A138793 -id:A138793 - cp's OEIS frontend

A138790 Numbers k such that A138793(k) is prime.

61, 946

Offset: 1

Artur Jasinski, Mar 30 2008, Mar 31 2008

There are no more primes for k <= 5000.

a(3) > 20000. - Robert Price, Mar 24 2015

			a(1) = 61 because the number 160695...654321 is prime.

Cf. A000422, A116504, A007908, A116505, A138793.

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, 1, Length[w]}]; p = FromDigits[Reverse[a]]; If[PrimeQ[p], Print[n]; AppendTo[b, p]], {n, 1, 2000}]; b (* Artur Jasinski, Mar 30 2008 *)
Select[Range[1, 1000], PrimeQ[lst = {}; Do[lst = Join[lst, IntegerDigits[n]], {n, 1, #}]; FromDigits[Reverse[lst]]] &] (* Robert Price, Mar 24 2015 *)

A138962 a(1) = 1, a(n) = the smallest prime divisor of A138793(n).

Original entry on oeis.org

1, 3, 3, 29, 3, 3, 19, 3, 3, 457, 3, 3, 16087, 3, 3, 35963, 3, 3, 167, 3, 3, 7, 3, 3, 13, 3, 3, 953, 3, 3, 7, 3, 3, 548636579, 3, 3, 19, 3, 3, 71, 3, 3, 13, 3, 3, 89, 3, 3, 114689, 3, 3, 17, 3, 3, 12037, 3, 3, 7, 3, 3

Offset: 1

Artur Jasinski, Apr 04 2008

nonn

a(61) > 10^11. - Robert Price, Mar 22 2015

Daniel Suteu, Table of n, a(n) for n = 1..333

Cf. A020639, A138793, A104759, A000422, A116504, A007908, A116505, A104759, A075019, A075020, A075021, A075022, A138789, A138790, A138960, A138962.

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, 1, Length[w]}]; p = FromDigits[Reverse[a]]; AppendTo[b, First[First[FactorInteger[p]]]], {n, 1, 31}]; b (* Artur Jasinski, Apr 04 2008 *)
lst = {}; Table[First[First[FactorInteger[FromDigits[Reverse[lst = Join[lst,IntegerDigits[n]]]]]]], {n, 1, 60}] (* Robert Price, Mar 22 2015 *)

f(n) = my(D = Vec(concat(apply(s->Str(s), [1..n])))); eval(concat(vector(#D, k, D[#D-k+1]))); \\ A138793
a(n) = my(k=f(n)); forprime(p=2, 10^6, if(k%p == 0, return(p))); if(n == 1, 1, vecmin(factor(k)[,1])); \\ Daniel Suteu, May 27 2022

a(n) = A020639(A138793(n)). - Daniel Suteu, May 27 2022

a(32)-a(60) from Robert Price, Mar 22 2015

A138794 a(n) = A138793(n+1)-A138793(n).

Original entry on oeis.org

20, 300, 4000, 50000, 600000, 7000000, 80000000, 900000000, 1000000000, 1100000000000, 210000000000000, 31000000000000000, 4100000000000000000, 510000000000000000000, 61000000000000000000000

Offset: 1

Artur Jasinski, Mar 30 2008

First differences of A138793

Cf. A000422, A116504, A007908, A116505, A104759, A138789, A138790, A138793, A075019, A075020, A075021, A075022.

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, 1, Length[w]}]; p = FromDigits[Reverse[a]]; AppendTo[b, p], {n, 1, 31}]; c = {}; Do[AppendTo[c, b[[n + 1]] - b[[n]]], {n, 1, Length[b] - 1}]; c (*Artur Jasinski*)

a(n) = A138793(n+1)-A138793(n)

A138963 a(1) = 1, a(n) = the largest prime divisor of A138793(n).

Original entry on oeis.org

1, 7, 107, 149, 953, 218107, 402859, 4877, 379721, 4349353, 169373, 182473, 1940144339383, 2184641, 437064932281, 5136696159619, 67580875919190833, 1156764458711, 464994193118899, 4617931439293, 1277512103328491957510030561, 8177269604099

Offset: 1

Artur Jasinski, Apr 04 2008

For the smallest prime divisors of A138793 see A138962.

Daniel Suteu and Robert Price, Table of n, a(n) for n = 1..63 (terms a(1)..a(45) from Robert Price)

Cf. A006530, A138793, A138962.

Cf. A104759, A000422, A116504, A007908, A116505, A104759, A075019, A075020, A075021, A075022, A138789, A138790, A138958, A138959, A138960.

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, 1, Length[w]}]; p = FromDigits[Reverse[a]]; AppendTo[b, First[Last[FactorInteger[p]]]], {n, 1, 31}]; b (* Artur Jasinski, Apr 04 2008 *)
lst = {}; Table[First[Last[FactorInteger[FromDigits[Reverse[lst = Join[lst,IntegerDigits[n]]]]]]], {n, 1, 10}] (* Robert Price, Mar 22 2015 *)

f(n) = my(D = Vec(concat(apply(s->Str(s), [1..n])))); eval(concat(vector(#D, k, D[#D-k+1]))); \\ A138793
a(n) = if(n == 1, 1, vecmax(factor(f(n))[,1])); \\ Daniel Suteu, May 26 2022

a(n) = A006530(A138793(n)). - Daniel Suteu, May 26 2022

A138795 a(n) = (A138793(n+1)-A138793(n))/10^n.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 1, 110, 2100, 31000, 410000, 5100000, 61000000, 710000000, 8100000000, 91000000000, 20000000000, 1200000000000, 22000000000000, 320000000000000, 4200000000000000, 52000000000000000, 620000000000000000

Offset: 1

Artur Jasinski, Mar 30 2008

First differences of A138793 divided by 10^n

Cf. A000422, A116504, A007908, A116505, A104759, A138789, A138790, A138793, A075019, A075020, A075021, A075022, A138794.

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[AppendTo[a, w[[k]]], {k, 1, Length[w]}]; p = FromDigits[Reverse[a]]; AppendTo[b, p], {n, 1, 61}]; c = {}; Do[AppendTo[c, (b[[n + 1]] - b[[n]])/(10^n)], {n, 1, Length[b] - 1}]; c (*Artur Jasinski*)

a(n) = A138793(n+1)-A138793(n)

A000422 Concatenation of numbers from n down to 1.

Original entry on oeis.org

1, 21, 321, 4321, 54321, 654321, 7654321, 87654321, 987654321, 10987654321, 1110987654321, 121110987654321, 13121110987654321, 1413121110987654321, 151413121110987654321, 16151413121110987654321, 1716151413121110987654321, 181716151413121110987654321

Offset: 1

R. Muller

The first prime term in this sequence is a(82) (see A176024). - Artur Jasinski, Mar 30 2008

For n < 10^4, a(n)/A000217(n) is an integer for n = 1, 2, and 18. The integers are 1, 7 (prime), and 1062667552123515268933651, respectively. - Derek Orr, Sep 04 2014

F. Smarandache, "Properties of the Numbers", University of Craiova Archives, 1975; Arizona State University Special Collections, Tempe, AZ

T. D. Noe, Table of n, a(n) for n = 1..150
R. W. Stephan, Factors and primes in two Smarandache sequences
Bertrand Teguia Tabuguia, Explicit formulas for concatenations of arithmetic progressions, arXiv:2201.07127 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Consecutive Number Sequences
Index entries for sequences related to Most Wanted Primes video

Cf. A007908, A058183, A104759, A116504, A116505, A138789, A138790, A138793.

See A176024 for primes.

a[1]:= 1:
for n from 2 to 100 do
a[n]:= n*10^(1+ilog10(a[n-1])) + a[n-1]
od:
seq(a[n],n=1..100); # Robert Israel, Sep 05 2014
# second Maple program:
a:= proc(n) a(n):= `if`(n=1, 1, parse(cat(n, a(n-1)))) end:
seq(a(n), n=1..22);  # Alois P. Heinz, Jan 12 2021

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[PrependTo[a, w[[Length[w] - k + 1]]], {k, 1, Length[w]}]; p = FromDigits[a]; AppendTo[b, p], {n, 1, 30}]; b (* Artur Jasinski, Mar 30 2008 *)
Table[FromDigits[Flatten[IntegerDigits/@Range[n,1,-1]]],{n,20}] (* Harvey P. Dale, Jul 06 2019 *)

a(n)=my(t=n);forstep(k=n-1,1,-1,t=t*10^#Str(k)+k);t \\ Charles R Greathouse IV, Jul 15 2011

A000422(n,p=1,L=1)=sum(k=1,n,k*p*=L+(k==L&&!L*=10)) \\ M. F. Hasler, Nov 02 2016

def a(n): return int("".join(map(str, range(n, 0, -1))))
print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Dec 08 2021

a(n+1) = (n+1)*10^len(a(n)) + a(n), where len(k) = number of digits in k.
a(n) = Sum_{k=1..n} k*10^(A058183(k) - (1+floor(log10(k)))). - Alexander Goebel, Mar 07 2020
From Serge Batalov, Dec 08 2021: (Start)
a(n) = ((n*9-1)*10^n+1)/9^2 for n < 10,
a(n) = ((n*99-1)*10^(2*n-19)-89)/99^2*10^10 + (8*10^10+1)/9^2 for 10 <= n < 100,
a(n) = ((n*999-1)*10^(3*n-299)-989)/999^2*10^191 + c2 for 10^2 <= n < 10^3,
a(n) = ((n*9999-1)*10^(4*n-3999)-9989)/9999^2*10^2892 + c3 for 10^3 <= n < 10^4,
a(n) = ((n*99999-1)*10^(5*n-49999)-99989)/99999^2*10^38893 + c4 for 10^4 <= n < 10^5,
a(n) = ((n*999999-1)*10^(6*n-599999)-999989)/999999^2*10^488894 + c5 for 10^5 <= n < 10^6,
where
c2 = (98*10^191 + 879*10^10 + 121)/99^2 = a(99),
c3 = (998*10^2701 - 989)/999^2*10^191 + c2 = a(999),
c4 = (9998*10^36001 - 9989)/9999^2*10^2892 + c3 = a(9999),
c5 = (99998*10^450001 - 99989)/99999^2*10^38893 + c4 = a(99999).
(End)

Edited by N. J. A. Sloane, Dec 03 2021

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

A104759 Concatenation of digits of natural numbers from n down to 1.

Original entry on oeis.org

1, 21, 321, 4321, 54321, 654321, 7654321, 87654321, 987654321, 1987654321, 1987654321, 101987654321, 1101987654321, 11101987654321, 211101987654321, 1211101987654321, 31211101987654321, 131211101987654321, 4131211101987654321, 14131211101987654321, 514131211101987654321

Offset: 1

Alexandre Wajnberg & Juliette Bruyndonckx, Apr 23 2005

			a(11) = a(10) because no number may begin with 0.
a(9)= [123456789]101112131415...=987654321
a(10)=[1234567891]01112131415...=1987654321
a(11)=[12345678910]1112131415...=01987654321=1987654321
a(12)=[123456789101]112131415...=101987654321
a(13)=[1234567891011]12131415...=1101987654321
a(14)=[12345678910111]2131415...=11101987654321
a(15)=[123456789101112]131415...=211101987654321

Michael De Vlieger, Table of n, a(n) for n = 1..1000 (last term has 1000 decimal digits)
Eric Weisstein's World of Mathematics, Consecutive numbers sequences.

Cf. A014925, A000422, A057138, A060554, A138793.

f[n_] := Block[{t = Reverse@ Flatten@ IntegerDigits@ Range@ n, k}, Reap@ For[k = 1, k <= Length@ t, k++, Sow[FromDigits@ Take[t, -k]]] // Flatten // Rest]; f@ 14 (* Michael De Vlieger, Mar 23 2015 *)
lst = {}; Do[lst = Join[lst, IntegerDigits[n]], {n, 1, 100}]; Table[FromDigits[Reverse[lst[[Range[1, n]]]]], {n, 1, Length[lst]}] (* Robert Price, Mar 24 2015 *)

a(n) = A138793(n) mod 10^(n-1). - R. J. Mathar, Sep 17 2011

A116504 Number of distinct prime divisors of the concatenation of n,...,1.

Original entry on oeis.org

0, 2, 2, 2, 3, 2, 2, 3, 3, 3, 2, 3, 3, 4, 5, 4, 6, 8, 4, 5, 4, 5, 4, 5, 6, 7, 5, 5, 7, 8, 3, 6, 5, 7, 8, 6, 4, 3, 6, 5, 8, 6, 3, 7, 6, 5, 7, 7, 3, 6, 3, 7, 9, 9, 3, 4, 4, 6, 3, 3, 5, 8, 5, 6, 7, 7, 4, 8, 8, 4, 8, 4, 7, 8, 10, 3, 7, 6, 4, 7, 7, 1, 3, 8, 3, 8, 5, 4, 5, 7, 11, 9, 6

Offset: 1

Parthasarathy Nambi, Mar 20 2006

			87654321 = 3*3*1997*4877, distinct prime divisors are 3, 1997 and 4877, hence a(8) = 3.

Sean A. Irvine, Table of n, a(n) for n = 1..106

Cf. A116505.

Cf. A000422, A116504, A007908, A116505, A104759, A138789, A138790, A138793.

b = {}; a = {}; Do[w = RealDigits[n]; w = First[w]; Do[PrependTo[a, w[[Length[w] - k + 1]]], {k, 1, Length[w]}]; p = FromDigits[a]; m = FactorInteger[p]; AppendTo[b, Length[m]], {n, 1, 30}]; b (* Artur Jasinski, Mar 30 2008 *)
Table[PrimeNu[FromDigits[Flatten[IntegerDigits/@Range[n,1,-1]]]],{n,95}] (* Harvey P. Dale, Oct 03 2015 *)

{a="";for(n=1,58,a=concat(n,a);print1(omega(eval(a)),","))}

Edited and extended by Klaus Brockhaus, Mar 29 2006
Terms a(59)-a(93) from Sean A. Irvine, Nov 04 2009
a(90) corrected by Sean A. Irvine, Nov 02 2024

A116505 Number of distinct prime divisors of the concatenation of 1..n.

Original entry on oeis.org

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

Offset: 1

Parthasarathy Nambi, Mar 20 2006

Dario Alpern's factorization program was used for n > 43.

			123456 = 2*2*2*2*2*2*3*643, with distinct prime divisors 2, 3 and 643. Hence, a(6) = 3.

Dario Alpern, Factorization using the Elliptic Curve Method

Cf. A000422, A116504, A007908, A116505, A104759, A138789, A138790, A138793.

Table[PrimeNu[FromDigits[Flatten[IntegerDigits[Range[n]]]]], {n, 30}] (* Jan Mangaldan, Jul 07 2020 *)

{a="";for(n=1,43,a=concat(a,n);print1(omega(eval(a)),", "))}

Edited and extended by Klaus Brockhaus, Mar 29 2006

Terms 59-100 from Sean A. Irvine, Nov 04 2009

cp's OEIS Frontend

A138790 Numbers k such that A138793(k) is prime.

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A138962 a(1) = 1, a(n) = the smallest prime divisor of A138793(n).

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A138794 a(n) = A138793(n+1)-A138793(n).

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A138963 a(1) = 1, a(n) = the largest prime divisor of A138793(n).

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A138795 a(n) = (A138793(n+1)-A138793(n))/10^n.

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A000422 Concatenation of numbers from n down to 1.

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

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