A055642 -id:A055642 - cp's OEIS frontend

A046760 Wasteful numbers.

4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 108, 110, 114

Offset: 1

N. J. A. Sloane

Write n as product of primes raised to powers, let D(n) = number of digits in product, l(n) = number of digits in n; sequence gives n such that D(n)>l(n).

A050252(a(n)) > A055642(a(n)). - Reinhard Zumkeller, Jun 21 2011

			For n = 125 = 5^3, l(n) = 3 but D(n) = 2. So 125 is not a term of this sequence. [clarified by _Derek Orr_, Jan 30 2015]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J.-P. Delahaye, Les chasseurs de nombres premiers.
R. G. E. Pinch, Economical numbers., arXiv:math/9802046 [math.NT], 1998.
Eric Weisstein's World of Mathematics, Wasteful Number.
Wikipedia, Extravagant number.

Cf. A046758, A046759.

a046760 n = a046760_list !! (n-1)
a046760_list = filter (\n -> a050252 n > a055642 n) [1..]
-- Reinhard Zumkeller, Aug 02 2013

Cases[Range[115], n_ /; Length[Flatten[IntegerDigits[FactorInteger[n] /. 1 -> Sequence[]]]] > Length[IntegerDigits[n]]] (* Jean-François Alcover, Mar 21 2011 *)

for(n=1,100,s="";F=factor(n);for(i=1,#F[,1],s=concat(s,Str(F[i,1]));s=concat(s,Str(F[i,2])));c=0;for(j=1,#F[,2],if(F[j,2]==1,c++));if(#digits(n)<#s-c,print1(n,", "))) \\ Derek Orr, Jan 30 2015

from itertools import count, islice
from sympy import factorint
def A046760_gen(): # generator of terms
    return (n for n in count(1) if len(str(n)) < sum(len(str(p))+(len(str(e)) if e > 1 else 0) for p, e in factorint(n).items()))
A046760_list = list(islice(A046760_gen(),20)) # Chai Wah Wu, Feb 18 2022

A075167 Number of edges in each rooted plane tree produced with the unranking algorithm presented in A075166, which is based on prime factorization.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 3, 4, 5, 4, 6, 5, 4, 3, 7, 4, 8, 5, 5, 6, 9, 4, 4, 7, 4, 6, 10, 5, 11, 4, 6, 8, 5, 5, 12, 9, 7, 5, 13, 6, 14, 7, 5, 10, 15, 5, 5, 5, 8, 8, 16, 5, 6, 6, 9, 11, 17, 6, 18, 12, 6, 4, 7, 7, 19, 9, 10, 6, 20, 5, 21, 13, 5, 10, 6, 8, 22, 6, 4, 14, 23, 7, 8, 15, 11, 7, 24, 6, 7, 11

Offset: 1

Antti Karttunen, Sep 13 2002

nonn

Each n occurs A000108(n) times in total.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Permutation of A072643 and A106457.
A253782 gives the positions where this sequence differs from A252464 (first time at n=16).
Cf. A000108, A000120, A029837, A055642, A051119, A061395, A071178, A075165, A075166, A106442, A253783.
Cf. also A106490.

a(n) = A106457(A106442(n)). - Antti Karttunen, May 09 2005
From Antti Karttunen, Jan 16 2015: (Start)
a(1) = 0; for n>1: a(n) = a(A071178(n)) + (A061395(n) - A061395(A051119(n))) + A253783(A051119(n)).
Other identities.
For all n >= 2, a(n) = A055642(A075166(n))/2. [Half of the number of decimal digits in A075166(n).]
For all n >= 2, a(n) = A029837(1+A075165(n))/2. [Half of the binary width of A075165(n).]
For all n >= 1, a(n) = A000120(A075165(n)). [Thus also the binary weight of A075165(n), because half of the bits are zeros.]
(End)

More terms from Antti Karttunen, May 09 2005

A100910 Table of number of occurrences in n of each decimal digit from 0 to 9.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0

Offset: 0

Rick L. Shepherd, Nov 21 2004

Each row of this table has length 10 and corresponds to one term of A100909. n = 0 is normally represented as the single digit 0, so the first row here is 1, 0, 0, 0, 0, 0, 0, 0, 0, 0.

Robert Israel, Table of n, a(n) for n = 0..10009 (rows 0 to 1000, flattened)

Cf. A100909 (similar but each row of A100910 provides one A100909 term).
Cf. A055642 (row sums), A055641 (column 0), A268643 (column 1), A316863 (column 2), A316864 (column 3), A316865 (column 4), A316866 (column 5), A316867 (column 6), A316868 (column 7), A316869 (column 8), A102683 (column 9).
Cf. A059995, A010879.

seq(seq(numboccur(k, convert(n,base,10)),k=0..9),n=0..100); # Robert Israel, Jul 08 2016

A100910row[n_] := RotateRight[DigitCount[n]];
Array[A100910row, 10, 0] (* Paolo Xausa, Jul 16 2025 *)

T(n, k) = #select(x->x==k, digits(n))+!(n+k); \\ Jinyuan Wang, Mar 01 2020

From Robert Israel, Jul 08 2016: (Start)
a(n,k) = a(A059995(n),k) + (1 if A010879(n)=k, otherwise 0).
G.f. g(x,y) satisfies g(x,y) = ((1-x^10)/(1-x))*g(x^10,y) + (x^10-x)/(1-x) + x^10/(1-x^10) + x*y*(1-x^9*y^9)/((1-x^10)*(1-x*y)). (End)

A158232 Numbers which yield primes when "13" is prefixed or appended: N natural number is a member of the sequence, if P="13N" (prefixed 13) and A="N13" (appended 13) are prime.

Original entry on oeis.org

1, 19, 21, 27, 61, 103, 121, 127, 147, 159, 177, 183, 187, 217, 241, 259, 267, 327, 331, 337, 367, 381, 411, 477, 523, 553, 567, 577, 591, 633, 681, 687, 693, 709, 723, 759, 807, 829, 873, 903, 931, 997, 1009, 1011, 1041, 1059, 1129, 1149, 1213, 1231, 1251

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Mar 14 2009

It is conjectured and numerically examined that sequences of this type are infinite.

It is also conjectured that an infinite number of primes are terms of the sequence; first 20 primes are: 19, 61, 103, 127, 241, 331, 337, 367, 523, 577, 709, 829, 997, 1009, 1129, 1213, 1381, 1489, 1543, 1627.

			19: 1319 and 1913 are primes => a(2)=19;
7 is not a term: 137 is prime but 713=23 * 31 is not.

A. Weil, Number theory: an approach through history, Birkhäuser, 1984.
Richard E. Crandall, Carl Pomerance, Prime Numbers, Springer 2005.
Paulo Ribenboim, The New Book of Prime Number Records, Springer 1996.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A157772.

A055642 := proc(n) max(1,ilog10(n)+1) ; end proc: cat2 := proc(a,b) a*10^A055642(b)+b ; end proc: A158232 := proc(n) option remember; local a; if n = 1 then 1; else for a from procname(n-1)+1 do if isprime(cat2(13,a)) and isprime(cat2(a,13)) then return a ; end if ; end do ; end if; end proc: seq(A158232(n),n=1..80) ; # R. J. Mathar, Nov 11 2009

Select[Range[1300],And@@PrimeQ[{13 10^IntegerLength[#]+#,100#+13}]&] (* Harvey P. Dale, May 28 2012 *)

A004218 a(n) = log_10(n) rounded up.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3

Offset: 1

N. J. A. Sloane

nonn

a(n) is the number of terms in the sequence A011557 (Powers of 10) that are less than n. For n > 1, a(n) is the number of digits in n-1. - Tanya Khovanova, Jun 22 2007

			From _M. F. Hasler_, Dec 07 2018: (Start)
log_10(1) = 0, therefore a(1) = 0.
log_10(2) = 0.301..., therefore a(2) = 1.
log_10(9) = 0.954..., therefore a(9) = 1.
log_10(10) = 1, therefore a(10) = 1.
log_10(11) = 1.04..., therefore a(11) = 2.
log_10(99) = 1.9956..., therefore a(99) = 2.
log_10(100) = 2, therefore a(100) = 2.
log_10(101) = 2.004..., therefore a(101) = 3. (End)

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

Cf. A004216, A007953, A055642.

a004218 n = if n == 1 then 0 else 1 + a004216 (n - 1)

A004218 := proc(n)
    ceil(log[10](n)) ;
end proc:
seq(A004218(n),n=1..120) ; # R. J. Mathar, May 16 2023

Array[Ceiling[Log10[#]] &, 100] (* Amiram Eldar, Dec 08 2018 *)

A004218(n)=logint(n-(n>1),10)+1 \\ M. F. Hasler, Dec 07 2018

a(1) = 0, a(n) = 1 + A004216(n-1) for n > 1. - Reinhard Zumkeller, Dec 22 2012

a(n) = A055642(n-1) for all n > 1. a(n+1) is the number of decimal digits of n if 0 is considered to have 0 digits. - M. F. Hasler, Dec 07 2018

A005341 Length of n-th term in Look and Say sequences A005150 and A007651.

Original entry on oeis.org

1, 2, 2, 4, 6, 6, 8, 10, 14, 20, 26, 34, 46, 62, 78, 102, 134, 176, 226, 302, 408, 528, 678, 904, 1182, 1540, 2012, 2606, 3410, 4462, 5808, 7586, 9898, 12884, 16774, 21890, 28528, 37158, 48410, 63138, 82350, 107312, 139984, 182376, 237746, 310036, 403966, 526646, 686646

Offset: 1

Jeffrey Shallit

Row lengths of A034002 and of A220424. - Reinhard Zumkeller, Dec 15 2012

Satisfies a recurrence of order 72. The characteristic polynomial of this recurrence is a degree-72 polynomial that factors as (x-1)*q(x), where q(x) is a degree-71 polynomial. The unique positive real root of q is approximately 1.3036 and is called Conway's constant (A014715), which equals the limiting ratio a(n+1)/a(n). - Nathaniel Johnston, Apr 12 2018 [Corrected by Richard Stanley, Dec 26 2018]

J. H. Conway, The weird and wonderful chemistry of audioactive decay, in T. M. Cover and Gopinath, eds., Open Problems in Communication and Communications, Springer, NY 1987, pp. 173-188.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 452-455.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Peter J. C. Moses, Table of n, a(n) for n = 1..3000 (first 71 terms from Zak Seidov)
S. R. Finch, Conway's Constant.
S. R. Finch, Conway's Constant. [From the Wayback Machine]
Christoph Koutschan,Regular Languages and their Generating Functions: The Inverse Problem.
Eric Weisstein's World of Mathematics, Look and Say Sequence.

a005341 = length . a034002_row  -- Reinhard Zumkeller, Dec 15 2012

RunLengthEncode[ x_List ] := (Through[ {First, Length}[ #1 ] ] &) /@ Split[ x ]; LookAndSay[ n_, d_:1 ] := NestList[ Flatten[ Reverse /@ RunLengthEncode[ # ] ] &, {d}, n - 1 ]; F[ n_ ] := LookAndSay[ n, 1 ][ [ n ] ]; Table[ Length[ F[ n ] ], {n, 1, 51} ]
p = {12, -18, 18, -18, 18, -20, -22, 31, 15, -4, -4, -19, 62, -50, -21, -11, 41, 54, -56, -44, 15, -27, -15, 45, -8, 89, -64, -66, -25, 38, 126, -39, -32, -33, -65, 107, 14, 16, -13, -79, 7, 42, 12, 8, -26, -9, 35, -23, -20, -30, 34, 58, -1, -20, -36, -6, 13, 8, 6, 3, -1, -4, -1, -4, -5, -1, 8, 6, 0, -6, -4, 1, 0, 1, 1, 1, 1, -1, -1}; q = {-6, 9, -9, 18, -16, 11, -14, 8, -1, 5, -7, -2, -8, 14, 5, 5, -19, -3, 6, 7, 6, -16, 7, -8, 22, -17, 12, -7, -5, -7, 8, -4, 7, 9, -13, 4, 6, -14, 14, -19, 7, 13, -2, 4, -18, 0, 1, 4, 12, -8, 5, 0, -8, -1, -7, 8, 5, 2, -3, -3, 0, 0, 0, 0, 2, 1, 0, -3, -1, 1, 1, 1, -1}; gf = Fold[x #1 + #2 &, 0, p]/Fold[x #1 + #2 &, 0, q]; CoefficientList[Series[gf, {x, 0, 99}], x] (* Peter J. C. Moses, Jun 23 2013 *)

print1(a=1);for(i=2,100,print1(",",#Str(a=A005150(2,a))))  \\ M. F. Hasler, Nov 08 2011

a(n) = A055642(A005150(n)) = A055642(A007651(n)). - Reinhard Zumkeller, Dec 15 2012

More terms from Mike Keith

A019523 Concatenation of Fibonacci(1) through Fibonacci(n).

Original entry on oeis.org

1, 11, 112, 1123, 11235, 112358, 11235813, 1123581321, 112358132134, 11235813213455, 1123581321345589, 1123581321345589144, 1123581321345589144233, 1123581321345589144233377, 1123581321345589144233377610, 1123581321345589144233377610987

Offset: 1

R. Muller

For n<=800, only a(2) and a(4) are primes. - Dmitry Kamenetsky, Feb 25 2009

a(n) has about kn(n+1) digits, where k = log phi/log 100 = 0.10449... - Charles R Greathouse IV, Sep 19 2012

S. Smarandoiu, Convergence of Smarandache continued fractions, Abstract 96T-11-195, Abstracts Amer. Math. Soc., 17 (No. 4, 1996), 680.

Reinhard Zumkeller, Table of n, a(n) for n = 1..96
F. Smarandache, Collected Papers, Vol. II
Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Consecutive Number Sequences
Eric Weisstein's World of Mathematics, Smarandache Sequences
Index entries for sequences related to Most Wanted Primes video

Cf. A000045, A038399.

a019523 n = read $ concatMap show $ take n $ tail a000045_list :: Integer
-- Reinhard Zumkeller, Mar 01 2014

[Seqint(Reverse(&cat[Reverse(Intseq(Fibonacci(k))): k in [1..n]])): n in [1..20]]; // Vincenzo Librandi, Dec 18 2016

Table[FromDigits[Flatten[IntegerDigits[Fibonacci[Range[n]]]]], {n,25}] (* G. C. Greubel, Nov 30 2016 *)

a(n) = a(n-1)*10^A055642(A000045(n)) + A000045(n). - José de Jesús Camacho Medina, Dec 16 2016

A030466 Squares that are concatenations of two consecutive nonzero numbers.

Original entry on oeis.org

183184, 328329, 528529, 715716, 60996100, 1322413225, 4049540496, 106755106756, 453288453289, 20661152066116, 29752082975209, 2214532822145329, 2802768328027684, 110213248110213249, 110667555110667556, 147928995147928996, 178838403178838404, 226123528226123529

Offset: 1

Patrick De Geest

British Mathematical Olympiad, 1993, Round 1, Question 1: "Find, showing your method, a six-digit integer n with the following properties: (i) n is a perfect square, (ii) the number formed by the last three digits of n is exactly one greater than the number formed by the first three digits of n. (Thus n might look like 123124, although this is not a square.)"
Steve Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 1 of the British Mathematical Olympiad 1993, page 164.

Iain Fox, Table of n, a(n) for n = 1..10471.
British Mathematical Olympiad 1993, Round 1, Problem 1.
Michael Penn, British Mathematics Olympiad 1993 Round 1 Question 1, YouTube, Apr 24, 2020.
Pante Stanica, Squares as concatenation of consecutive integers, Slides, West Coast Number Theory, Dec 17 2017.
Index to sequences related to Olympiads.

Cf. A000993, A030465, A030467, A054214, A054215, A054216, A020339, A020340.

fQ[n_] := IntegerQ[Sqrt[n*10^Floor[1 + Log10[n + 1]] + n + 1]]; (* Robert G. Wilson v, Dec 27 2017 *)

lista(nn) = forstep(n=183, nn, [3, 5, 7, 5, 3, 1, 4, 7, 5, 3, 5, 7, 5, 3, 5, 7, 5, 3, 5, 7, 4, 1], my(s = eval(concat(Str(n), Str(n+1)))); if(issquare(s), print1(s, ", "))) \\ Iain Fox, Dec 27 2017

eea(x, y) = my(a=max(x,y), b=min(x,y), s=0, so=1, st, r=b, ro=a, rt, q, t); while(r, q=ro\r; rt=r; r=ro-q*r; ro=rt; st=s; s=so-q*s; so=st); t=(ro-so*a)\b; if(x>y, [so, t], [t, so]) \\ Extended Euclidean Algorithm
lista(nn) = my(res=Set(), b, f2, c, s); for(d=3, nn, b=10^d+1; fordiv(b, f, if(f!=1 && f!=b, f2=b/f; if(gcd(f, f2)==1, c=eea(f, f2); if(c[1]<0, s=f*(f2+2*c[1])*f2*(f-2*c[2])+1, s=f*(2*c[1])*f2*(-2*c[2])+1); if(#digits(s)==d*2, res=setunion(res, Set(s))))))); Vec(res) \\ (Will find all values of length nn*2 or shorter) Iain Fox, Oct 16 2021

a(n) = A030465(n)*(10^A055642(A030465(n))+1)+1. - Iain Fox, Oct 16 2021

a(15)-a(17) from Arkadiusz Wesolowski, Apr 02 2014

a(18) from Iain Fox, Dec 27 2017

A050252 Number of digits in the prime factorization of n (counting terms of the form p^1 as p).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 4, 2, 3, 3, 3, 2, 3, 2, 4, 3, 3, 2, 3, 2, 3, 3, 4, 2, 3, 3, 3, 3, 3, 2, 4, 2, 3, 3, 2, 3, 4, 2, 4, 3, 3, 2, 4, 2, 3, 3, 4, 3, 4, 2, 3, 2, 3, 2, 4, 3, 3, 3, 4, 2, 4, 3, 4, 3, 3, 3, 3, 2, 3, 4, 4, 3, 4

Offset: 1

Eric W. Weisstein

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization.

Cf. A046758, A073048, A055642, A020639, A027748, A124010, A110475.

Cf. also A377369 (analog for base 2).

a050252 1 = 1
a050252 n = sum $ map a055642 $
            (a027748_row n) ++ (filter (> 1) $ a124010_row n)
-- Reinhard Zumkeller, Aug 03 2013, Jun 21 2011

nd[n_]:=Total@IntegerLength@Select[Flatten@FactorInteger[n],#>1&];Table[If[n==1,1,nd[n]],{n,102}] (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)

from sympy import factorint
def a(n): return 1 if n == 1 else sum(len(str(p))+(len(str(e)) if e>1 else 0) for p, e in factorint(n).items())
print([a(n) for n in range(1, 103)]) # Michael S. Branicky, Dec 27 2024

a(A192010(n)) = n and a(m) != n for m < A192010(n);

a(A046759(n)) < A055642(A046759(n)); a(A046758(n)) = A055642(A046758(n)); a(A046760(n)) > A055642(A046760(n)). [Reinhard Zumkeller, Jun 21 2011]

A082776 a(1) = 1, a(n) = smallest palindromic multiple of a(n-1) obtained by inserting digits anywhere in a(n-1).

Original entry on oeis.org

1, 11, 121, 12221, 12233221, 122344443221, 15223441414432251, 15223441429655692414432251

Offset: 1

Amarnath Murthy, Apr 19 2003

a(n)<=(10^A055642(a(n-1))+1)*a(n-1). If a(n-1) > 10 and the last digit of a(n-1) <= 4, then a(n)<=(10^(A055642(a(n-1))-1)+1)*a(n-1). - Chai Wah Wu, Mar 06 2021

Cf. A055642, A082777, A082778, A082779, A082780.

Corrected by Ray Chandler, Oct 13 2003

a(8) from Sean A. Irvine, Apr 19 2010

cp's OEIS Frontend

A046760 Wasteful numbers.

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A075167 Number of edges in each rooted plane tree produced with the unranking algorithm presented in A075166, which is based on prime factorization.

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A100910 Table of number of occurrences in n of each decimal digit from 0 to 9.

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A158232 Numbers which yield primes when "13" is prefixed or appended: N natural number is a member of the sequence, if P="13N" (prefixed 13) and A="N13" (appended 13) are prime.

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A004218 a(n) = log_10(n) rounded up.

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A005341 Length of n-th term in Look and Say sequences A005150 and A007651.

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A019523 Concatenation of Fibonacci(1) through Fibonacci(n).

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