A001020 -id:A001020 - cp's OEIS frontend

A272269 Numbers n such that 11^n does not contain all ten decimal digits.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 25, 27, 28, 34, 38, 41

Offset: 1

Altug Alkan, Apr 24 2016

Inspiration was the simple form of 11 that is concatenation of 1 and 1. With similar motivation, A130696 focuses on the values of 2^n = (1 + 1)^n. Since this sequence exists in base 10, 11^n*10 is simply concatenation of 11^n and 0. So 11^(n+1) = concat(11^n, 0) + 11^n while 2^(n+1) = 2^n + 2^n.

A030706 is a subsequence. So note that if there is currently no proof of finiteness of A030706, then there is no proof yet of the finiteness of this sequence.

			25 is a term because 11^25 = 108347059433883722041830251 that does not contain digit 6.
26 is not a term because 11^26 = 11^25*10 + 11^25 = 1083470594338837220418302510 + 108347059433883722041830251 = 1191817653772720942460132761 that contains all ten decimal digits.

Cf. A001020, A030706, A130696, A171102.

Select[Range[0, 120], AnyTrue[DigitCount[11^#], # == 0 &] &] (* Michael De Vlieger, Apr 24 2016, Version 10 *)

isA171102(n) = 9<#vecsort(Vecsmall(Str(n)), , 8);
lista(nn) = for(n=0, nn, if(!isA171102(11^n), print1(n, ", ")));

select( is_A272269(n)=#Set(digits(11^n))<10 ,[0..100]) \\ M. F. Hasler, May 18 2017

A319074 a(n) is the sum of the first n nonnegative powers of the n-th prime.

Original entry on oeis.org

1, 4, 31, 400, 16105, 402234, 25646167, 943531280, 81870575521, 15025258332150, 846949229880161, 182859777940000980, 23127577557875340733, 1759175174860440565844, 262246703278703657363377, 74543635579202247026882160, 21930887362370823132822661921, 2279217547342466764922495586798

Offset: 1

Omar E. Pol, Sep 11 2018

nonn

			For n = 4 the 4th prime is 7 and the sum of the first four nonnegative powers of 7 is 7^0 + 7^1 + 7^2 + 7^3 = 1 + 7 + 49 + 343 = 400, so a(4) = 400.

Main diagonal of A319076.
Cf. A000040, A000203, A006093, A062457, A069459, A093360, A319075.
Cf. A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Cf. A126646, A003462, A003463, A023000, A016123, A091030, A091045, A218722, A218726, A218732, A218734, A218740, A218744, A218746, A218750.

a(n) = sum(k=0, n-1, prime(n)^k); \\ Michel Marcus, Sep 13 2018

a(n) = Sum_{k=0..n-1} A000040(n)^k.
a(n) = Sum_{k=0..n-1} A319075(k,n).
a(n) = (A000040(n)^n - 1)/(A000040(n) - 1).
a(n) = (A062457(n) - 1)/A006093(n).
a(n) = A069459(n)/A006093(n).
a(n) = A000203(A000040(n)^(n-1)).
a(n) = A000203(A093360(n)).

A319076 Square array T(n,k) read by antidiagonal upwards in which column k lists the partial sums of the powers of the k-th prime, n >= 0, k >= 1.

Original entry on oeis.org

1, 3, 1, 7, 4, 1, 15, 13, 6, 1, 31, 40, 31, 8, 1, 63, 121, 156, 57, 12, 1, 127, 364, 781, 400, 133, 14, 1, 255, 1093, 3906, 2801, 1464, 183, 18, 1, 511, 3280, 19531, 19608, 16105, 2380, 307, 20, 1, 1023, 9841, 97656, 137257, 177156, 30941, 5220, 381, 24, 1, 2047, 29524, 488281, 960800, 1948717

Offset: 0

Omar E. Pol, Sep 09 2018

T(n,k) is also the sum of the divisors of the n-th nonnegative power of the k-th prime, n >= 0, k >= 1.

			The corner of the square array is as follows:
         A126646 A003462 A003463  A023000    A016123    A091030     A091045
A000012        1,      1,      1,       1,         1,         1,          1, ...
A008864        3,      4,      6,       8,        12,        14,         18, ...
A060800        7,     13,     31,      57,       133,       183,        307, ...
A131991       15,     40,    156,     400,      1464,      2380,       5220, ...
A131992       31,    121,    781,    2801,     16105,     30941,      88741, ...
A131993       63,    364,   3906,   19608,    177156,    402234,    1508598, ...
.......      127,   1093,  19531,  137257,   1948717,   5229043,   25646167, ...
.......      255,   3280,  97656,  960800,  21435888,  67977560,  435984840, ...
.......      511,   9841, 488281, 6725601, 235794769, 883708281, 7411742281, ...
...

Rows 0-5: A000012, A008864, A060800, A131991, A131992, A131993.
Columns 1-15: A126646, A003462, A003463, A023000, A016123, A091030, A091045, A218722, A218726, A218732, A218734, A218740, A218744, A218746, A218750.
Main diagonal gives A319074.
Cf. A000079, A000244, A000351, A000420, A001020, A001022, A001026, A001029, A009967, A009973, A009975, A009981, A009985, A009987, A009991.
Cf. A000040, A000203, A006093, A319075.

T(n, k) = sigma(prime(k)^n); \\ Michel Marcus, Sep 13 2018

T(n,k) = A000203(A000040(k)^n).
T(n,k) = Sum_{j=0..n} A000040(k)^j.
T(n,k) = Sum_{j=0..n} A319075(j,k).
T(n,k) = (A000040(k)^(n+1) - 1)/(A000040(k) - 1).
T(n,k) = (A000040(k)^(n+1) - 1)/A006093(k).

A338784 a(n) is the smallest number with exactly n divisors such that all its divisors end with the same digit (which is necessarily 1).

Original entry on oeis.org

1, 11, 121, 341, 14641, 3751, 1771561, 13981, 116281, 453871, 25937424601, 153791, 3138428376721, 54918391, 14070001, 852841, 45949729863572161, 4767521, 5559917313492231481, 18608711, 1702470121, 804060162631, 81402749386839761113321, 9381251, 13521270961, 97291279678351, 195468361

Offset: 1

Bernard Schott, Nov 09 2020

As 1 is a divisor for each number, all the divisors must end with 1.

			121 is the smallest number whose 3 divisors (1, 11, 121) end with 1, hence a(3) = 121.
3751 is the smallest number whose 6 divisors (1, 11, 31, 121, 341, 3751) end with 1, hence a(6) = 121.
a(18) = 4767521 = 11^2 * 31^2 * 41 as it has 18 divisors all of which end in 1. - _David A. Corneth_, Nov 09 2020

David A. Corneth, Table of n, a(n) for n = 1..966
Project Euler, Problem 474: Last digits of divisors.

Cf. A001020, A030430, A092609, A330348.

Subsequence of A004615.

a(n) = {my(pr); if(n==1, return(1)); if(isprime(n), return(11^(n-1))); forstep(i = 1, oo, 10, f = factor(i); if(numdiv(f) == n, pr = 1; for(j = 1, #f~, if(f[j, 1]%10 != 1, pr = 0; next(2) ) ) ); if(pr, return(i)); ) } \\ David A. Corneth, Nov 09 2020

If n is prime p, then a(p) = 11^(p-1) = A001020(p-1).

For k>=1, a(2^k) = {Product_m=1..k} A030430(m) = A092609(k).

Data corrected by David A. Corneth, Nov 09 2020

A339794 a(n) is the least integer k satisfying rad(k)^2 < sigma(k) and whose prime factors set is the same as the prime factors set of A005117(n+1).

Original entry on oeis.org

4, 9, 25, 18, 49, 80, 121, 169, 112, 135, 289, 361, 441, 352, 529, 416, 841, 360, 961, 891, 1088, 875, 1369, 1216, 1053, 1681, 672, 1849, 1472, 2209, 2601, 2809, 3025, 3249, 1856, 3481, 3721, 1984, 4225, 1584, 4489, 4761, 1960, 5041, 5329, 4736, 5929, 2496, 6241

Offset: 1

Michel Marcus, Dec 17 2020

nonn

Equivalently, subsequence of terms of A339744 excluding terms whose prime factor set has already been encountered.
a(n) = A005117(n + 1)^2 when A005117(n + 1) is prime. Proof: if A005117(n + 1) is a prime p then rad(A005117(n + 1))^2 = rad(p)^2 = p^2 and so integers whose prime factors set is the same as the prime factors set of A005117(n + 1) = p are p^m where m >= 1. p^2 > sigma(p^1) = p + 1 but p^2 < sigma(p^2) = p^2 + p + 1. Q.E.D. - David A. Corneth, Dec 19 2020
From Bernard Schott, Jan 19 2021: (Start)
Indeed, a(n) satisfies the double inequality A005117(n+1) < a(n) <= A005117(n+1)^2.
It is also possible that a(n) = A005117(n+1)^2, even when A005117(n+1) is not prime; the smallest such example is for a(13) = 441 = 21^2 = A005117(14)^2. (End)

			   n  a(n) prime factor set
   1    4  [2]           A000079
   2    9  [3]           A000244
   3   25  [5]           A000351
   4   18  [2, 3]        A033845
   5   49  [7]           A000420
   6   80  [2, 5]        A033846
   7  121  [11]          A001020
   8  169  [13]          A001022
   9  112  [2, 7]        A033847
  10  135  [3, 5]        A033849
  11  289  [17]          A001026
  12  361  [19]          A001029
  13  441  [3, 7]        A033850
  14  352  [2, 11]       A033848
  15  529  [23]          A009967
  16  416  [2, 13]       A288162
  17  841  [29]          A009973
  18  360  [2, 3, 5]     A143207

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Michel Marcus)

Cf. A000203 (sigma), A007947 (rad).
Cf. A005117 (squarefree numbers), A027748, A265668, A339744.
Subsequence: A001248 (squares of primes).

u(n) = {my(fn=factor(n)[,1]); for (k = n, n^2, my(fk = factor(k)); if (fk[,1] == fn, if (factorback(fk[,1])^2 < sigma(fk), return (k));););}
lista(nn) = {for (n=2, nn, if (issquarefree(n), print1(u(n), ", ");););}

a(n) <= A005117(n+1)^2. - David A. Corneth, Dec 19 2020

A348553 Number of digits in 11^n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 0

Seiichi Manyama, Oct 22 2021

			a(24) = 25 because 11^24 = 9849732675807611094711841, which has 25 digits.
a(25) = 27 because 11^25 = 108347059433883722041830251, which has 27 digits.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Number of digits in b^n: A034887 (b=2), A034888 (b=3), A210434 (b=4), A210435 (b=5), A210436 (b=6), A210062 (b=7), this sequence (b=11).

Cf. A001020, A055642.

a[n_] := IntegerLength[11^n]; Array[a, 100, 0] (* Amiram Eldar, Oct 22 2021 *)

PARI
```
a(n) = #Str(11^n);
```

def a(n): return len(str(11**n))
print([a(n) for n in range(98)]) # Michael S. Branicky, Oct 22 2021

a(n) = A055642(A001020(n)) = A055642(11^n).

A013618 Triangle of coefficients in expansion of (1+11x)^n.

Original entry on oeis.org

1, 1, 11, 1, 22, 121, 1, 33, 363, 1331, 1, 44, 726, 5324, 14641, 1, 55, 1210, 13310, 73205, 161051, 1, 66, 1815, 26620, 219615, 966306, 1771561, 1, 77, 2541, 46585, 512435, 3382071, 12400927, 19487171, 1, 88, 3388, 74536, 1024870, 9018856, 49603708, 155897368, 214358881

Offset: 0

N. J. A. Sloane

T(n,k) equals the number of n-length words on {0,1,...,11} having n-k zeros. - Milan Janjic, Jul 24 2015

Seiichi Manyama, Rows n = 0..139, flattened

Cf. A001020 (right edge).

T:= n-> (p-> seq(coeff(p, x, k), k=0..n))((1+11*x)^n):
seq(T(n), n=0..10);  # Alois P. Heinz, Jul 24 2015

G.f.: 1 / (1 - x(1+11y)).

T(n,k) = 11^k*C(n,k) = Sum_{i=n-k..n} C(i,n-k)*C(n,i)*10^(n-i). Row sums are 12^n = A001021. - Mircea Merca, Apr 28 2012

A013795 a(n) = 11^(4*n+3).

Original entry on oeis.org

1331, 19487171, 285311670611, 4177248169415651, 61159090448414546291, 895430243255237372246531, 13109994191499930367061460371, 191943424957750480504146841291811, 2810243684806424785061213903353404851

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (14641).

Subsequence of A001020. - Michel Marcus, Apr 08 2017

[11^(4*n+3): n in [0..10]]; // Vincenzo Librandi, Jun 28 2011

A013795:=n->11^(4*n+3): seq(A013795(n), n=0..10); # Wesley Ivan Hurt, Apr 08 2017

11^(4*Range[0,10]+3) (* or *) NestList[14641#&,1331,10] (* Harvey P. Dale, May 26 2016 *)
CoefficientList[Series[1331/(1- 14641x), {x, 0, 10}], x] (* Indranil Ghosh, Apr 09 2017 *)

a(n) = 11^(4*n + 3) \\ Indranil Ghosh, Apr 09 2017

def a(n): return 11**(4*n + 3) # Indranil Ghosh, Apr 09 2017

a(n) = 14641*a(n-1). - Harvey P. Dale, May 26 2016

G.f.: 1331/(1 - 14641x). - Indranil Ghosh, Apr 09 2017

A013861 a(n) = 11^(5*n+4).

Original entry on oeis.org

14641, 2357947691, 379749833583241, 61159090448414546291, 9849732675807611094711841, 1586309297171491574414436704891, 255476698618765889551019445759400441, 41144777789250865278081232758997200423491

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (161051).

Cf. A001020 (11^n), A016897 (5*n+4).

[11^(5*n+4): n in [0..10]]; // Vincenzo Librandi, Jul 08 2011

A013861:=n->11^(5*n+4): seq(A013861(n), n=0..10); # Wesley Ivan Hurt, May 02 2017

11^(5Range[0,10]+4) (* Harvey P. Dale, Mar 21 2011 *)

A038259 Triangle whose (i,j)-th entry is binomial(i,j)6^(i-j)5^j.

Original entry on oeis.org

1, 6, 5, 36, 60, 25, 216, 540, 450, 125, 1296, 4320, 5400, 3000, 625, 7776, 32400, 54000, 45000, 18750, 3125, 46656, 233280, 486000, 540000, 337500, 112500, 15625, 279936, 1632960, 4082400, 5670000, 4725000, 2362500, 656250, 78125

Offset: 0

N. J. A. Sloane

			   1
   6    5
  36   60   25
216  540  450  125
1296 4320 5400 3000  625
7776 32400 54000 45000 18750 3125

B. N. Cyvin et al., Isomer enumeration of unbranched catacondensed polygonal systems with pentagons and heptagons, Match, No. 34 (Oct 1996), pp. 109-121.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A001020 (row sums).

Table[Binomial[k,m]6^(k-m) 5^m,{k,0,10},{m,0,k}]//Flatten (* Harvey P. Dale, Aug 26 2020 *)

cp's OEIS Frontend

A272269 Numbers n such that 11^n does not contain all ten decimal digits.

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A319074 a(n) is the sum of the first n nonnegative powers of the n-th prime.

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A319076 Square array T(n,k) read by antidiagonal upwards in which column k lists the partial sums of the powers of the k-th prime, n >= 0, k >= 1.

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A338784 a(n) is the smallest number with exactly n divisors such that all its divisors end with the same digit (which is necessarily 1).

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A339794 a(n) is the least integer k satisfying rad(k)^2 < sigma(k) and whose prime factors set is the same as the prime factors set of A005117(n+1).

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A348553 Number of digits in 11^n.

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A013618 Triangle of coefficients in expansion of (1+11x)^n.

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A013795 a(n) = 11^(4*n+3).

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A038259 Triangle whose (i,j)-th entry is binomial(i,j)6^(i-j)5^j.