A011557 -id:A011557 - cp's OEIS frontend

A006880 Number of primes < 10^n.

0, 4, 25, 168, 1229, 9592, 78498, 664579, 5761455, 50847534, 455052511, 4118054813, 37607912018, 346065536839, 3204941750802, 29844570422669, 279238341033925, 2623557157654233, 24739954287740860, 234057667276344607, 2220819602560918840, 21127269486018731928, 201467286689315906290

Offset: 0

N. J. A. Sloane and Simon Plouffe

Number of primes with at most n digits; or pi(10^n).
Partial sums of A006879. - Lekraj Beedassy, Jun 25 2004
Also omega( (10^n)! ), where omega(x): number of distinct prime divisors of x. - Cino Hilliard, Jul 04 2007
This sequence also gives a good approximation for the sum of primes less than 10^(n/2). This is evident from the fact that the number of primes less than 10^2n closely approximates the sum of primes less than 10^n. See link on Sum of Primes for the derivation. - Cino Hilliard, Jun 08 2008
It appears that (10^n)/log((n+3)!) is a lower bound close to a(n), see A025201. - Eric Desbiaux, Jul 20 2010, edited by M. F. Hasler, Dec 03 2018

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 143, 146.
Richard Crandall and Carl B. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; p. 11.
Keith Devlin, Mathematics: The New Golden Age, new and revised edition. New York: Columbia University Press (1993): p. 6, Table 1.
Marcus du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; p. 48.
Calvin T. Long, Elementary Introduction to Number Theory. Prentice-Hall, Englewood Cliffs, NJ, 1987, p. 77.
Paulo Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 179.
H. Riesel, "Prime numbers and computer methods for factorization," Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, page 38.
D. Shanks, Solved and Unsolved Problems in Number Theory. Chelsea, NY, 2nd edition, 1978, p. 15.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See entry 455052511 at p. 190.

David Baugh, Table of n, a(n) for n = 0..29 (terms n = 1..27 from Charles R Greathouse IV).
Chris K. Caldwell, How Many Primes Are There?
Chris K. Caldwell, Mark Deleglise's work
Muhammed Hüsrev Cilasun, An Analytical Approach to Exponent-Restricted Multiple Counting Sequences, arXiv preprint arXiv:1412.3265 [math.NT], 2014.
Muhammed Hüsrev Cilasun, Generalized Multiple Counting Jacobsthal Sequences of Fermat Pseudoprimes, Journal of Integer Sequences, Vol. 19, 2016, #16.2.3.
Jens Franke, Thorsten Kleinjung, Jan Büthe, and Alexander Jost, A practical analytic method for calculating pi(x), Math. Comp. 86 (2017), 2889-2909.
Xavier Gourdon, a(22) found by pi(x) project
Xavier Gourdon and Pascal Sebah, The pi(x) project : results and current computations
Andrew Granville and Greg Martin, Prime number races, Amer. Math. Monthly, 113 (No. 1, 2006), 1-33.
Andrew Granville and Greg Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Cino Hilliard, Sum of primes [unusable link]
Ronald K. Hoeflin, Titan Test
D. S. Kluk and N. J. A. Sloane, Correspondence, 1979, [see p. 6 of the pdf].
Rishi Kumar, Kepler Sets of Second-Order Linear Recurrence Sequences Over Q_p, arXiv:2406.05890 [math.NT], 2024. See p. 7.
J. C. Lagarias, V. S. Miller, and A. M. Odlyzko, Computing pi(x): The Meissel-Lehmer method, Math. Comp., 44 (1985), pp. 537-560.
J. C. Lagarias and Andrew M. Odlyzko, Computing pi(x): An analytic method, J. Algorithms, 8 (1987), pp. 173-191.
Pieter Moree, Izabela Petrykiewicz, and Alisa Sedunova, A computational history of prime numbers and Riemann zeros, arXiv:1810.05244 [math.NT], 2018. See Table 1 p. 6.
Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)
Tomás Oliveira e Silva, Computing pi(x): the combinatorial method, Revista do Detua, Vol. 4, N 6, March 2006.
David J. Platt, Computing pi(x) analytically, arXiv:1203.5712 [math.NT], 2012-2013.
Vladimir Pletser, Conjecture on the value of Pi(10^26), the number of primes less than 10^26, arXiv:1307.4444 [math.NT], 2013.
Vladimir Pletser, Global Generalized Mersenne Numbers: Definition, Decomposition, and Generalized Theorems, Preprints.org, 2024. See p. 20.
Douglas B. Staple, The combinatorial algorithm for computing pi(x), arXiv:1503.01839 [math.NT], 2015.
M. R. Watkins, The distribution of prime numbers
Eric Weisstein's World of Mathematics, Prime Counting Function
Wikipedia, Prime number theorem
Robert G. Wilson v, Letter to N. J. A. Sloane, Jan. 1989
Index entries for sequences related to numbers of primes in various ranges

Cf. A000720, A006879, A006988, A007053, A011557, A025201, A040014.

a006880 = a000720 . (10 ^)  -- Reinhard Zumkeller, Mar 17 2015

Table[PrimePi[10^n], {n, 0, 14}] (* Jean-François Alcover, Nov 08 2016, corrected Sep 29 2020, a(14) being the maximum computable with certain implementations *)

a(n)=primepi(10^n) \\ Charles R Greathouse IV, Nov 08 2011

a(n) = A000720(10^n). - M. F. Hasler, Dec 03 2018

Limit_{n->oo} a(n)/a(n-1) = 10. - Stefano Spezia, Aug 31 2025

Lehmer gave the incorrect value 455052512 for the 10th term. More terms May 1996. Jud McCranie points out that the 11th term is not 4188054813 but rather 4118054813.
a(22) from Robert G. Wilson v, Sep 04 2001
a(23) (see Gourdon and Sebah) has yet to be verified and the assumed error is +-1. - Robert G. Wilson v, Jul 10 2002 [The actual error was 14037804. - N. J. A. Sloane, Nov 28 2007]
a(23) corrected by N. J. A. Sloane from the web page of Tomás Oliveira e Silva, Nov 28 2007
a(25) from J. Buethe, J. Franke, A. Jost, T. Kleinjung, Jun 01 2013, who said: "We have calculated pi(10^25) = 176846309399143769411680 unconditionally, using an analytic method based on Weil's explicit formula".
a(26) from Douglas B. Staple, Dec 02 2014
a(27) in the b-file from David Baugh and Kim Walisch via Charles R Greathouse IV, Jun 01 2016
a(28) in the b-file from David Baugh and Kim Walisch, Oct 26 2020
a(29) in the b-file from David Baugh and Kim Walisch, Feb 28 2022

A138148 Cyclops numbers with binary digits only.

Original entry on oeis.org

0, 101, 11011, 1110111, 111101111, 11111011111, 1111110111111, 111111101111111, 11111111011111111, 1111111110111111111, 111111111101111111111, 11111111111011111111111, 1111111111110111111111111, 111111111111101111111111111, 11111111111111011111111111111

Offset: 0

Omar E. Pol, Mar 18 2008

All members are palindromes A002113. The first five members are mentioned in A129868.
Also, binary representation of A129868.
a(A090748(n)) is equal to A138831(n), the n-th perfect number minus 1, written in base 2.
Except for the first term (replace 0 with 1) the binary representation of the n-th iteration of the elementary cellular automaton, Rule 219 starting with a single ON (black) cell. - Robert Price, Feb 21 2016
a(1) = 101 is only prime number in this sequence since a(n) = (10^(n+1)+1)*(10^n-1)/9. - Altug Alkan, May 11 2016

			n ........ a(n) .... A129868(n): value of a(n) read in base 2.
0 ......... 0 ......... 0
1 ........ 101 ........ 5
2 ....... 11011 ....... 27
3 ...... 1110111 ...... 119
4 ..... 111101111 ..... 495
5 .... 11111011111 .... 2015
6 ... 1111110111111 ... 8127

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 9, 14.
Brady Haran and Simon Pampena, Glitch Primes and Cyclops Numbers, Numberphile video, video (2015).
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cyclops numbers: A134808. Cf. A002113, A129868.
Cf. A000396, A090748, A135627, A138831.
Cf. A002275 (repunits R_n = (10^n-1)/9), A011557 (10^n).

[(-1 - 9*10^n + 10^(1 + 2*n))/9 : n in [0..15]]; // Wesley Ivan Hurt, Dec 08 2015

A138148:=n->(-1-9*10^n+10^(1+2*n))/9: seq(A138148(n), n=0..15); # Wesley Ivan Hurt, Dec 08 2015

Table[(-1 - 9*10^n + 10^(1 + 2*n))/9, {n, 0, 15}] (* Wesley Ivan Hurt, Dec 08 2015 *)

for(n=1, 20, if(n%2==1, c=((10^n-1)/9)-10^((n-1)/2); print1(c, ", "))) \\ Felix Fröhlich, Jul 07 2014

apply( {A138148(n)=10^(n*2+1)\9-10^n}, [0..15]) \\ M. F. Hasler, Feb 08 2020

From Colin Barker, Feb 21 2013: (Start)
a(n) = (-1-9*10^n+10^(1+2*n))/9.
G.f.: x*(200*x-101) / ((x-1)*(10*x-1)*(100*x-1)). (End)
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>2. - Wesley Ivan Hurt, Dec 08 2015
a(n) = A000533(n+1)*A002275(n). - Altug Alkan, May 12 2016
E.g.f.: (-1 - 9*exp(9*x) + 10*exp(99*x))*exp(x)/9. - Ilya Gutkovskiy, May 12 2016
a(n) = A002275(2n+1) - A011557(n). - M. F. Hasler, Feb 08 2020

More terms from Omar E. Pol, Feb 09 2020

A225546 Tek's flip: Write n as the product of distinct factors of the form prime(i)^(2^(j-1)) with i and j integers, and replace each such factor with prime(j)^(2^(i-1)).

Original entry on oeis.org

1, 2, 4, 3, 16, 8, 256, 6, 9, 32, 65536, 12, 4294967296, 512, 64, 5, 18446744073709551616, 18, 340282366920938463463374607431768211456, 48, 1024, 131072, 115792089237316195423570985008687907853269984665640564039457584007913129639936, 24, 81, 8589934592, 36, 768

Offset: 1

Paul Tek, May 10 2013

This is a multiplicative self-inverse permutation of the integers.
A225547 gives the fixed points.
From Antti Karttunen and Peter Munn, Feb 02 2020: (Start)
This sequence operates on the Fermi-Dirac factors of a number. As arranged in array form, in A329050, this sequence reflects these factors about the main diagonal of the array, substituting A329050[j,i] for A329050[i,j], and this results in many relationships including significant homomorphisms.
This sequence provides a relationship between the operations of squaring and prime shift (A003961) because each successive column of the A329050 array is the square of the previous column, and each successive row is the prime shift of the previous row.
A329050 gives examples of how significant sets of numbers can be formed by choosing their factors in relation to rows and/or columns. This sequence therefore maps equivalent derived sets by exchanging rows and columns. Thus odd numbers are exchanged for squares, squarefree numbers for powers of 2 etc.
Alternative construction: For n > 1, form a vector v of length A299090(n), where each element v[i] for i=1..A299090(n) is a product of those distinct prime factors p(i) of n whose exponent e(i) has the bit (i-1) "on", or 1 (as an empty product) if no such exponents are present. a(n) is then Product_{i=1..A299090(n)} A000040(i)^A048675(v[i]). Note that because each element of vector v is squarefree, it means that each exponent A048675(v[i]) present in the product is a "submask" (not all necessarily proper) of the binary string A087207(n).
This permutation effects the following mappings:
A000035(a(n)) = A010052(n), A010052(a(n)) = A000035(n). [Odd numbers <-> Squares]
A008966(a(n)) = A209229(n), A209229(a(n)) = A008966(n). [Squarefree numbers <-> Powers of 2]
(End)
From Antti Karttunen, Jul 08 2020: (Start)
Moreover, we see also that this sequence maps between A016825 (Numbers of the form 4k+2) and A001105 (2*squares) as well as between A008586 (Multiples of 4) and A028983 (Numbers with even sum of the divisors).
(End)

			  7744  = prime(1)^2^(2-1)*prime(1)^2^(3-1)*prime(5)^2^(2-1).
a(7744) = prime(2)^2^(1-1)*prime(3)^2^(1-1)*prime(2)^2^(5-1) = 645700815.

Cf. A225547 (fixed points) and the subsequences listed there.
Transposes A329050, A329332.
An automorphism of positive integers under the binary operations A059895, A059896, A059897, A306697, A329329.
An automorphism of A059897 subgroups: A000379, A003159, A016754, A122132.
Permutes lists where membership is determined by number of Fermi-Dirac factors: A000028, A050376, A176525, A268388.
Also permutes: A036554, A059404, A066427, A066428, A072774, A331593.
Sequences f that satisfy f(a(n)) = f(n): A048675, A064179, A064547, A097248, A302777, A331592.
Pairs of sequences (f,g) that satisfy a(f(n)) = g(a(n)): (A000265,A008833), (A000290,A003961), (A005843,A334747), (A006519,A007913), (A008586,A334748).
Pairs of sequences (f,g) that satisfy a(f(n)) = g(n), possibly with offset change: (A000040,A001146), (A000079,A019565).
Pairs of sequences (f,g) that satisfy f(a(n)) = g(n), possibly with offset change: (A000035, A010052), (A008966, A209229), (A007814, A248663), (A061395, A299090), (A087207, A267116), (A225569, A227291).
Cf. A331287 [= gcd(a(n),n)].
Cf. A331288 [= min(a(n),n)], see also A331301.
Cf. A331309 [= A000005(a(n)), number of divisors].
Cf. A331590 [= a(a(n)*a(n))].
Cf. A331591 [= A001221(a(n)), number of distinct prime factors], see also A331593.
Cf. A331740 [= A001222(a(n)), number of prime factors with multiplicity].
Cf. A331733 [= A000203(a(n)), sum of divisors].
Cf. A331734 [= A033879(a(n)), deficiency].
Cf. A331735 [= A009194(a(n))].
Cf. A331736 [= A000265(a(n)) = a(A008833(n)), largest odd divisor].
Cf. A335914 [= A038040(a(n))].
A self-inverse isomorphism between pairs of A059897 subgroups: (A000079,A005117), (A000244,A062503), (A000290\{0},A005408), (A000302,A056911), (A000351,A113849 U {1}), (A000400,A062838), (A001651,A252895), (A003586,A046100), (A007310,A000583), (A011557,A113850 U {1}), (A028982,A042968), (A053165,A065331), (A262675,A268390).
A bijection between pairs of sets: (A001248,A011764), (A007283,A133466), (A016825, A001105), (A008586, A028983).
Cf. also A336321, A336322 (compositions with another involution, A122111).

Array[If[# == 1, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[#]]] &, 28] (* Michael De Vlieger, Jan 21 2020 *)

A019565(n) = factorback(vecextract(primes(logint(n+!n, 2)+1), n));
a(n) = {my(f=factor(n)); for (i=1, #f~, my(p=f[i,1]); f[i,1] = A019565(f[i,2]); f[i,2] = 2^(primepi(p)-1);); factorback(f);} \\ Michel Marcus, Nov 29 2019

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };
A225546(n) = if(1==n,1,my(f=factor(n),u=#binary(vecmax(f[, 2])),prods=vector(u,x,1),m=1,e); for(i=1,u,for(k=1,#f~, if(bitand(f[k,2],m),prods[i] *= f[k,1])); m<<=1); prod(i=1,u,prime(i)^A048675(prods[i]))); \\ Antti Karttunen, Feb 02 2020

from math import prod
from sympy import prime, primepi, factorint
def A225546(n): return prod(prod(prime(i) for i, v in enumerate(bin(e)[:1:-1],1) if v == '1')**(1<Chai Wah Wu, Mar 17 2023

Multiplicative, with a(prime(i)^j) = A019565(j)^A000079(i-1).
a(prime(i)) = 2^(2^(i-1)).
From Antti Karttunen and Peter Munn, Feb 06 2020: (Start)
a(A329050(n,k)) = A329050(k,n).
a(A329332(n,k)) = A329332(k,n).
Equivalently, a(A019565(n)^k) = A019565(k)^n. If n = 1, this gives a(2^k) = A019565(k).
a(A059897(n,k)) = A059897(a(n), a(k)).
The previous formula implies a(n*k) = a(n) * a(k) if A059895(n,k) = 1.
a(A000040(n)) = A001146(n-1); a(A001146(n)) = A000040(n+1).
a(A000290(a(n))) = A003961(n); a(A003961(a(n))) = A000290(n) = n^2.
a(A000265(a(n))) = A008833(n); a(A008833(a(n))) = A000265(n).
a(A006519(a(n))) = A007913(n); a(A007913(a(n))) = A006519(n).
A007814(a(n)) = A248663(n); A248663(a(n)) = A007814(n).
A048675(a(n)) = A048675(n) and A048675(a(2^k * n)) = A048675(2^k * a(n)) = k + A048675(a(n)).
(End)
From Antti Karttunen and Peter Munn, Jul 08 2020: (Start)
For all n >= 1, a(2n) = A334747(a(n)).
In particular, for n = A003159(m), m >= 1, a(2n) = 2*a(n). [Note that A003159 includes all odd numbers]
(End)

Name edited by Peter Munn, Feb 14 2020

"Tek's flip" prepended to the name by Antti Karttunen, Jul 08 2020

A031347 Multiplicative digital root of n (keep multiplying digits of n until reaching a single digit).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2, 8, 4, 0, 4, 8, 2, 6, 0, 8, 6, 6, 8, 0, 5, 0, 5, 0, 0, 0, 5, 0, 0, 0, 6, 2, 8, 8, 0, 8, 8, 6, 0, 0, 7, 4, 2, 6, 5, 8, 8, 0, 8, 0, 8, 6, 8, 6, 0, 6

Offset: 0

Eric W. Weisstein

a(n) = 0 for almost all n. - Charles R Greathouse IV, Oct 02 2013

More precisely, a(n) = 0 asymptotically almost surely, namely, among others, for all numbers n which have a digit '0', and as n has more and more digits, it becomes increasingly less probable that no digit is equal to zero. (The set A011540 has density 1.) Thus the density of numbers for which a(n) > 0 is zero, although this happens for infinitely many numbers, for example all repunits n = (10^k - 1)/9 = A002275(k). - M. F. Hasler, Oct 11 2015

T. D. Noe, Table of n, a(n) for n = 0..10000
Shyam Sunder Gupta, Digital Root Wonders, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 1, 1-28.
Eric Weisstein's World of Mathematics, Multiplicative Digital Root.
Index entries for Colombian or self numbers and related sequences

Cf. A007954, A007953, A003001, A010888 (additive digital root of n), A031286 (additive persistence of n), A031346 (multiplicative persistence of n).

Numbers having multiplicative digital roots 0-9: A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056.

a031347 = until (< 10) a007954
-- Reinhard Zumkeller, Oct 17 2011, Sep 22 2011

A007954 := proc(n) return mul(d, d=convert(n,base,10)): end: A031347 := proc(n) local m: m:=n: while(length(m)>1)do m:=A007954(m): od: return m: end: seq(A031347(n),n=0..100); # Nathaniel Johnston, May 04 2011

mdr[n_] := NestWhile[Times @@ IntegerDigits@# &, n, UnsameQ, All]; Table[ mdr[n], {n, 0, 104}] (* Robert G. Wilson v, Aug 04 2006 *)
Table[NestWhile[Times@@IntegerDigits[#] &, n, # > 9 &], {n, 0, 90}] (* Harvey P. Dale, Mar 10 2019 *)

A031347(n)=local(resul); if(n<10, return(n) ); resul = n % 10; n = (n - n%10)/10; while( n > 0, resul *= n %10; n = (n - n%10)/10; ); return(A031347(resul))
for(n=1,80, print1(A031347(n),",")) \\ R. J. Mathar, May 23 2006

A031347(n)={while(n>9,n=prod(i=1,#n=digits(n),n[i]));n} \\ M. F. Hasler, Dec 07 2014

from operator import mul
from functools import reduce
def A031347(n):
    while n > 9:
       n = reduce(mul, (int(d) for d in str(n)))
    return n
# Chai Wah Wu, Aug 23 2014

from math import prod
def A031347(n):
    while n > 9: n = prod(map(int, str(n)))
    return n
print([A031347(n) for n in range(100)]) # Michael S. Branicky, Apr 17 2024

def iterDigitProd(n: Int): Int = n.toString.length match {
  case 1 => n
  case  => iterDigitProd(n.toString.toCharArray.map( - 48).scanRight(1)( * ).head)
}
(0 to 99).map(iterDigitProd) // Alonso del Arte, Apr 11 2020

a(n) = d in {1, ..., 9} if (but not only if) n = (10^k - 1)/9 + (d - 1)*10^m = A002275(k) + (d - 1)*A011557(m) for some k > m >= 0. - M. F. Hasler, Oct 11 2015

A052216 Sums of two powers of 10.

Original entry on oeis.org

2, 11, 20, 101, 110, 200, 1001, 1010, 1100, 2000, 10001, 10010, 10100, 11000, 20000, 100001, 100010, 100100, 101000, 110000, 200000, 1000001, 1000010, 1000100, 1001000, 1010000, 1100000, 2000000, 10000001, 10000010, 10000100, 10001000, 10010000, 10100000, 11000000, 20000000

Offset: 1

Henry Bottomley, Feb 01 2000

Numbers whose digit sum is 2.
A007953(a(n)) = 2; number of repdigits = #{2,11} = A242627(2) = 2. - Reinhard Zumkeller, Jul 17 2014
By extension, numbers k such that digitsum(k)^2 - 1 is prime. (PROOF: For any number k whose digit sum d > 2, d^2 - 1 = (d+1)*(d-1) and thus is not prime.) - Christian N. K. Anderson, Apr 22 2024

			From _Bruno Berselli_, Mar 07 2013: (Start)
The triangular array starts (see formula):
        2;
       11,      20;
      101,     110,     200;
     1001,    1010,    1100,    2000;
    10001,   10010,   10100,   11000,   20000;
   100001,  100010,  100100,  101000,  110000,  200000;
  1000001, 1000010, 1000100, 1001000, 1010000, 1100000, 2000000;
  ...
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (terms 1..48 from Vincenzo Librandi, terms 49..1036 from T. D. Noe)

Subsequence of A069263 and A107679. A038444 is a subsequence.
Sums of n powers of 10: A011557 (1), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225(14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Cf. A002262, A003056, A007953, A242614, A242627, A237424.

a052216 n = a052216_list !! (n-1)
a052216_list = 2 : f [2] 9 where
   f xs@(x:_) z = ys ++ f ys (10 * z) where
                  ys = (x + z) : map (* 10) xs
-- Reinhard Zumkeller, Jan 28 2015, Jul 17 2014

[n: n in [1..10100000] | &+Intseq(n) eq 2]; // Vincenzo Librandi, Mar 07 2013

/* As a triangular array: */ [[10^n+10^m: m in [0..n]]: n in [0..8]]; // Bruno Berselli, Mar 07 2013

t = 10^Range[0, 9]; Select[Union[Flatten[Table[i + j, {i, t}, {j, t}]]], # <= t[[-1]] + 1 &] (* T. D. Noe, Oct 09 2011 *)
With[{nn=7},Sort[Join[Table[FromDigits[PadRight[{2},n,0]],{n,nn}], FromDigits/@Flatten[Table[Table[Insert[PadRight[{1},n,0],1,i]],{n,nn},{i,2,n+1}],1]]]] (* Harvey P. Dale, Nov 15 2011 *)
Select[Range[10^9], Total[IntegerDigits[#]] == 2&] (* Vincenzo Librandi, Mar 07 2013 *)
T[n_,k_]:=10^(n-1)+10^(k-1); Table[T[n,k],{n,8},{k,n}]//Flatten (* Stefano Spezia, Nov 03 2023 *)

a(n)=my(d=(sqrtint(8*n)-1)\2,t=n-d*(d+1)/2-1); 10^d + 10^t \\ Charles R Greathouse IV, Dec 19 2016

from itertools import count, islice
def agen(): yield from (10**i + 10**j for i in count(0) for j in range(i+1))
print(list(islice(agen(), 34))) # Michael S. Branicky, May 15 2022

from math import isqrt
def A052216(n): return 10**(a:=(k:=isqrt(m:=n<<1))+(m>k*(k+1))-1)+10**(n-1-(a*(a+1)>>1)) # Chai Wah Wu, Apr 08 2025

def A052216(n,k): return 10^(n-1) + 10^(k-1)
flatten([[A052216(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Feb 22 2024

T(n,k) = 10^(n-1) + 10^(k-1) with 1 <= k <= n.
a(n) = 3*A237424(n) - 1. - Reinhard Zumkeller, Jan 28 2015
a(n) = 10^A003056(n-1) + 10^A002262(n-1). - Chai Wah Wu, Apr 08 2025

A052224 Numbers whose sum of digits is 10.

Original entry on oeis.org

19, 28, 37, 46, 55, 64, 73, 82, 91, 109, 118, 127, 136, 145, 154, 163, 172, 181, 190, 208, 217, 226, 235, 244, 253, 262, 271, 280, 307, 316, 325, 334, 343, 352, 361, 370, 406, 415, 424, 433, 442, 451, 460, 505, 514, 523, 532, 541, 550, 604, 613, 622, 631, 640

Offset: 1

Henry Bottomley, Feb 01 2000

Proper subsequence of A017173. - Rick L. Shepherd, Jan 12 2009
Subsequence of A227793. - Michel Marcus, Sep 23 2013
A007953(a(n)) = 10; number of repdigits = #{55,22222,1^10} = A242627(10) = 3. - Reinhard Zumkeller, Jul 17 2014
a(n) = A094677(n) for n = 1..28. - Reinhard Zumkeller, Nov 08 2015
The number of terms having <= m digits is the coefficient of x^10 in sum(i=0,9,x^i)^m = ((1-x^10)/(1-x))^m. - David A. Corneth, Jun 04 2016
In general, the set of numbers with sum of base-b digits equal to b is a subset of { (b-1)*k + 1; k = 2, 3, 4, ... }. - M. F. Hasler, Dec 23 2016

Rémy Sigrist, Table of n, a(n) for n = 1..10000 (first 3921 terms from T. D. Noe)

Cf. A007953, A017173, A228915, A254524.
Cf. A011557 (1), A052216 (2), A052217 (3), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A166311 (11), A235151 (12), A143164 (13), A235225 (14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Cf. A242614, A242627.
Cf. A094677.
Cf. A018900, A187813.
Sum of base-b digits equal b: A226636 (b = 3), A226969 (b = 4), A227062 (b = 5), A227080 (b = 6), A227092 (b = 7), A227095 (b = 8), A227238 (b = 9).

a052224 n = a052224_list !! (n-1)
a052224_list = filter ((== 10) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..1000] | &+Intseq(n) eq 10 ]; // Vincenzo Librandi, Mar 10 2013

sd := proc (n) options operator, arrow: add(convert(n, base, 10)[j], j = 1 .. nops(convert(n, base, 10))) end proc: a := proc (n) if sd(n) = 10 then n else end if end proc: seq(a(n), n = 1 .. 800); # Emeric Deutsch, Jan 16 2009

Union[Flatten[Table[FromDigits /@ Permutations[PadRight[s, 7]], {s, Rest[IntegerPartitions[10]]}]]] (* T. D. Noe, Mar 08 2013 *)
Select[Range[1000], Total[IntegerDigits[#]] == 10 &] (* Vincenzo Librandi, Mar 10 2013 *)

isok(n) = sumdigits(n) == 10; \\ Michel Marcus, Dec 28 2015

\\ This algorithm needs a modified binomial.
C(n, k)=if(n>=k, binomial(n, k), 0)
\\ ways to roll s-q with q dice having sides 0 through n - 1.
b(s, q, n)=if(s<=q*(n-1), s+=q; sum(i=0, q-1, (-1)^i*C(q, i)*C(s-1-n*i, q-1)), 0)
\\ main algorithm; this program applies to all sequences of the form "Numbers whose sum of digits is m."
a(n,{m=10}) = {my(q); q = 2; while(b(m, q, 10) < n, q++); q--; s = m; os = m; r=0; while(q, if(b(s, q, 10) < n, n-=b(s, q, 10); s--, r+=(os-s)*10^(q); os = s; q--)); r+= s; r}
\\ David A. Corneth, Jun 05 2016

from sympy.utilities.iterables import multiset_permutations
def auptodigs(maxdigits, b=10, sod=10): # works for any base, sum-of-digits
    alst = [sod] if 0 <= sod < b else []
    nzdigs = [i for i in range(1, b) if i <= sod]
    nzmultiset = []
    for d in range(1, b):
        nzmultiset += [d]*(sod//d)
    for d in range(2, maxdigits + 1):
        fullmultiset = [0]*(d-1-(sod-1)//(b-1)) + nzmultiset
        for firstdig in nzdigs:
            target_sum, restmultiset = sod - int(firstdig), fullmultiset[:]
            restmultiset.remove(firstdig)
            for p in multiset_permutations(restmultiset, d-1):
              if sum(p) == target_sum:
                  alst.append(int("".join(map(str, [firstdig]+p)), b))
                  if p[0] == target_sum:
                      break
    return alst
print(auptodigs(4)) # Michael S. Branicky, Sep 14 2021

def A052224(N = 19):
    """Return a generator of the sequence of all integers >= N with the same
    digit sum as N."""
    while True:
        yield N
        N = A228915(N) # skip to next larger integer with the same digit sum
a = A052224(); [next(a) for  in range(50)] # _M. F. Hasler, Mar 16 2022

a(n+1) = A228915(a(n)) for any n > 0. - Rémy Sigrist, Jul 10 2018

Incorrect formula deleted by N. J. A. Sloane, Jan 15 2009
Extended by Emeric Deutsch, Jan 16 2009
Offset changed by Bruno Berselli, Mar 07 2013

A052217 Numbers whose sum of digits is 3.

Original entry on oeis.org

3, 12, 21, 30, 102, 111, 120, 201, 210, 300, 1002, 1011, 1020, 1101, 1110, 1200, 2001, 2010, 2100, 3000, 10002, 10011, 10020, 10101, 10110, 10200, 11001, 11010, 11100, 12000, 20001, 20010, 20100, 21000, 30000, 100002, 100011, 100020, 100101

Offset: 1

Henry Bottomley, Feb 01 2000

From Joshua S.M. Weiner, Oct 19 2012: (Start)
Sequence is a representation of the "energy states" of "multiplex" notation of 3 quantum of objects in a juggling pattern.
0 = an empty site, or empty hand. 1 = one object resides in the site. 2 = two objects reside in the site. 3 = three objects reside in the site. (See A038447.) (End)
A007953(a(n)) = 3; number of repdigits = #{3,111} = A242627(3) = 2. - Reinhard Zumkeller, Jul 17 2014
Can be seen as a table whose n-th row holds the n-digit terms {10^(n-1) + 10^m + 10^k, 0 <= k <= m < n}, n >= 1. Row lengths are then (1, 3, 6, 10, ...) = n*(n+1)/2 = A000217(n). The first and the n last terms of row n are 10^(n-1) + 2 resp. 2*10^(n-1) + 10^k, 0 <= k < n. - M. F. Hasler, Feb 19 2020

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms 1..84 from Vincenzo Librandi, terms 85..1140 from T. D. Noe)

Cf. A069521 to A069530, A069532 to A069537.
Cf. A007953, A218043 (subsequence).
Row n=3 of A245062.
Other digit sums: A011557 (1), A052216 (2), A052218 (4), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225(14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Other bases: A014311 (binary), A226636 (ternary), A179243 (Zeckendorf).
Cf. A242614, A242627.
Cf. A003056, A002262 (triangular coordinates), A056556, A056557, A056558 (tetrahedral coordinates).

a052217 n = a052217_list !! (n-1)
a052217_list = filter ((== 3) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..100101] | &+Intseq(n) eq 3 ]; // Vincenzo Librandi, Mar 07 2013

Union[FromDigits/@Select[Flatten[Table[Tuples[Range[0,3],n],{n,6}],1],Total[#]==3&]] (* Harvey P. Dale, Oct 20 2012 *)
Select[Range[10^6], Total[IntegerDigits[#]] == 3 &] (* Vincenzo Librandi, Mar 07 2013 *)
Union[Flatten[Table[FromDigits /@ Permutations[PadRight[s, 18]], {s, IntegerPartitions[3]}]]] (* T. D. Noe, Mar 08 2013 *)

isok(n) = sumdigits(n) == 3; \\ Michel Marcus, Dec 28 2015

apply( {A052217_row(n,s,t=-1)=vector(n*(n+1)\2,k,t++>s&&t=!s++;10^(n-1)+10^s+10^t)}, [1..5]) \\ M. F. Hasler, Feb 19 2020

from itertools import count, islice
def agen(): yield from (10**i + 10**j + 10**k for i in count(0) for j in range(i+1) for k in range(j+1))
print(list(islice(agen(), 40))) # Michael S. Branicky, May 14 2022

from math import comb, isqrt
from sympy import integer_nthroot
def A052217(n): return 10**((m:=integer_nthroot(6*n,3)[0])-(a:=n<=comb(m+2,3)))+10**((k:=isqrt(b:=(c:=n-comb(m-a+2,3))<<1))-((b<<2)<=(k<<2)*(k+1)+1))+10**(c-1-comb(k+(b>k*(k+1)),2)) # Chai Wah Wu, Dec 11 2024

T(n,k) = 10^(n-1) + 10^A003056(k) + 10^A002262(k) when read as a table with row lengths n*(n+1)/2, n >= 1, 0 <= k < n*(n+1)/2. - M. F. Hasler, Feb 19 2020

a(n) = 10^A056556(n-1) + 10^A056557(n-1) + 10^A056558(n-1). - Kevin Ryde, Apr 17 2021

Offset changed from 0 to 1 by Vincenzo Librandi, Mar 07 2013

A062397 a(n) = 10^n + 1.

Original entry on oeis.org

2, 11, 101, 1001, 10001, 100001, 1000001, 10000001, 100000001, 1000000001, 10000000001, 100000000001, 1000000000001, 10000000000001, 100000000000001, 1000000000000001, 10000000000000001, 100000000000000001

Offset: 0

Henry Bottomley, Jun 22 2001

The first three terms (indices 0, 1 and 2) are the only known primes. Moreover, the terms not of the form a(2^k) are all composite, except for a(0). Indeed, for all n >= 0, a(2n+1) is divisible by 11, a(4n+2) is divisible by 101, a(8n+4) is divisible by 73, a(16n+8) is divisible by 17, a(32n+16) is divisible by 353, a(64n+32) is divisible by 19841, etc. - M. F. Hasler, Nov 03 2018 [Edited based on the comment by Jeppe Stig Nielsen, Oct 17 2019]
This sequence also results when each term is generated by converting the previous term into a Roman numeral, then replacing each letter with its corresponding decimal value, provided that the vinculum is used and numerals are written in a specific way for integers greater than 3999, e.g., IV with a vinculum over the I and V for 4000. - Jamie Robert Creasey, Apr 14 2021
By Mihăilescu's theorem, a(n) can never be a perfect power (see "Catalan's conjecture" in Links). - Marco Ripà, Mar 10 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wikipedia, Catalan's conjecture.
Index entries for linear recurrences with constant coefficients, signature (11,-10).

Except for the initial term, essentially the same as A000533. Cf. A054977, A007395, A000051, A034472, A052539, A034474, A062394, A034491, A062395, A062396, A007689, A063376, A063481, A074600-A074624, A034524, A178248, A228081 for numbers one more than powers, i.e., this sequence translated from base n (> 2) to base 10.
Cf. A002283, A011557.
Cf. A038371 (smallest prime factor), A185121.

[10^n + 1: n in [0..35]]; // Vincenzo Librandi, Apr 30 2011

LinearRecurrence[{11, -10},{2, 11},18] (* Ray Chandler, Aug 26 2015 *)
10^Range[0,20]+1 (* Harvey P. Dale, Jan 21 2020 *)

a(n)=10^n+1 \\ Charles R Greathouse IV, Sep 24 2015

a(n) = 10*a(n-1) - 9 = A011557(n) + 1 = A002283(n) + 2.
From Mohammad K. Azarian, Jan 02 2009: (Start)
G.f.: 1/(1-x) + 1/(1-10*x).
E.g.f.: exp(x) + exp(10*x). (End)

A009964 Powers of 20.

Original entry on oeis.org

1, 20, 400, 8000, 160000, 3200000, 64000000, 1280000000, 25600000000, 512000000000, 10240000000000, 204800000000000, 4096000000000000, 81920000000000000, 1638400000000000000, 32768000000000000000

Offset: 0

N. J. A. Sloane

Same as Pisot sequences E(1, 20), L(1, 20), P(1, 20), T(1, 20). Essentially same as Pisot sequences E(20, 400), L(20, 400), P(20, 400), T(20, 400). See A008776 for definitions of Pisot sequences.
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 20-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011
a(n) gives the number of small cubes in the n-th iteration of the Menger sponge fractal. - Felix Fröhlich, Jul 09 2016
Equivalently, the number of vertices in the n-Menger sponge graph.

T. D. Noe, Table of n, a(n) for n = 0..100
Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Tanya Khovanova, Recursive Sequences
Eric Weisstein's World of Mathematics, Menger Sponge
Eric Weisstein's World of Mathematics, Menger Sponge Graph
Eric Weisstein's World of Mathematics, Vertex Count
Wikipedia, Menger sponge
Index entries for linear recurrences with constant coefficients, signature (20).

Cf. A291066 (edge count).
Cf. A000079, A011557; A000302, A000351; A000244, A159991.
Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.

List([0..20],n->20^n); # Muniru A Asiru, Nov 21 2018

[20^n: n in [0..100]] // Vincenzo Librandi, Nov 21 2010

[20^n$n=0..20]; # Muniru A Asiru, Nov 21 2018

20^Range[0, 10] (* or *) LinearRecurrence[{20}, {1}, 20] (* Eric W. Weisstein, Aug 17 2017 *)

makelist(20^n,n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=20^n \\ Charles R Greathouse IV, Jun 19 2015

powers(20,12) \\ Charles R Greathouse IV, Jun 19 2015

[20**n for n in range(21)] # Stefano Spezia, Nov 21 2018

[20^n for n in range(21)] # Zerinvary Lajos, Apr 29 2009

G.f.: 1/(1-20*x).
E.g.f.: exp(20*x).
a(n) = A159991(n)/A000244(n). - Reinhard Zumkeller, May 02 2009
From Vincenzo Librandi, Nov 21 2010: (Start)
a(n) = 20^n.
a(n) = 20*a(n-1) for n > 0, a(0) = 1. (End)
a(n) = A000079(n)*A011557(n) = A000302(n)*A000351(n). - Felix Fröhlich, Jul 09 2016

A052218 Numbers whose sum of digits is 4.

Original entry on oeis.org

4, 13, 22, 31, 40, 103, 112, 121, 130, 202, 211, 220, 301, 310, 400, 1003, 1012, 1021, 1030, 1102, 1111, 1120, 1201, 1210, 1300, 2002, 2011, 2020, 2101, 2110, 2200, 3001, 3010, 3100, 4000, 10003, 10012, 10021, 10030, 10102, 10111, 10120, 10201, 10210, 10300

Offset: 1

Henry Bottomley, Feb 01 2000

A007953(a(n)) = 4; number of repdigits = #{4,22,1111} = A242627(4) = 3. - Reinhard Zumkeller, Jul 17 2014

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1001 from Vincenzo Librandi and T. D. Noe, terms 1..201 from Vincenzo Librandi)

Cf. A007953.
Cf. A011557 (1), A052216 (2), A052217 (3), A052219 (5), A052220 (6), A052221 (7), A052222 (8), A052223 (9), A052224 (10), A166311 (11), A235151 (12), A143164 (13), A235225(14), A235226 (15), A235227 (16), A166370 (17), A235228 (18), A166459 (19), A235229 (20).
Cf. A242614, A242627.

a052218 n = a052218_list !! (n-1)
a052218_list = filter ((== 4) . a007953) [0..]
-- Reinhard Zumkeller, Jul 17 2014

[n: n in [1..10300] | &+Intseq(n) eq 4 ]; // Vincenzo Librandi, Mar 07 2013

Select[Range[10^5], Total[IntegerDigits[#]] == 4 &] (* Vincenzo Librandi, Mar 07 2013 *)
Union[Flatten[Table[FromDigits /@ Permutations[PadRight[s, 11]], {s, IntegerPartitions[4]}]]] (* T. D. Noe, Mar 08 2013 *)

isok(n) = sumdigits(n) == 4; \\ Michel Marcus, Dec 28 2015

from itertools import count, islice
def agen(): yield from (10**i + 10**j + 10**k + 10**m for i in count(0) for j in range(i+1) for k in range(j+1) for m in range(k+1))
print(list(islice(agen(), 45))) # Michael S. Branicky, May 15 2022

Offset changed from Bruno Berselli, Mar 07 2013

cp's OEIS Frontend

A006880 Number of primes < 10^n.

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A138148 Cyclops numbers with binary digits only.

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A225546 Tek's flip: Write n as the product of distinct factors of the form prime(i)^(2^(j-1)) with i and j integers, and replace each such factor with prime(j)^(2^(i-1)).

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A031347 Multiplicative digital root of n (keep multiplying digits of n until reaching a single digit).

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A052216 Sums of two powers of 10.

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