition[IntegerDigits[7#^7],7,1],{7,7,7,7,7,7,7}]&] (* Harvey P. Dale, Sep 01 2014 *)

is_A032747(n)=is_A014569(n, 7)
for(n=1,oo, is_A032747(n)&& print1(n", ")) \\ Quite slow, even to get the first few terms. - M. F. Hasler, Jul 16 2024

A032757 Palindromic Super-9 Numbers.

Original entry on oeis.org

351636153, 6260330626, 9999999999, 99999999999, 999999999999, 9971824281799, 9999999999999, 59999999999995, 89999999999998, 99999999999999, 499999999999994, 596038242830695, 599999999999995

Offset: 1

Patrick De Geest, May 15 1998; extended Feb 22 2004

The smallest palindromic super-9 prime is 15125111011152151.

C. A. Pickover, "Keys to Infinity", New York: Wiley, p. 7, 1995.

Eric Weisstein's World of Mathematics, Super-d Number.

Cf. A014569, A032749.

cp's OEIS Frontend

A014569 Super-3 Numbers (3n^3 contains substring '333' in its decimal expansion).

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A032743 Super-2 Numbers (2 * n^2 contains substring '22' in its decimal expansion).

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A032744 Super-4 Numbers (4 * n^4 contains substring '4444' in its decimal expansion).

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A032745 Super-5 Numbers (5 * n^5 contains substring '55555' in its decimal expansion).

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A032746 Super-6 Numbers (6 * n^6 contains substring '666666' in its decimal expansion).

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A032748 Super-8 Numbers (8 * n^8 contains substring '88888888' in its decimal expansion).

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