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A176642 Triangle T(n, k) = 8^(k*(n-k)), read by rows.

1, 1, 1, 1, 8, 1, 1, 64, 64, 1, 1, 512, 4096, 512, 1, 1, 4096, 262144, 262144, 4096, 1, 1, 32768, 16777216, 134217728, 16777216, 32768, 1, 1, 262144, 1073741824, 68719476736, 68719476736, 1073741824, 262144, 1, 1, 2097152, 68719476736, 35184372088832, 281474976710656, 35184372088832, 68719476736, 2097152, 1

Offset: 0

Roger L. Bagula, Apr 22 2010

			Triangle begins as:
  1;
  1,      1;
  1,      8,          1;
  1,     64,         64,           1;
  1,    512,       4096,         512,           1;
  1,   4096,     262144,      262144,        4096,          1;
  1,  32768,   16777216,   134217728,    16777216,      32768,      1;
  1, 262144, 1073741824, 68719476736, 68719476736, 1073741824, 262144, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. this sequence (q=2), A176643 (q=3), A176644 (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), this sequence (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26).
Cf. A000567, A004247.

[8^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 30 2021

T[n_, k_, q_]:= (q*(3*q-2))^(k*(n-k)); Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten
With[{m=6}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Jun 30 2021 *)

flatten([[8^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 30 2021

T(n, k, q) = c(n,q)/(c(k, q)*c(n-k, q)) where c(n, q) = (q*(3*q - 2))^binomial(n+1,2) and q = 2.
T(n, k, q) = (q*(3*q-2))^(k*(n-k)) with q = 2.
T(n, k) = 8^A004247(n,k), where A004247 is interpreted as a triangle. [relation detected by sequencedb.net]. - R. J. Mathar, Jun 30 2021
T(n, k, m) = (m+2)^(k*(n-k)) with m = 6. - G. C. Greubel, Jun 30 2021

Edited by R. J. Mathar and G. C. Greubel, Jun 30 2021

A241842 Number of simple connected graphs on n nodes that are non-integral.

Original entry on oeis.org

0, 0, 1, 4, 18, 106, 846, 11095, 261056, 11716488, 1006700452

Offset: 1

Travis Hoppe and Anna Petrone, Apr 29 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Travis Hoppe and Anna Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Integral Graph

Cf. A001349, A064731.

a(n) = A001349(n) - A064731(n).

A051003 Beastly (or hateful) numbers: numbers containing the string 666 in their decimal expansion.

Original entry on oeis.org

666, 1666, 2666, 3666, 4666, 5666, 6660, 6661, 6662, 6663, 6664, 6665, 6666, 6667, 6668, 6669, 7666, 8666, 9666, 10666, 11666, 12666, 13666, 14666, 15666, 16660, 16661, 16662, 16663, 16664, 16665, 16666, 16667, 16668, 16669, 17666, 18666

Offset: 1

Eric W. Weisstein

Jens Kruse Andersen, Table of n, a(n) for n = 1..1000
Tony Padilla and Brady Haran, The Most Evil Number, Numberphile video (2018)
Eric Weisstein's World of Mathematics, Beast Number

Cf. A131645, A057564, A002282, A138563, A043515, A046720, A186086.

Select[Range[18666], ! StringFreeQ[ToString[#], "666"] &] (* Arkadiusz Wesolowski, Sep 08 2011 *)

A052046 Squares whose digits occur with the same frequency.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 169, 196, 256, 289, 324, 361, 529, 576, 625, 729, 784, 841, 961, 1024, 1089, 1296, 1369, 1764, 1849, 1936, 2304, 2401, 2601, 2704, 2809, 2916, 3025, 3249, 3481, 3721, 4096, 4356, 4761, 5041, 5184, 5329, 5476, 6084, 6241

Offset: 1

Patrick De Geest, Dec 15 1999

Contains all members of A078255 that are < 10^10. 7744 is the first member that is not a member of A078255. - David Wasserman, Jun 27 2006

Robert Israel, Table of n, a(n) for n = 1..10000
P. De Geest, Numbers whose digits occur with same frequency

Cf. A045540, A052047, A052048, A052049, A052050, A052051, A052052.

Cf. A078255.

filter:= proc(n) local L,M;
  L:= convert(n,base,10);
  M:= {seq(numboccur(i,L),i=0..9)} minus {0};
  nops(M) = 1
end proc:
select(filter, [seq(i^2,i=0..200)]); # Robert Israel, Jan 08 2018

t={}; Do[If[Length[DeleteDuplicates[Transpose[Tally[IntegerDigits[n^2]]][[2]]]] == 1, AppendTo[t,n^2]], {n,0,80}]; t (* Jayanta Basu, May 10 2013 *)
sfQ[n_]:=Length[Union[Select[DigitCount[n],#!=0&]]]==1; Select[ Range[ 0,80]^2,sfQ] (* Harvey P. Dale, May 05 2019 *)

Offset corrected by Michel Marcus, Aug 12 2015

A114762 a(n) = floor(3^(1/2)*10^n)^2.

Original entry on oeis.org

1, 289, 29929, 2999824, 299982400, 29999972025, 2999997202500, 299999997378064, 29999999737806400, 2999999998029351249, 299999999976140205625, 29999999999692481531536, 2999999999996960966074624, 299999999999973224736673344

Offset: 0

Amarnath Murthy, Nov 17 2005

Largest square < 3*10^(2n).

			a(1) = floor(3^(1/2)*10)^2 = 17^2 = 289.

Ivan Panchenko, Table of n, a(n) for n = 0..100

Cf. A114761.

Table[Floor[3^(1/2)*10^n]^2, {n, 0, 20}] (* Stefan Steinerberger, Apr 14 2006 *)
Floor[Sqrt[3] 10^Range[0,20]]^2 (* Harvey P. Dale, Sep 25 2021 *)

More terms from Stefan Steinerberger, Apr 14 2006

A135654 Divisors of 8128 (the 4th perfect number), written in base 2.

Original entry on oeis.org

1, 10, 100, 1000, 10000, 100000, 1000000, 1111111, 11111110, 111111100, 1111111000, 11111110000, 111111100000, 1111111000000

Offset: 1

Omar E. Pol, Feb 23 2008, Mar 03 2008

The number of divisors of the 4th perfect number is equal to 2*A000043(4)=A061645(4)=14.

			The structure of divisors of 8128 (see A133024)
-------------------------------------------------------------------------
n ... Divisor . Formula ....... Divisor written in base 2 ...............
-------------------------------------------------------------------------
1)......... 1 = 2^0 ........... 1
2)......... 2 = 2^1 ........... 10
3)......... 4 = 2^2 ........... 100
4)......... 8 = 2^3 ........... 1000
5)........ 16 = 2^4 ........... 10000
6)........ 32 = 2^5 ........... 100000
7)........ 64 = 2^6 ........... 1000000 ... (The 4th superperfect number)
8)....... 127 = 2^7 - 2^0 ..... 1111111 ... (The 4th Mersenne prime)
9)....... 254 = 2^8 - 2^1 ..... 11111110
10)...... 508 = 2^9 - 2^2 ..... 111111100
11)..... 1016 = 2^10- 2^3 ..... 1111111000
12)..... 2032 = 2^11- 2^4 ..... 11111110000
13)..... 4064 = 2^12- 2^5 ..... 111111100000
14)..... 8128 = 2^13- 2^6 ..... 1111111000000 ... (The 4th perfect number)

Index entries for sequences related to divisors of numbers

For more information see A133024 (Divisors of 8128). Cf. A000043, A000079, A000396, A000668, A019279, A061645, A061652.

FromDigits[IntegerDigits[#,2]]&/@Divisors[8128] (* Harvey P. Dale, Jan 08 2014 *)

a(n)=A133024(n), written in base 2. Also, for n=1 .. 14: If n<=(A000043(4)=7) then a(n) is the concatenation of the digit "1" and n-1 digits "0" else a(n) is the concatenation of A000043(4)=7 digits "1" and (n-1-A000043(4)) digits "0".

A135655 Divisors of 33550336 (the 5th perfect number), written in base 2.

Original entry on oeis.org

1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000, 10000000000, 100000000000, 1000000000000, 1111111111111, 11111111111110, 111111111111100, 1111111111111000, 11111111111110000

Offset: 1

Omar E. Pol, Feb 23 2008, Mar 01 2008, Mar 03 2008

The number of divisors of the 5th perfect number is equal to 2*A000043(5)=A061645(5)=26.

			The structure of divisors of 33550336 (see A133025)
------------------------------------------------------------------------
n ...... Divisor . Formula ....... Divisor written in base 2 ...........
------------------------------------------------------------------------
1)............ 1 = 2^0 ........... 1
2)............ 2 = 2^1 ........... 10
3)............ 4 = 2^2 ........... 100
4)............ 8 = 2^3 ........... 1000
5)........... 16 = 2^4 ........... 10000
6)........... 32 = 2^5 ........... 100000
7)........... 64 = 2^6 ........... 1000000
8).......... 128 = 2^7 ........... 10000000
9).......... 256 = 2^8 ........... 100000000
10)......... 512 = 2^9 ........... 1000000000
11)........ 1024 = 2^10 .......... 10000000000
12)........ 2048 = 2^11 .......... 100000000000
13) ....... 4096 = 2^12 .......... 1000000000000 ... (The 5th superperfect number)
14) ....... 8191 = 2^13 - 2^0 .... 1111111111111 ... (The 5th Mersenne prime)
15) ...... 16382 = 2^14 - 2^1 .... 11111111111110
16) ...... 32764 = 2^15 - 2^2 .... 111111111111100
17) ...... 65528 = 2^16 - 2^3 .... 1111111111111000
18) ..... 131056 = 2^17 - 2^4 .... 11111111111110000
19) ..... 262112 = 2^18 - 2^5 .... 111111111111100000
20) ..... 524224 = 2^19 - 2^6 .... 1111111111111000000
21) .... 1048448 = 2^20 - 2^7 .... 11111111111110000000
22) .... 2096896 = 2^21 - 2^8 .... 111111111111100000000
23) .... 4193792 = 2^22 - 2^9 .... 1111111111111000000000
24) .... 8387584 = 2^23 - 2^10 ... 11111111111110000000000
25) ... 16775168 = 2^24 - 2^11 ... 111111111111100000000000
26) ... 33550336 = 2^25 - 2^12 ... 1111111111111000000000000 ... (The 5th perfect number)

Omar E. Pol, Table of n, a(n) for n = 1..26 (shows all terms).
Index entries for sequences related to divisors of numbers

For more information see A133025 (Divisors of 33550336). Cf. A000043, A000079, A000396, A000668, A019279, A061645, A061652.

a(n)=A133025(n), written in base 2. Also, for n=1 .. 26: If n<=(A000043(5)=13) then a(n) is the concatenation of the digit "1" and n-1 digits "0" else a(n) is the concatenation of A000043(5)=13 digits "1" and (n-1-A000043(5)) digits "0".

A137756 Nontrivial elements in writing the numerator of an element first and then the denominator of that element (left to right) of Leibniz's harmonic-like triangle. That is, the nontrivial elements of A137752.

Original entry on oeis.org

2, 2, 3, 5, 6, 3, 4, 7, 12, 7, 12, 4, 5, 9, 20, 31, 30, 9, 20, 5, 6, 11, 30, 49, 60, 49, 60, 11, 30, 6, 7, 13, 42, 71, 105, 209, 140, 71, 105, 13, 42, 7, 8, 15, 56, 97, 168, 351, 280, 351, 280, 97, 168, 15, 56, 8, 9, 17, 72, 127, 252, 545, 504, 1471

Offset: 1

Mohammad K. Azarian, Feb 10 2008

			1/1; -->
1/2, 1/2; --> 2 2
1/3, 5/6, 1/3; --> 3 5 6 3
1/4, 7/12, 7/12, 1/4; --> ...
1/5, 9/20, 31/30, 9/20, 1/5;

Cf. A003506; A007622; A046201; A046204; A046205; A046206; A046208; A046212; A137752; A137753; A137754; A137755.

A166579 Prime numbers containing the string 17.

Original entry on oeis.org

17, 173, 179, 317, 617, 1117, 1171, 1217, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 2017, 2179, 2417, 2617, 2917, 3217, 3517, 3617, 3917, 4177, 4217, 4517, 4817, 5171, 5179, 5417, 5717, 6173, 6217, 6317, 6917, 7177, 7417

Offset: 1

Vincenzo Librandi, Nov 01 2009

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A166571 - A166573, A166580, A166581, A166582.

isA166579 := proc(n) local dgs,wrks; if isprime(n) then dgs := convert(n,base,10) ; wrks := false; for i from 1 to nops(dgs)-1 do if op(i,dgs) = 7 and op(i+1,dgs) = 1 then return true; end if; od: return false; else false; end if; end proc: for n from 1 to 8000 do if isA166579(n) then printf("%d,",n) ; end if; od: # R. J. Mathar, Nov 30 2009

p17Q[n_] := Module[{idn = IntegerDigits[n]}, MemberQ[Partition[idn, 2, 1], {1, 7}]]; Select[Prime[Range[1000]], p17Q] (* Vincenzo Librandi Sep 14 2012 *)
Select[Prime[Range[1000]],SequenceCount[IntegerDigits[#],{1,7}]>0&] (* Harvey P. Dale, Apr 18 2022 *)

contains(n,k)=my(N=digits(n),K=digits(k)); for(i=0,#N-#K, for(j=1,#K,if(N[i+j]!=K[j],next(2))); return(1)); 0
is(n)=isprime(n) && contains(n,17) \\ Charles R Greathouse IV, Jun 20 2014

a(n) ~ n log n. - Charles R Greathouse IV, Jun 20 2014

A216148 Primes of the form 2*k^k + 1 = A216147(k).

Original entry on oeis.org

3, 17832200896513, 78692816150593075150849

Offset: 1

M. F. Hasler, Sep 02 2012

The sequence should be extended through A110932, which lists the corresponding values of k: The next term, 2*251^251 + 1 = A216147(A110932(4)) ~ 4.16*10^602, is too large to include here.

M. F. Hasler, Table of n, a(n) for n = 1..4
C. Caldwell, G.L. Honaker (Eds), Prime Curios!: 78692816150593075150849.
"Jim", Topic: The Lost Proof of Fermat, on mathforum.org, Jan 31 2004

Cf. A110932.

A subsequence of A133663, with b=a and c=1.

Select[Table[2n^n+1,{n,20}],PrimeQ] (* Harvey P. Dale, Mar 27 2016 *)

for(n=1,999, ispseudoprime(p=n^n*2+1) & print1(p","))

a(2) = A216147(12) = A005109(95) = A070855(12) = A058383(89) = A133663(18).

a(3) = A216147(18) = A005109(183)= A070855(18) = A058383(177)= A133663(36).

cp's OEIS Frontend

A176642 Triangle T(n, k) = 8^(k*(n-k)), read by rows.

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Magma

Mathematica

Sage

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A241842 Number of simple connected graphs on n nodes that are non-integral.

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A051003 Beastly (or hateful) numbers: numbers containing the string 666 in their decimal expansion.

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Mathematica

A052046 Squares whose digits occur with the same frequency.

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Maple

Mathematica

Extensions

A114762 a(n) = floor(3^(1/2)*10^n)^2.

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Mathematica

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A135654 Divisors of 8128 (the 4th perfect number), written in base 2.

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Mathematica

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A135655 Divisors of 33550336 (the 5th perfect number), written in base 2.

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A137756 Nontrivial elements in writing the numerator of an element first and then the denominator of that element (left to right) of Leibniz's harmonic-like triangle. That is, the nontrivial elements of A137752.

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A166579 Prime numbers containing the string 17.

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