A000030 -id:A000030 - cp's OEIS frontend

A267939 Number x = concat(MSD(x),b), where MSD = A000030 stands for Most Significant Digit, such that MSD(x)*b is equal to the reverse of x.

351, 621, 886, 920781, 3524751, 338752611, 35247524751, 920780120781, 920879219781, 3387524752611, 3526124738751, 338738752612611, 352475247524751, 33875247524752611, 35247387526124751, 35261247524738751, 920780120780120781, 920780219879120781, 920879120780219781, 920879219879219781

Offset: 1

Paolo P. Lava, Jan 22 2016

If we consider numbers x = concat(a,b), where a has two digits, such that a*b is equal to the reverse of x, the first terms are 425322, 44235301, 119910901, ...
Terms of the form 3(5247)*51, i.e. 351, 3524751, 35247524751, ..., form an infinite subsequence. - Robert Israel, Jan 28 2016
Other infinite sequences of terms include 92078(012078)*1 and 33875(2475)*2611. - Robert Israel, Jan 31 2016

			3*51 = 153;
6*21 = 126;
3*524751 = 1574253.

Robert Israel, Table of n, a(n) for n = 1..415

Cf. A000030, A004086.

T:=proc(w) local x, y, z; x:=w; y:=0;
for z from 1 to ilog10(x)+1 do y:=10*y+(x mod 10); x:=trunc(x/10); od; y; end:
P:=proc(q) local a,b,n; for n from 1 to q do a:=n mod 10; b:=trunc(n/10^ilog10(n));
if (a=1 and b>1) or (a=6 and (b=2 or b=4 or b=6 or b=8)) or (b=5 and (a=3 or a=5 or a=7 or a=9)) then
if T(n)=b*(n mod 10^ilog10(n)) then print(n); fi; fi; od; end: P(10^10);
# alternative:
N:= 20: # to get all terms with at most N digits.
extend:= proc(d,psol,eqs)
  local peqs, cvars, bvars, ncs, res,T, cs, ceqs, sol, svals;
  peqs:= subs(psol, eqs);
  cvars,bvars:= selectremove(t -> op(0,t) = 'c',indets(peqs));
  ncs:= nops(cvars);
  res:= NULL;
  if ncs >= 1 then
    T:= combinat:-cartprod([[$0..d-1]$ncs]);
    while not T[finished] do
      cs:= T[nextvalue]();
      cs:= seq(cvars[i]=cs[i],i=1..ncs);
      ceqs:= subs(cs,peqs);
      sol:= solve(ceqs,bvars); svals:= map(rhs,sol);
      if indets(svals) <> {} then error("Oops: %1",svals) fi;
      if svals::set(nonnegint) and max(svals) <= 9 then
        res:= res, [op(psol), cs, op(sol)];
      fi
    od
  else
    sol:= solve(peqs,bvars);
    svals:= map(rhs,sol);
    if indets(svals) <> {} then error("Oops: %1",svals) fi;
    if svals::set(nonnegint) and max(svals) <= 9 then
        res:= [op(psol), op(sol)];
    fi
  fi;
  [res]
end proc:
G:= proc(d,n)
     local eqs, i, rs, b0s;
     eqs:= [d*b[0] - d - 10*c[0],
            seq(d*b[i]+c[i-1] - b[n-i] - 10*c[i], i=1..n-2),
            d*b[n-1] + c[n-2] - b[1] - 10*b[0]];
     b0s:= [msolve(eqs[1] mod 10,10)];
     rs:= select(t -> (map(rhs,t))::set(nonnegint),
         map(t -> t union solve(eval(eqs[1],t),{c[0]}),b0s));
     for i from 1 to floor(n/2) do
        rs:= map(s -> op(extend(d,s,{eqs[i+1],eqs[-i]})), rs);
     od;
     sort(map(s -> d*10^n + subs(s, add(10^i*b[i],i=0..n-1)), rs));
end proc:
A:= NULL;
for n from 2 to N-1 do
  for d from 3 to 9 do
    res:= G(d,n);
    if res <> [] then
      A:= A, op(res);
    fi
  od
od:
A; # Robert Israel, Feb 01 2016

Select[Range@ 4000000, First[#] FromDigits@ Rest@ # == FromDigits@ Reverse@ # &@ IntegerDigits@ # &] (* Michael De Vlieger, Jan 29 2016 *)

a(7) to a(20) from Robert Israel, Feb 01 2016

A031298 Triangle T(n,k): write n in base 10, reverse order of digits.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

The length of n-th row is given in A055642(n). - Reinhard Zumkeller, Jul 04 2012

According to the formula for T(n,1), columns are numbered starting with 1. One might also number columns starting with the offset 0, as to have the coefficient of 10^k in column k. - M. F. Hasler, Jul 21 2013

Reinhard Zumkeller, Rows n = 0..2500 of triangle, flattened

Cf. A030308, A030341, A030386, A031235, A030567, A031007, A031045, A031087 for the base-2 to base-9 analogs.

a031298 n k = a031298_tabf !! n !! k
a031298_row n = a031298_tabf !! n
a031298_tabf = iterate succ [0] where
   succ []     = [1]
   succ (9:ds) = 0 : succ ds
   succ (d:ds) = (d + 1) : ds
-- Reinhard Zumkeller, Jul 04 2012

Table[Reverse[IntegerDigits[n]],{n,0,50}]//Flatten (* Harvey P. Dale, Mar 07 2023 *)

T(n,k)=n\10^(k-1)%10 \\ M. F. Hasler, Jul 21 2013

T(n,1) = A010879(n); T(n,A055642(n)) = A000030(n). - Reinhard Zumkeller, Jul 04 2012

Initial 0 and better name by Philippe Deléham, Oct 20 2011

Edited by M. F. Hasler, Jul 21 2013

A037123 a(n) = a(n-1) + sum of digits of n.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 46, 48, 51, 55, 60, 66, 73, 81, 90, 100, 102, 105, 109, 114, 120, 127, 135, 144, 154, 165, 168, 172, 177, 183, 190, 198, 207, 217, 228, 240, 244, 249, 255, 262, 270, 279, 289, 300, 312, 325, 330, 336, 343, 351, 360, 370, 381

Offset: 0

Vasiliy Danilov (danilovv(AT)usa.net), Jun 15 1998

Sum of digits of A007908(n). - Franz Vrabec, Oct 22 2007
Also digital sum of A138793(n) for n > 0. - Bruno Berselli, May 27 2011
Sum of the digital sum of i for i from 0 to n. - N. J. A. Sloane, Nov 13 2013

N. Agronomof, Sobre una función numérica, Revista Mat. Hispano-Americana 1 (1926), 267-269.
Maurice d'Ocagne, Sur certaines sommations arithmétiques, J. Sciencias Mathematicas e Astronomicas 7 (1886), 117-128.

David A Corneth, Table of n, a(n) for n = 0..10008
P.-H. Cheo and S.-C. Yien, A problem on the k-adic representation of positive integers, Acta Math. Sinica 5, 433-438 (1955).
J. Coquet, Power sums of digital sums, J. Number Theory 22 (1986), no. 2, 161-176.
H. Delange, Sur la fonction sommatoire de la fonction "somme des chiffres", Enseignement Math. (2) 21 (1975), 31-47.
P. J. Grabner, P. Kirschenhofer, H. Prodinger and R. F. Tichy, On the moments of the sum-of-digits function, Applications of Fibonacci numbers, Vol. 5 (St. Andrews, 1992), Kluwer Acad. Publ., Dordrecht, 1993, 263-271.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
J.-L. Mauclaire and Leo Murata, On q-additive functions, I. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 6, 274-276.
J.-L. Mauclaire and Leo Murata, On q-additive functions, II. Proc. Japan Acad. Ser. A Math. Sci. 59 (1983), no. 9, 441-444.
H. Riede, Asymptotic estimation of a sum of digits, Fibonacci Q. 36, No. 1, 72-75 (1998).
J. R. Trollope, An explicit expression for binary digital sums, Math. Mag. 41 1968 21-25.

Cf. A004207, A016052, A131383, A131384, A131451.
Cf. also A074784, A231688, A231689.
Partial sums of A007953.

[ n eq 0 select 0 else &+[&+Intseq(k): k in [0..n]]: n in [0..56] ];  // Bruno Berselli, May 27 2011

# From N. J. A. Sloane, Nov 13 2013:
digsum:=proc(n,B) local a; a := convert(n, base, B):
add(a[i], i=1..nops(a)): end;
f:=proc(n,k,B) global digsum; local i;
add( digsum(i,B)^k,i=0..n); end;
lprint([seq(digsum(n,10),n=0..100)]); # A007953
lprint([seq(f(n,1,10),n=0..100)]); #A037123
lprint([seq(f(n,2,10),n=0..100)]); #A074784
lprint([seq(f(n,3,10),n=0..100)]); #A231688
lprint([seq(f(n,4,10),n=0..100)]); #A231689

Table[Plus@@Flatten[IntegerDigits[Range[n]]], {n, 0, 200}] (* Enrique Pérez Herrero, Oct 12 2015 *)
a[0] = 0; a[n_] := a[n - 1] + Plus @@ IntegerDigits@ n; Array[a, 70, 0] (* Robert G. Wilson v, Jul 06 2018 *)

a(n)=n*(n+1)/2-9*sum(k=1,n,sum(i=1,ceil(log(k)/log(10)),floor(k/10^i)))

a(n)={n++;my(t,i,s);c=n;while(c!=0,i++;c\=10);for(j=1,i,d=(n\10^(i-j))%10;t+=(10^(i-j)*(s*d+binomial(d,2)+d*9*(i-j)/2));s+=d);t} \\ David A. Corneth, Aug 16 2013

for $i (0..100){ @j = split "", $i; for (@j){ $sum += $; } print "$sum,"; } __END_ # gamo(AT)telecable.es

a(n) = Sum_{k=0..n} s(k) = Sum_{k=0..n} A007953(k), where s(k) denote the sum of the digits of k in decimal representation. Asymptotic expression: a(n-1) = Sum_{k=0..n-1} s(k) = 4.5*n*log_10(n) + O(n). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
a(n) = n*(n+1)/2 - 9*Sum_{k=1..n} Sum_{i=1..ceiling(log_10(k))} floor(k/10^i). - Benoit Cloitre, Aug 28 2003
From Hieronymus Fischer, Jul 11 2007: (Start)
G.f.: Sum_{k>=1} ((x^k - x^(k+10^k) - 9x^(10^k))/(1-x^(10^k)))/(1-x)^2.
a(n) = (1/2)*((n+1)*(n - 18*Sum_{k>=1} floor(n/10^k)) + 9*Sum_{k>=1} (1 + floor(n/10^k))*floor(n/10^k)*10^k).
a(n) = (1/2)*((n+1)*(2*A007953(n)-n) + 9*Sum_{k>=1} (1+floor(n/10^k))*floor(n/10^k)*10^k). (End)
a(n) = A007953(A053064(n)). - Reinhard Zumkeller, Oct 10 2008
From Wojciech Raszka, Jun 14 2019: (Start)
a(10^k - 1) = 10*a(10^(k - 1) - 1) + 45*10^(k - 1) for k > 0.
a(n) = a(n mod m) + MSD*a(m - 1) + (MSD*(MSD - 1)/2)*m + MSD*((n mod m) + 1), where m = 10^(A055642(n) - 1), MSD = A000030(n). (End)

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002

A008905 Leading digit of n!.

Original entry on oeis.org

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

Offset: 0

Russ Cox

Kunoff proved that the distribution of terms of this sequence follows Benford's law, i.e., the asymptotic density of terms with value d (between 1 and 9) is log_10(1+1/d). - Amiram Eldar, Sep 23 2019

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Sharon Kunoff, N! has the first digit property, The Fibonacci Quarterly, Vol. 25, No. 4 (1987), pp. 365-367.
Index entries for sequences related to factorial numbers

Cf. A000966, A000142, A018799, A202021 (leading digit of (10^n)!), A213201.

a008905 = a000030 . a000142  -- Reinhard Zumkeller, Apr 08 2012

f[n_] := Quotient[n!, 10^Floor@ Log[10, n!]]; Array[f, 105, 0]

a(n) = A000030(A000142(n)). - Reinhard Zumkeller, Apr 08 2012

Two less-efficient Mathematica codings removed by Robert G. Wilson v, Nov 05 2010

A087097 Lunar primes (formerly called dismal primes) (cf. A087062).

Original entry on oeis.org

19, 29, 39, 49, 59, 69, 79, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 109, 209, 219, 309, 319, 329, 409, 419, 429, 439, 509, 519, 529, 539, 549, 609, 619, 629, 639, 649, 659, 709, 719, 729, 739, 749, 759, 769, 809, 819, 829, 839, 849, 859, 869, 879, 901, 902, 903, 904, 905, 906, 907, 908, 909, 912, 913, 914, 915, 916, 917, 918, 919, 923, 924, 925, 926, 927, 928, 929, 934, 935, 936, 937, 938, 939, 945, 946, 947, 948, 949, 956, 957, 958, 959, 967, 968, 969, 978, 979, 989

Offset: 1

Marc LeBrun, Oct 20 2003

9 is the multiplicative unit. A number is a lunar prime if it is not a lunar product (see A087062 for definition) r*s where neither r nor s is 9.
All lunar primes must contain a 9, so this is a subsequence of A011539.
Also, numbers k such that the lunar sum of the lunar prime divisors of k is k. - N. J. A. Sloane, Aug 23 2010
We have changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing. - N. J. A. Sloane, Aug 06 2014
(Lunar) composite numbers are not necessarily a product of primes. (For example 1 = 1*x for any x in {1, ..., 9} is not a prime but can't be written as the product of primes.) Therefore, to establish primality, it is not sufficient to consider only products of primes; one has to consider possible products of composite numbers as well. - M. F. Hasler, Nov 16 2018

			8 is not prime since 8 = 8*8. 9 is not prime since it is the multiplicative unit. 10 is not prime since 10 = 10*8. Thus 19 is the smallest prime.

David Applegate and N. J. A. Sloane, Table of n, a(n) for n = 1..22095 [all primes with at most 5 digits]
D. Applegate, C program for lunar arithmetic and number theory
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011.
Brady Haran and N. J. A. Sloane, Primes on the Moon (Lunar Arithmetic), Numberphile video, Nov 2018.
Index entries for sequences related to dismal (or lunar) arithmetic

Cf. A087019, A087061, A087062, A087636, A087638, A087984.

A87097=select( is_A087097(n)={my(d); if( n<100, n>88||(n%10==9&&n>9), vecmax(d=digits(n))<9, 0, #d<5, vecmin(d)A087062(m,k)==n&&return))))}, [1..999]) \\ M. F. Hasler, Nov 16 2018

The set { m in A011539 | 9A054054(m) < min(A000030(m),A010879(m)) } (9ish numbers A011539 with 2 digits or such that the smallest digit is strictly smaller than the first and the last digit) is equal to this sequence up to a(1656) = 10099. The next larger 9ish number 10109 is also in that set but is the lunar square of 109, thus not in this sequence of primes. - M. F. Hasler, Nov 16 2018

A037904 Greatest digit of n - least digit of n.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

a(n) = A054055(n)-A054054(n); a(A010785(n)) = 0; for k>0: a(n) = a(n*10^k + A000030(n)) = a(n*10^k + A010879(n)) = a(n*10^k + A054054(n)) = a(n*10^k + A054055(n)) . - Reinhard Zumkeller, Dec 14 2007; corrected by David Wasserman, May 21 2008

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

Cf. A040163, A064834.

a037904 = f 9 0 where
   f u v 0 = v - u
   f u v z = f (min u d) (max v d) z' where (z', d) = divMod z 10
-- Reinhard Zumkeller, Dec 16 2013

f:= n -> (max-min)(convert(n,base,10)):
map(f, [$1..1000]); # Robert Israel, Jul 07 2016

f[n_] := Block[{d = IntegerDigits[n]}, Max[d] - Min[d]]; Table[ f[n], {n, 1, 15}]

a(n)=my(d=digits(n)); vecmax(d)-vecmin(d) \\ Charles R Greathouse IV, Feb 07 2017

def A037904(n): return int(max(s:=str(n)))-int(min(s)) # Chai Wah Wu, Nov 10 2023

Incorrect comments deleted by Robert Israel, Jul 07 2016

A008963 Initial digit of Fibonacci number F(n).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Benford's law applies since the Fibonacci sequence is of exponential growth: P(d)=log_10(1+1/d), in fact among first 5000 values the digit d=1 appears 1505 times, while 5000*P(1) is about 1505.15. - Carmine Suriano, Feb 14 2011

Wlodarski observed and Webb proved that the distribution of terms of this sequence follows Benford's law. - Amiram Eldar, Sep 23 2019

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
William Webb, Distribution of the first digits of Fibonacci numbers, The Fibonacci Quarterly, Vol. 13, No. 4 (1975), pp. 334-336.
Wikipedia, Benford's law.
J. Wlodarski, Fibonacci and Lucas Numbers Tend to Obey Benford's Law, The Fibonacci Quarterly, Vol. 9, No. 1 (1971), pp. 87-88.
Index entries for sequences related to Benford's law.

Cf. A000045, A003893 (final digit).

Cf. A000030, A261607, A213201.

a008963 = a000030 . a000045  -- Reinhard Zumkeller, Sep 09 2015

F:= combinat[fibonacci]:
a:= n-> parse(""||(F(n))[1]):
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 22 2023

Table[IntegerDigits[Fibonacci[n]][[1]], {n, 0, 100}] (* T. D. Noe, Sep 23 2011 *)

vector(10001,n,f=fibonacci(n-1);f\10^(#Str(f)-1))

a(n) = A000030(A000045(n)). - Amiram Eldar, Sep 23 2019
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{d=1..9} d*log(1+1/d)/log(10) = 3.440236... (A213201). - Amiram Eldar, Jan 12 2023
For n>5, a(n) = floor(10^{alpha*n-beta}), where alpha=log_10(phi), beta=log_10(5)/2, {x}=x-floor(x) denotes the fractional part of x, log_10(phi) = A097348, and phi = (1+sqrt(5))/2 = A001622. - Hans J. H. Tuenter, Aug 20 2025

A131835 Numbers starting with 1.

Original entry on oeis.org

1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142

Offset: 1

Andrew Good (yipes_stripes(AT)yahoo.com), Jul 20 2007

The lower and upper asymptotic densities of this sequence are 1/9 and 5/9, respectively. - Amiram Eldar, Feb 27 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Bryan Brown, Michael Dairyko, Stephan Ramon Garcia, Bob Lutz and Michael Someck, Four quotient set gems, The American Mathematical Monthly, Vol. 121, No. 7 (2014), pp. 590-598; arXiv preprint, arXiv:1312.1036 [math.NT], 2013.
Index entries for 10-automatic sequences.

Subsequence of A011531.
Disjoint union of A045707 and A206286.
Cf. A000030, A000027, A002275, A262390 (permutation).
Cf. A217394, A217395, A217397, A217398, A217399, A217400, A217401, A217402.

a131835 n = a131835_list !! (n-1)
a131835_list = concat $
               iterate (concatMap (\x -> map (+ 10 * x) [0..9])) [1]
-- Reinhard Zumkeller, Jul 16 2014

isA131835 := proc(n) if op(-1,convert(n,base,10)) = 1 then true; else false ; fi ; end: for n from 1 to 300 do if isA131835(n) then printf("%d, ",n) ; fi ; od : # R. J. Mathar, Jul 24 2007

Select[Range[150], IntegerDigits[#][[1]] == 1 &] (* Amiram Eldar, Feb 27 2021 *)

a(n, {base=10}) = my (o=1); while (n>o, n-=o; o*=base); return (o+n-1) \\ Rémy Sigrist, Jun 23 2017

a(n) = n--; s = #digits(9*n+1); n + 8 * (10^(s-1))/9 + 1/9 \\ David A. Corneth, Jun 23 2017

nxt(n) = my(d = digits(n+1)); if(d[1]==1, n+1, 10^#d) \\ David A. Corneth, Jun 23 2017

def A131835(n): return n+(10**(len(str(9*n-8))-1)<<3)//9 # Chai Wah Wu, Dec 07 2024

A000030(a(n)) = 1. - Reinhard Zumkeller, Jul 16 2014
a(A002275(n)+1) = 10^n for any n >= 0. - Rémy Sigrist, Jun 23 2017
a(n) = n + (8*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 16 2023

More terms from R. J. Mathar, Jul 24 2007

A045708 Primes with first digit 2.

Original entry on oeis.org

2, 23, 29, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221

Offset: 1

Felice Russo

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040.
For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509
Cf. A000030, subsequence of A208272.
Column k=2 of A262369.

a045708 n = a045708_list !! (n-1)
a045708_list = filter ((== 2) . a000030) a000040_list
-- Reinhard Zumkeller, Mar 16 2012

[p: p in PrimesUpTo(2300) | Intseq(p)[#Intseq(p)] eq 2]; // Vincenzo Librandi, Aug 08 2014

Select[Table[Prime[n], {n, 3000}], First[IntegerDigits[#]]==2 &] (* Vincenzo Librandi, Aug 08 2014 *)

from sympy import isprime
def agen(limit=float('inf')):
  yield 2
  digits, adder = 1, 20
  while True:
    for i in range(1, 10**digits, 2):
      test = adder + i
      if test > limit: return
      if isprime(test): yield test
    digits, adder = digits+1, adder*10
agento = lambda lim: agen(limit=lim)
print(list(agento(2222))) # Michael S. Branicky, Feb 23 2021

from sympy import primepi
def A045708(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+primepi(min(((m:=10**(l:=len(str(x))-1))<<1)-1,x))-primepi(min(3*m-1,x))+sum(primepi(((m:=10**i)<<1)-1)-primepi(3*m-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 07 2024

See A045707 for comments on density of these sequences.

More terms from Erich Friedman.

Offset fixed by Reinhard Zumkeller, Mar 15 2012

A008952 Leading digit of 2^n.

Original entry on oeis.org

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

Offset: 0

Leonid Broukhis

Statistically, sequence obeys Benford's law, i.e. digit d occurs with probability log_10(1 + 1/d); thus 1 appears about 6.6 times more often than 9. - Lekraj Beedassy, May 04 2005
The most significant digits of the n-th powers of 2 are not cyclic and in the first 1000000 terms, 1 appears 301030 times, 2 appears 176093, 3 appears 124937, 4 appears 96911, 5 appears 79182, 6 appears 66947, 7 appears 57990, 8 appears 51154 and 9 appears 45756 times. - Robert G. Wilson v, Feb 03 2008
In fact the sequence follows Benford's law precisely by the equidistribution theorem. - Charles R Greathouse IV, Oct 11 2015

Robert G. Wilson v, Table of n, a(n) for n = 0..100000.
Brady Haran and Dmitry Kleinbock, Powers of 2, Numberphile video (2015). More footage.
Jaap Spies, A Bit of Math, The Art of Problem Solving, Jaap Spies Publishers (2019).
Wikipedia, Benford's law.
Wikipedia, Zipf's law.
Index entries for sequences related to Benford's law

Cf. A000030, A000079.

a:= n-> parse(""||(2^n)[1]):
seq(a(n), n=0..100);  # Alois P. Heinz, Aug 06 2021

a[n_] := First@ IntegerDigits[2^n]; Array[a, 105, 0] (* Robert G. Wilson v, Feb 03 2008 and corrected Nov 24 2014 *)

a(n)=digits(2^n)[1] \\ Charles R Greathouse IV, Oct 11 2015

def A008952(n): return int(str(1<Chai Wah Wu, Jul 07 2022

a(n) = [2^n / 10^([log_10(2^n)])] = [2^n / 10^([n*log_10(2)])].

a(n) = A000030(A000079(n)). - Omar E. Pol, Jul 04 2019

cp's OEIS Frontend

A267939 Number x = concat(MSD(x),b), where MSD = A000030 stands for Most Significant Digit, such that MSD(x)*b is equal to the reverse of x.

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A031298 Triangle T(n,k): write n in base 10, reverse order of digits.

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A037123 a(n) = a(n-1) + sum of digits of n.

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A008905 Leading digit of n!.

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A087097 Lunar primes (formerly called dismal primes) (cf. A087062).

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A037904 Greatest digit of n - least digit of n.

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