A055642 -id:A055642 - cp's OEIS frontend

A164771 Numbers k such that the average digit of k^2 is 1.

1, 1049, 1490, 10002, 10005, 10011, 10020, 10050, 10101, 10110, 10149, 10200, 10500, 11001, 11010, 11100, 11490, 12000, 14499, 15000, 17610, 18000, 20001, 20010, 20100, 21000, 24900, 30000, 33200, 35000, 36100, 44900, 44990, 45100

Offset: 1

Zak Seidov, Aug 26 2009

There are 117 such n's < 10^7: 1, 1049, 1490, 10002, 10005, 10011, 10020, 10050, 10101, 10110, 10149, 10200, 10500, 11001, 11010, 11100, 11490, 12000, 14499, 15000, 17610, 18000, 20001, 20010, 20100, 21000, 24900, 30000, 33200, 35000, 36100, 44900, 44990, 45100, 46000, 54800, 55000, 64900, 71000, 80000, 1000006, 1000015, 1000051, 1000055, 1000060, 1000105, 1000150, 1000501, 1000510, 1000550, 1000600, 1001005, 1001050, 1001500, 1005001, 1005010, 1005100, 1005500, 1006000, 1006490, 1009951, 1010005, 1010050, 1010149, 1010500, 1011490, 1015000, 1024900, 1050001, 1050010, 1050100, 1051000, 1055000, 1060000, 1064900, 1095500, 1096000, 1100005, 1100050, 1100500, 1105000, 1114900, 1145000, 1150000, 1190000, 1224749, 1244990, 1249000, 1414249, 1415000, 1420000, 1424900, 1429000, 1451000, 1460000, 1484251, 1500001, 1500010, 1500100, 1501000, 1510000, 1550000, 1600000, 1735000, 1739000, 1789000, 1820000, 2000005, 2000050, 2000500, 2005000, 2050000, 2239000, 2261000, 2450000, 2500000, 2900000.

Or: Numbers k such that k^2 is in A061384, i.e., square root of squares in A061384. - M. F. Hasler, Dec 05 2010

			1049 is a term because 1049^2 = 1100401 and (1 + 1 + 0 + 0 + 4 + 0 + 1)/7 = 1.

Subsequence of A164817.
Average of digits of n^2 = s: A164771 (s=1), A164770 (s=2), A164782 (s=3), A164776 (s=4), A164774 (s=5), A164778 (s=6), A164773 (s=7), A164772 (s=8).
Cf. A007953, A055642, A061384.

Select[Range[50000],Mean[IntegerDigits[#^2]]==1&] (* Harvey P. Dale, Dec 15 2014 *)

{for(d=1,9, for(n=sqrtint(10^(d-1)-1)+1, sqrtint(10^d-1), my(n2=divrem(n^2,10)); sum( k=2,d, (n2=divrem(n2[1],10))[2],n2[2])/d==1 & print1(n",")))}  \\ M. F. Hasler, Dec 05 2010

A055642(a(n)^2) = A007953(a(n)^2). - M. F. Hasler, Dec 05 2010

Terms up to a(117) checked with given PARI code by M. F. Hasler, Dec 05 2010

A178361 Numbers with rounded up arithmetic mean of digits = 1.

Original entry on oeis.org

1, 10, 11, 20, 100, 101, 102, 110, 111, 120, 200, 201, 210, 300, 1000, 1001, 1002, 1003, 1010, 1011, 1012, 1020, 1021, 1030, 1100, 1101, 1102, 1110, 1111, 1120, 1200, 1201, 1210, 1300, 2000, 2001, 2002, 2010, 2011, 2020, 2100, 2101, 2110, 2200, 3000, 3001

Offset: 1

Reinhard Zumkeller, May 27 2010

A004427(a(n)) = 1;

A000027 = union of A178362, A178363, A178364, A178365, A178366, A178367, A178368, A178369, and this sequence.

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

Cf. A007953, A055642.

a178361 n = a178361_list !! (n-1)
a178361_list = [x | x <- [1..], a007953 x <= a055642 x]
-- Reinhard Zumkeller, May 06 2015

Select[Range[4000],Ceiling[Mean[IntegerDigits[#]]]==1&] (* Harvey P. Dale, Dec 31 2019 *)

A007953(a(n)) <= A055642(a(n)). - Reinhard Zumkeller, May 06 2015

A317716 Square array A(n, k), read by antidiagonals downwards: k-th prime p such that cyclic digit shifts produce exactly n different primes.

Original entry on oeis.org

2, 3, 13, 5, 17, 113, 7, 31, 131, 1193, 11, 37, 197, 1931, 11939, 19, 71, 199, 3119, 19391, 193939, 23, 73, 311, 3779, 19937, 199933, 17773937, 29, 79, 337, 7793, 37199, 319993, 39371777, 119139133, 41, 97, 373, 7937, 39119, 331999, 71777393, 133119139

Offset: 1

Felix Fröhlich, Aug 05 2018

k-th prime p such that A262988(p) = n.
Are all rows of the array infinite?
A term q of A270083 occurs in row A055642(q) - 1 in this array.
A term r of A293663 occurs in row A055642(r) in this array.
Row 1 is a supersequence of A004022.
Column 1 is A247153.

			Array starts
          2,         3,         5,         7,        11,        19,        23, ...
         13,        17,        31,        37,        71,        73,        79, ...
        113,       131,       197,       199,       311,       337,       373, ...
       1193,      1931,      3119,      3779,      7793,      7937,      9311, ...
      11939,     19391,     19937,     37199,     39119,     71993,     91193, ...
     193939,    199933,    319993,    331999,    391939,    393919,    919393, ...
   17773937,  39371777,  71777393,  73937177,  77393717,  77739371,  93717773, ...
  119139133, 133119139, 139133119, 191391331, 311913913, 331191391, 913311913, ...
...

Robert G. Wilson v, Antidiagonals n = 1..13, flattened

Cf. A004022, A055642, A247153, A262988, A270083, A293663.

eva(n) = subst(Pol(n), x, 10)
rot(n) = if(#Str(n)==1, v=vector(1), v=vector(#n-1)); for(i=2, #n, v[i-1]=n[i]); u=vector(#n); for(i=1, #n, u[i]=n[i]); v=concat(v, u[1]); v
count_primes(n) = my(d=digits(n), i=0); while(1, if(ispseudoprime(eva(d)), i++); d=rot(d); if(d==digits(n), return(i)))
row(n, terms) = my(i=0); forprime(p=1, , if(count_primes(p)==n, print1(p, ", "); i++); if(i==terms, print(""); break))
array(rows, cols) = for(x=1, rows, row(x, cols))
array(7, 7) \\ print initial 7 rows and 7 columns of array

A053816 Another version of the Kaprekar numbers (A006886): n such that n = q+r and n^2 = q*10^m+r, for some m >= 1, q >= 0 and 0 <= r < 10^m, with n != 10^a, a >= 1 and n an m-digit number.

Original entry on oeis.org

1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, 7777, 9999, 17344, 22222, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170, 538461, 609687, 643357

Offset: 1

Robert Munafo

Consider an m-digit number n. Square it and add the right m digits to the left m or m-1 digits. If the resultant sum is n, then n is a term of the sequence.
4879 and 5292 are in A006886 but not in this version.
Shape of plot (see links) seems to consist of line segments whose lengths along the x-axis depend on the number of unitary divisors of 10^m-1 which is equal to 2^w if m is a multiple of 3 or 2^(w+1) otherwise, where w is the number of distinct prime factors of the repunit of length m (A095370). w for m = 60 is 20, whereas w <= 15 for m < 60. This leads to the long segment corresponding to m = 60. - Chai Wah Wu, Jun 02 2016
If n*(n-1) is divisible by 10^m-1 then n is a term where m is the number of decimal digits in n. - Giorgos Kalogeropoulos, Mar 27 2025

			703 is Kaprekar because 703 = 494 + 209, 703^2 = 494209.

D. R. Kaprekar, On Kaprekar numbers, J. Rec. Math., 13 (1980-1981), 81-82.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 151.

Chai Wah Wu, Table of n, a(n) for n = 1..50000
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.
Douglas E. Iannucci, The Kaprekar numbers, J. Integer Sequences, Vol. 3, 2000, #1.2.
Robert Munafo, Kaprekar Sequences.
Eric Weisstein's World of Mathematics, Kaprekar Number.
Chai Wah Wu, Semilog plot of A053816(n), n = 1..1003371 (all terms with m <= 60).
Chai Wah Wu, Plot of A053816(n), n = 1..1003371 (all terms with m <= 60).
Index entries for Colombian or self numbers and related sequences

Cf. A006886, A037042, A053394, A053395, A053396, A053397, A045913, A003052, A055642, A095370.

Fixed points of A380585.

a053816 n = a053816_list !! (n-1)
a053816_list = 1 : filter f [4..] where
   f x = length us - length vs <= 1 &&
         read (reverse us) + read (reverse vs) == x
         where (us, vs) = splitAt (length $ show x) (reverse $ show (x^2))
-- Reinhard Zumkeller, Oct 04 2014

kapQ[n_]:=Module[{idn2=IntegerDigits[n^2],len},len=Length[idn2];FromDigits[ Take[idn2,Floor[len/2]]]+FromDigits[Take[idn2, -Ceiling[len/2]]]==n]; Select[Range[540000],kapQ] (* Harvey P. Dale, Aug 22 2011 *)
ktQ[n_] := ((x = n^2) - (z = FromDigits[Take[IntegerDigits[x], y = -IntegerLength[n]]]))*10^y + z == n; Select[Range[540000], ktQ] (* Jayanta Basu, Aug 04 2013 *)
Select[Range[540000],Total[FromDigits/@TakeDrop[IntegerDigits[#^2], Floor[ IntegerLength[ #^2]/2]]] ==#&] (* The program uses the TakeDrop function from Mathematica version 10 *) (* Harvey P. Dale, Jun 03 2016 *)
maxDigits=6; Flatten[Table[lst={};sub=Subsets@FactorInteger[v=10^d-1]; Do[a=Times@@Power@@@s; n=ChineseRemainder[{0,1},{a,v/a},1]; If[10^(d-1)<=n<10^d,AppendTo[lst,n]],{s,sub}];Union@lst,{d,maxDigits}]] (* Giorgos Kalogeropoulos, Mar 27 2025 *)

isok(n) = n == vecsum(divrem(n^2, 10^(1+logint(n, 10)))); \\ Ruud H.G. van Tol, Jun 02 2024

def is_A053816(n): return n==sum(divmod(n**2,10**len(str(n)))) and n
print(upto_1e5:=list(filter(is_A053816, range(10**5)))) # M. F. Hasler, Mar 28 2025

More terms from Michel ten Voorde, Apr 11 2001

A066022 Number of digits in n^n.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 15, 17, 18, 20, 21, 23, 25, 27, 28, 30, 32, 34, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 88, 90, 92, 94, 96, 98, 101, 103, 105, 107, 109, 112, 114, 116, 118, 121, 123, 125

Offset: 1

Robert A. Stump (bee_ess107(AT)yahoo.com), Dec 11 2001

This is almost certainly the same as the number of decimal digits of the sum of the n-th powers of the divisors of n (a sequence submitted by Labos Elemer on Jan 14 2002). Although no formal proof for this is known, Jon E. Schoenfield has verified it for n up to 10^8 and has given a plausible heuristic argument that it is true for all n.

Harry J. Smith, Table of n, a(n) for n=1..1000

Cf. A000312, A054382, A055642.

[ #Intseq(n^n): n in [1..80] ]; // Vincenzo Librandi, Mar 08 2015

[seq(length(n^n), n=1..55)]; # Zerinvary Lajos, Mar 10 2007

Table[IntegerLength[n^n], {n, 80}] (* Vincenzo Librandi, Mar 08 2015 *)

a(n)=#digits(n^n);  \\ Joerg Arndt, Apr 30 2020

a(n) = A055642(A000312(n)). - Michel Marcus, Dec 05 2019

Edited by N. J. A. Sloane Jan 03 2009 at the suggestion of Jon E. Schoenfield.

A066686 Array T(i,j) read by antidiagonals, where T(i,j) is the concatenation of i and j (1<=i, 1<=j).

Original entry on oeis.org

11, 12, 21, 13, 22, 31, 14, 23, 32, 41, 15, 24, 33, 42, 51, 16, 25, 34, 43, 52, 61, 17, 26, 35, 44, 53, 62, 71, 18, 27, 36, 45, 54, 63, 72, 81, 19, 28, 37, 46, 55, 64, 73, 82, 91, 110, 29, 38, 47, 56, 65, 74, 83, 92, 101, 111, 210, 39, 48, 57, 66, 75, 84, 93, 102, 111

Offset: 1

Robert G. Wilson v, Jan 11 2002

The element at T(i,j) is the {(i+j-1)(i+j-2)/2 + i}-th element read in the sequence.

			The array begins
11 12 13 14 15 16 17 18 19 110 ...
21 22 23 24 25 26 27 28 29 210 ...
31 32 33 34 35 36 37 38 39 310 ...
41 42 43 44 45 46 47 48 49 410 ...

Alexander Bogomolny, What is a number?
Boris Putievskiy, Transformations [Of] Integer Sequences And Pairing Functions, arXiv preprint arXiv:1212.2732 [math.CO], 2012.

Cf. A055642, A067574, A084854.

a = {}; Do[ a = Append[a, ToExpression[ StringJoin[ ToString[k], ToString[n - k]]]], {n, 2, 13}, {k, 1, n - 1} ]; a

def T(i, j): return int(str(i) + str(j))
def auptodiag(maxd):
    return [T(i, d+1-i) for d in range(1, maxd+1) for i in range(1, d+1)]
print(auptodiag(12)) # Michael S. Branicky, Nov 21 2021

T(i, j) = i*10^A055642(i) + j. - Michael S. Branicky, Nov 21 2021

A095407 Total number of decimal digits of all distinct prime factors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 1, 3, 1, 2, 2, 3, 2, 1, 3, 3, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 1, 2, 3, 3, 2, 2, 3, 2, 3, 3, 2, 3, 2, 3, 2, 1, 3, 4, 2, 3, 3, 3, 2, 2, 2, 3, 2, 3, 3, 4, 2, 2, 1, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 2, 2, 2, 3, 2, 3, 4, 3, 3, 3

Offset: 1

Labos Elemer, Jun 21 2004

a(n) <= A055642(n) + A001221(n) - 1 since the product of a k digit number and an m digit number has at least k+m-1 digits. - Chai Wah Wu, Nov 03 2019

			n=22: prime set={2,11}, a[22]=1+2=3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A055642, A001221.

Prepend[(Total [ Length[IntegerDigits[#]] & /@ ( #[[1]] & /@ FactorInteger[#])] & /@ Range[2, 105]), 0] (* Zak Seidov, Aug 26 2016 *)

a(n)=vecsum(apply(p->#Str(p), factor(n)[,1])) \\ Charles R Greathouse IV, Aug 26 2016

A097944 Number of digits in n-th prime.

Original entry on oeis.org

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, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Cino Hilliard, Sep 05 2004

For primes p <= n sum(a(n)) -> n/2 and n-> inf.

			The first 4 primes are 2,3,5,7. These are 1-digit numbers so the first 4 entries in the table are 1's.

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

Cf. A060417, A068670 (partial sums).

a097944 = a055642 . a000040 -- Reinhard Zumkeller, Apr 08 2012

a[n_]:=StringLength[ToString[Prime[n]]]; (* Vladimir Joseph Stephan Orlovsky, Dec 03 2008 *)
IntegerLength[Prime[Range[110]]] (* Harvey P. Dale, Oct 04 2012 *)

PARI
```
a(n)=#Str(prime(n))
```

A097944(n)=logint(prime(n),10)+1 \\ M. F. Hasler, Oct 24 2019

a(n) = (log n + log log n)/(log 10) + O(1).
a(n) = A055642(A000040(n)). - Reinhard Zumkeller, Apr 08 2012
a(n) = A068670(n) - A068670(n-1). - M. F. Hasler, Oct 24 2019

A175420 Sequence of numbers after 1st step of iteration defined in A175419.

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, 1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 0, 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 0, 1, 32, 243, 1024, 3125, 7776, 16807, 32768, 59049, 0, 1, 64, 729, 4096, 15625, 46656

Offset: 0

Jaroslav Krizek, May 09 2010

			For n = 33: a(33) = 27 because for the number 33 there are 4 steps of defined iteration: {3^3 = 27}, {7^2 = 49}, {9^4 = 6561}, {((1^6)^5)^6 = 1} and the 1st step of the iteration ends with 27.

Stefano Spezia, Table of n, a(n) for n = 0..10000

Cf. A175419, A175421, A175422, A175423, A175424, A175425, A175426, A175427.

Cf. A055642.

A175420 := proc(n) local dgs,a,i ; if n = 0 then return 0 ; end if; dgs := convert(n,base,10) ; a := op(1,dgs) ; for i from 2 to nops(dgs) do a := a^ op(i,dgs) ; end do: a ; end proc: seq(A175420(n),n=0..120) ; # R. J. Mathar, May 12 2010

Unprotect[Power]; Power[0, 0] = 1; Protect[Power]; a[0]=0; a[n_]:=If[(len=IntegerLength[n])==1, n, Last[list=IntegerDigits[n]]^Product[Part[Drop[list, -1], i], {i, len-1}]]; Array[a, 67, 0] (* Stefano Spezia, Feb 25 2024 *)

a(n) = if (n, my(d=digits(n), r=d[#d]); forstep (k=#d-1, 1, -1, r = r^d[k];); r); \\ Michel Marcus, Jan 20 2022

a(n) = (((D_k^D_(k-1))^D_(k-2))^...)^D_1, where D_k = k-th digit D of number n and k = the number of digits of number n in decimal expansion of n (A055642).

More terms from R. J. Mathar, May 12 2010

A349194 a(n) is the product of the sum of the first i digits of n, as i goes from 1 to the total number of digits of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 49, 56, 63, 70, 77

Offset: 1

Malo David, Nov 10 2021

The only primes in the sequence are 2, 3, 5 and 7. - Bernard Schott, Nov 23 2021

			For n=256, a(256) = 2*(2+5)*(2+5+6) = 182.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Malo David, About the Product Sum Digit Sequence

Cf. A055642, A284001 (binary analog), A349190 (fixed points).
Cf. A007953 (sum of digits), A059995 (floor(n/10)).
Cf. A349278 (similar, with the last digits).
Cf. A349898, A349899.

f:=func; [f(n):n in [1..100]]; // Marius A. Burtea, Nov 23 2021

Table[Product[Sum[Part[IntegerDigits[n],j],{j,i}],{i,Length[IntegerDigits[n]]}],{n,74}] (* Stefano Spezia, Nov 10 2021 *)

a(n) = my(d=digits(n)); prod(i=1, #d, sum(j=1, i, d[j])); \\ Michel Marcus, Nov 10 2021

first(n)=if(n<9,return([1..n])); my(v=vector(n)); for(i=1,9,v[i]=i); for(i=10,n, v[i]=sumdigits(i)*v[i\10]); v \\ Charles R Greathouse IV, Dec 04 2021

from math import prod
from itertools import accumulate
def a(n): return prod(accumulate(map(int, str(n))))
print([a(n) for n in range(1, 100)]) # Michael S. Branicky, Nov 10 2021

For n>10: a(n) = a(A059995(n))*A007953(n) where A059995(n) = floor(n/10).
In particular, for n<100: a(n) = floor(n/10)*A007953(n)
From Bernard Schott, Nov 23 2021: (Start)
a(n) = 1 iff n = 10^k, k >= 0 (A011557).
a(n) = 2 iff n = 10^k + 1, k >= 0 (A000533 \ {1}).
a(n) = 3 iff n = 10^k + 2, k >= 0 (A133384).
a(n) = 5 iff n = 10^k + 4, k >= 0.
a(n) = 7 iff n = 10^k + 6, k >= 0. (End)
From Marius A. Burtea, Nov 23 2021: (Start)
a(A002275(n)) = n! = A000142(n), n >= 1.
a(A090843(n - 1)) = (2*n - 1)!! = A001147(n), n >= 1.
a(A097166(n)) = (3*n - 2)!!! = A007559(n).
a(A093136(n)) = 2^n = A000079(n).
a(A093138(n)) = 3^n = A000244(n). (End)

cp's OEIS Frontend

A164771 Numbers k such that the average digit of k^2 is 1.

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A178361 Numbers with rounded up arithmetic mean of digits = 1.

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A317716 Square array A(n, k), read by antidiagonals downwards: k-th prime p such that cyclic digit shifts produce exactly n different primes.

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A053816 Another version of the Kaprekar numbers (A006886): n such that n = q+r and n^2 = q*10^m+r, for some m >= 1, q >= 0 and 0 <= r < 10^m, with n != 10^a, a >= 1 and n an m-digit number.

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A066022 Number of digits in n^n.

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A066686 Array T(i,j) read by antidiagonals, where T(i,j) is the concatenation of i and j (1<=i, 1<=j).

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A095407 Total number of decimal digits of all distinct prime factors of n.

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