A055642 -id:A055642 - cp's OEIS frontend

A270083 Near-miss circular primes: Primes p where all but one of the numbers obtained by cyclically permuting the digits of p are prime.

19, 23, 29, 41, 43, 47, 53, 59, 61, 67, 83, 89, 101, 103, 107, 127, 149, 157, 163, 173, 181, 191, 271, 277, 307, 313, 317, 331, 359, 367, 379, 397, 419, 479, 491, 571, 577, 593, 617, 631, 673, 701, 709, 727, 739, 757, 761, 787, 797, 811, 839, 877, 907, 911

Offset: 1

Felix Fröhlich, Mar 10 2016

Prime p is a term of the sequence iff A262988(p) = A055642(p) - 1.
If a(512) exists, it is larger than 10^16. - Giovanni Resta, Apr 27 2017
If one of the digits is even or 5, the miss occurs when that digit is permuted to the ones place. Avoiding that simple obstruction, this sequence intersected with A091633 is 19, 173, 191, 313, 317, 331, 379, 397, 739, 797, 911, 937, 977, 1319, 1777, 1913, 1979, 1993, 3191, 3373, 3719, 3733, 3793, ... . Is this an infinite subsequence? - Danny Rorabaugh, May 15 2017

Felix Fröhlich and Giovanni Resta, Table of n, a(n) for n = 1..511 (first 487 terms from Felix Fröhlich)

Cf. A045978, A068652, A262988.

NearCircPrmsUpTo10powerK[k_]:= Union @ Flatten[ Table[ParallelMap[If[(Count[FromDigits /@ NestList[RotateLeft, IntegerDigits[#], IntegerLength[#]-1], ?PrimeQ] == IntegerLength[#]-1), #, Nothing] &, Select[FromDigits /@ Tuples[Range[0, 9], n], PrimeQ]], {n, k}], 1]; NearCircPrmsUpTo10powerK[7] (* _Mikk Heidemaa, Apr 26 2017 *)

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
eva(n) = subst(Pol(n), x, 10)
is(n) = my(r=rot(digits(n)), i=0); while(r!=digits(n), if(ispseudoprime(eva(r)), i++); r=rot(r)); if(ispseudoprime(eva(r)), i++); if(n==1 || n==11, return(0)); if(i==#Str(n)-1, 1, 0)
forprime(p=1, 1e3, if(is(p), print1(p, ", ")))

A293663 Circular primes that are not repunits.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, 199, 311, 337, 373, 719, 733, 919, 971, 991, 1193, 1931, 3119, 3779, 7793, 7937, 9311, 9377, 11939, 19391, 19937, 37199, 39119, 71993, 91193, 93719, 93911, 99371, 193939, 199933, 319993, 331999, 391939

Offset: 1

Felix Fröhlich, Dec 30 2017

Relative complement of A004022 in A068652.
Conjecture: The sequence is finite.
From Michael De Vlieger, Dec 30 2017: (Start)
Primes > 5 in this sequence must only have digits that are in the reduced residue system modulo 10, i.e., {1, 3, 7, 9}.
There are 54 terms that have 6 or fewer decimal digits, the largest of which is 999331.
a(55) must be larger than 10^11. (End) [Corrected by Felix Fröhlich, Mar 15 + 24 2019]
From Felix Fröhlich, Mar 16 2019: (Start)
a(55) > 10^23 if it exists (cf. De Geest link).
Numbers k such that A262988(k) = A055642(k). (End)

			The numbers resulting from cyclic permutations of the digits of 1193 are 1931, 9311 and 3119, respectively and all those numbers are prime, so 1193, 1931, 3119 and 9311 are terms of the sequence.

Felix Fröhlich, Table of n, a(n) for n = 1..54
P. De Geest, Circular Primes

Cf. A004022, A055642, A068652, A262988, A293142.

Cf. base-b nonrepunit circular primes: A293657 (b=4), A293658 (b=5), A293659 (b=6), A293660 (b=7), A293661 (b=8), A293662 (b=9).

Select[Prime@ Range[10^5], Function[w, And[AllTrue[Array[FromDigits@ RotateRight[w, #] &, Length@ w - 1], PrimeQ], Union@ w != {1} ]]@ IntegerDigits@ # &] (* or *)
Select[Flatten@ Array[FromDigits /@ Most@ Rest@ Tuples[{1, 3, 7, 9}, #] &, 9, 2], Function[w, And[AllTrue[Array[FromDigits@ RotateRight[w, #] &, Length@ w], PrimeQ], Union@ w != {1} ]]@ IntegerDigits@ # &] (* Michael De Vlieger, Dec 30 2017 *)

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
is_circularprime(p) = my(d=digits(p), r=rot(d)); if(vecmin(d)==0, return(0), while(1, if(!ispseudoprime(eva(r)), return(0)); r=rot(r); if(r==d, return(1))))
forprime(p=1, , if(vecmax(digits(p)) > 1, if(is_circularprime(p), print1(p, ", "))))

/* The following is a much faster program that only tests numbers whose decimal expansion consists of digits from the set {1, 3, 7, 9}. */
eva(n) = subst(Pol(n), x, 10)
next_v(vec) = my(k=#vec); if(vecmin(vec)==9, vec=concat(vector(#vec, t, 1), [3]); return(vec)); while(k > 0, if(vec[k]==9, vec[k]=1, if(vec[k]==3, vec[k]=7; return(vec), vec[k]=vec[k]+2, return(vec))); k--)
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
search(n) = my(d=digits(n), e=[], ed=0); while(1, e=rot(d); while(1, if(!ispseudoprime(eva(e)), break, e=rot(e); if(e==d && ispseudoprime(eva(e)), print1(eva(d), ", "); break))); d=next_v(d))
searchfrom(n) = if(n < 12, forprime(p=n, 10, print1(p, ", ")); search(13), my(d=digits(n)); for(k=1, #d, if(d[k]%2==0, d[k]++, if(d[k]==5, d[k]=7))); search(eva(d)))
/* Start a search from 1 upwards as follows: */
searchfrom(1) \\ Felix Fröhlich, Mar 23 2019

A001637 Numbers with an even number of digits.

Original entry on oeis.org

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 1000, 1001

Offset: 1

N. J. A. Sloane

The lower and upper asymptotic densities of this sequence are 1/11 and 10/11, respectively. - Amiram Eldar, Feb 01 2021

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001633 (complement), A055642.

a001637 n = a001637_list !! (n-1)
a001637_list = filter (even . a055642) [0..]
-- Reinhard Zumkeller, Jul 14 2014

Select[Range[0, 1001], EvenQ[Length[IntegerDigits[#]]] &] (* T. D. Noe, Aug 09 2012 *)

is(n)=#Str(n)%2==0 \\ Charles R Greathouse IV, Nov 26 2018

a(n) = n + 100^logint(110*n,100) \ 11; \\ Kevin Ryde, Nov 10 2022

def a(n): return n + (100**((len(str(110*n))-1)//2) - 1)//11
print([a(n) for n in range(1, 100)]) # Michael S. Branicky, Nov 10 2022 after Kevin Ryde

A055642(a(n)) mod 2 = 0. - Reinhard Zumkeller, Jul 14 2014

a(n) = n + (100^floor(log_100(110*n)) - 1)/11. - Kevin Ryde, Nov 10 2022

A046758 Equidigital numbers.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 105, 106, 107, 109, 111, 112, 113, 115, 118, 119, 121, 122, 123, 127, 129, 131, 133, 134, 135, 137, 139

Offset: 1

N. J. A. Sloane

Write n as product of primes raised to powers, let D(n) = A050252 = total number of digits in product representation (number of digits in all the primes plus number of digits in all the exponents that are greater than 1) and l(n) = number of digits in n; sequence gives n such that D(n)=l(n).

The term "equidigital number" was coined by Recamán (1995). - Amiram Eldar, Mar 10 2024

			For n = 125 = 5^3, l(n) = 3 but D(n) = 2. So 125 is not a member of this sequence.

Bernardo Recamán Santos, Equidigital representation: problem 2204, J. Rec. Math., Vol. 27, No. 1 (1995), pp. 58-59.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. P. Delahaye, "Primes Hunters", Economical and Prodigal Numbers (Text in French). [Wayback Machine link]
R. G. E. Pinch, Economical numbers, arXiv:math/9802046 [math.NT], 1998.
Eric Weisstein's World of Mathematics, Equidigital Number..
Wikipedia, Equidigital number.

Cf. A046759, A046760, A050252, A055642, A073048.

a046758 n = a046758_list !! (n-1)
a046758_list = filter (\n -> a050252 n == a055642 n) [1..]
-- Reinhard Zumkeller, Jun 21 2011

edQ[n_] := Total[IntegerLength[DeleteCases[Flatten[FactorInteger[n]], 1]]] == IntegerLength[n]; Join[{1}, Select[Range[140], edQ]] (* Jayanta Basu, Jun 28 2013 *)

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 A046758_gen(): # generator of terms
    return (n for n in count(1) if n == 1 or len(str(n)) == sum(len(str(p))+(len(str(e)) if e > 1 else 0) for p, e in factorint(n).items()))
A046758_list = list(islice(A046758_gen(),20)) # Chai Wah Wu, Feb 18 2022

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

More terms from Eric W. Weisstein

A091762 Last n digits of concatenation of first n primes.

Original entry on oeis.org

2, 23, 235, 2357, 35711, 571113, 7111317, 11131719, 113171923, 1317192329, 31719232931, 171923293137, 7192329313741, 19232931374143, 923293137414347, 2329313741434753, 32931374143475359, 293137414347535961, 9313741434753596167, 31374143475359616771, 137414347535961677173, 3741434753596167717379

Offset: 1

Reinhard Zumkeller, Feb 04 2004

a(n) = A019518(n) mod 10^n; a(n) mod 10^A055642(A000040(n)) = A000040(n); for the primes in this sequence see A091763.

			The first 5 primes are 2, 3, 5, 7, 11. The last 5 digits concatenated are 35711 so a(5) = 35711. - _David A. Corneth_, Sep 15 2019

David A. Corneth, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Consecutive Number Sequences

Table[FromDigits[Take[Flatten[IntegerDigits/@Prime[Range[n]]],-n]],{n,20}] (* Harvey P. Dale, Sep 15 2019 *)

a(n) = {my(p = prime(n), v = digits(p)); while(#v < n, p = precprime(p - 1); v = concat(digits(p), v)); fromdigits(vector(n, i, v[#v - n + i]))} \\ David A. Corneth, Sep 15 2019

More terms from David A. Corneth, Sep 15 2019

A171797 A modified Sisyphus function: a(n) = concatenation of (number of digits in n) (number of even digits) (number of odd digits).

Original entry on oeis.org

110, 101, 110, 101, 110, 101, 110, 101, 110, 101, 211, 202, 211, 202, 211, 202, 211, 202, 211, 202, 220, 211, 220, 211, 220, 211, 220, 211, 220, 211, 211, 202, 211, 202, 211, 202, 211, 202, 211, 202, 220, 211, 220, 211, 220, 211, 220, 211, 220, 211, 211, 202

Offset: 0

N. J. A. Sloane, Oct 15 2010

Start with n, repeatedly apply the map i -> a(i). Then every number converges to 312. - Eric Angelini and Alexandre Wajnberg, Oct 15 2010

			11 has 2 digits, both odd, so a(11) = 202.
12 has 2 digits, one even and one odd, so a(12)=211. Then a(211) = 312.

M. E. Coppenbarger, Iterations of a modified Sisyphus function, Fib. Q., 56 (No. 2, 2018), 130-141.

Reinhard Zumkeller, Table of n, a(n) for n = 0..25000

Cf. A073053 (Sisyphus), A171798, A171813, A055642, A196563, A196564, A308002, A308003 (another version).

A100961 gives steps to reach 312.

a171797 n = read $ concatMap (show . ($ n))
                   [a055642, a196563, a196564] :: Integer
-- Reinhard Zumkeller, Feb 22 2012, Oct 15 2010

nevenDgs := proc(n) local a, d; a := 0 ; for d in convert(n,base,10) do if type(d,'even') then a :=a +1 ; end if; end do; a ; end proc:
cat2 := proc(a,b) local ndigsb; ndigsb := max(ilog10(b)+1,1) ; a*10^ndigsb+b ; end:
catL := proc(L) local a, i; a := op(1,L) ; for i from 2 to nops(L) do a := cat2(a,op(i,L)) ; end do; a; end proc:
A055642 := proc(n) max(1,ilog10(n)+1) ; end proc:
A171797 := proc(n) local n1,n2 ; n1 := A055642(n) ; n2 := nevenDgs(n) ; catL([n1,n2,n1-n2]) ; end proc:
seq(A171797(n),n=1..80) ; # R. J. Mathar, Oct 15 2010 and Oct 18 2010

def a(n):
    s = str(n); e = sum(d in "02468" for d in s)
    return int("".join(map(str, (len(s), e, len(s)-e))))
print([a(n) for n in range(52)]) # Michael S. Branicky, Jun 15 2021

More terms from R. J. Mathar, Oct 15 2010

a(0) added by N. J. A. Sloane, May 12 2019

A227362 Distinct digits of n arranged in decreasing order.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 21, 31, 41, 51, 61, 71, 81, 91, 20, 21, 2, 32, 42, 52, 62, 72, 82, 92, 30, 31, 32, 3, 43, 53, 63, 73, 83, 93, 40, 41, 42, 43, 4, 54, 64, 74, 84, 94, 50, 51, 52, 53, 54, 5, 65, 75, 85, 95, 60, 61, 62, 63, 64, 65, 6, 76, 86

Offset: 0

Reinhard Zumkeller, Jul 09 2013

a(n) <= 9876543210; a(a(n)) = a(n);
A055642(a(n)) <= 10;
A055642(a(n)) <= A055642(n), A055642(a(n)) = A055642(n) iff A178788(n) = 1;
a(A109303(n)) < A109303(n); a(A009995(n)) = A009995(n); a(A071589(n)) > A071589(n);
a(n) = A151949(n) + A180410(n).

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A004186, A230959.

import Data.List (nub, sort)
a227362 = read . reverse . sort . nub . show :: Integer -> Integer

a:= n-> parse(cat(sort([{convert(n, base, 10)[]}[]], `>`)[])):
seq(a(n), n=0..68);  # Alois P. Heinz, Sep 21 2022

f[n_] := FromDigits[Reverse@ Union@ IntegerDigits@ n]; f /@ Range[0, 68] (* Michael De Vlieger, Apr 16 2015, corrected by Robert G. Wilson v *)

a(n) = {if (n == 0, d = [0], d = digits(n)); eval(subst(Pol(vecsort(d,,12)), x, 10));} \\ Michel Marcus, Apr 16 2015

a(n)=fromdigits(vecsort(digits(n),,12)) \\ Charles R Greathouse IV, Apr 16 2015

def A227362(n): return int(''.join(sorted(set(str(n)),reverse=True))) # Chai Wah Wu, Nov 23 2022

A293142 Largest nonrepunit base-n circular prime (conjectured).

Original entry on oeis.org

7, 1013, 3121, 211

Offset: 3

Felix Fröhlich, Oct 01 2017

A circular prime is a prime where all numbers produced by cyclic permutations of the digits are also prime.
No nonrepunit circular prime exists in base 2, since any nonrepunit prime contains at least one 0 digit in its base-2 representation that yields an even number and thus a composite when permuted to the least significant place, so the offset of the sequence is 3.
a(3)-a(6) were found via a brute-force approach searching from the largest prime with 12 base-n digits backwards. The number of base-n digits in a(n) for n = 3, 4, 5, 6 is 2, 5, 5, 3, respectively. Since this is much shorter than 12 digits, it is conjectured that the terms are the maximal circular primes for those bases. This also verifies that no circular primes with a length between A055642(a(n)) and 13 digits exist in bases 3, 4, 5 and 6.
Candidates for a(7), a(8) and a(9) are 13143449029, 16244441 and 4717103, respectively.
a(10) is probably 999331. If not, it must have more than 23 digits (cf. De Geest link).

			1013 written in base 4 is 33311. The base-4 numbers 33311, 33113, 31133, 11333, 13331 written in base 10 are 1013, 983, 863, 383 and 509, respectively. All those base-10 numbers are prime and since there is no larger prime up to 12 base-4 digits where all cyclic permutations of base-4 digits are primes, 1013 is conjectured to be the largest nonrepunit circular prime in base 4.

P. De Geest, Circular Primes, World!Of Numbers.

Cf. A016114, A068652.

Cf. base-b nonrepunit circular primes: A293657 (b=4), A293658 (b=5), A293659 (b=6), A293660 (b=7), A293661 (b=8), A293662 (b=9), A293663 (b=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
decimal(v, base) = my(w=[]); for(k=0, #v-1, w=concat(w, v[#v-k]*base^k)); sum(i=1, #w, w[i])
is_circularprime(p, base) = my(db=digits(p, base), r=rot(db), i=0); if(vecmin(db)==0, return(0), while(1, dec=decimal(r, base); if(!ispseudoprime(dec), return(0)); r=rot(r); if(r==db, return(1))))
a(base, maxlength) = my(p=precprime(base^maxlength)); while(p > 2, if(vecmin(digits(p, base))!=vecmax(digits(p, base)), if(is_circularprime(p, base), return(p))); p=precprime(p-1))
for(n=3, 6, print1(a(n, 12), ", ")) \\ start searching a(n) from largest prime with 12 base-n digits backwards

A034888 Number of digits in 3^n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 32, 33, 33

Offset: 0

N. J. A. Sloane

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A000244 (powers of 3), A055642 (number of digits of n).

[#Intseq(3^n): n in [0..100] ]; // Vincenzo Librandi, Jun 23 2015

Maple
```
seq(floor(n*ln(3)/ln(10))+1, n=0..100);
```

Table[Length[IntegerDigits[3^n]], {n, 0, 100}] (* T. D. Noe, Mar 20 2012 *)
IntegerLength[3^Range[0,70]] (* Harvey P. Dale, Jan 28 2015 *)

a(n)=#Str(3^n) \\ Charles R Greathouse IV, Jul 03 2013

def a(n): return len(str(3**n))
print([a(n) for n in range(70)]) # Michael S. Branicky, Dec 23 2022

a(n) = floor(n*log(3)/log(10)) + 1. E.g., a(10)=5 because 3^10 = 59049 and floor(10*log(3)/log(10)) + 1 = 4 + 1 = 5. - Jaap Spies, Dec 15 2003

a(n) = A055642(3^n) = A055642(A000244(n)). - Michel Marcus, Jun 23 2015

A036299 Binary Fibonacci (or rabbit) sequence.

Original entry on oeis.org

1, 10, 101, 10110, 10110101, 1011010110110, 101101011011010110101, 1011010110110101101011011010110110, 1011010110110101101011011010110110101101011011010110101

Offset: 0

N. J. A. Sloane

A055642(a(n)) = A000045(n+2). - Reinhard Zumkeller, Jul 06 2014

N. G. De Bruijn, (1989, January). Updown generation of Beatty sequences. Koninklijke Nederlandsche Akademie van Wetenschappen (Indationes Math.), Proc., Ser. A, 92:4 (1968), 385-407. See Fig. 3.
J. Kappraff, D. Blackmore and G. Adamson, Phyllotaxis as a dynamical system: a study in number, Chap. 17 of Jean and Barabe, eds., Symmetry in Plants, World Scientific, Studies in Math. Biology and Medicine, Vol. 4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..14
M. S. El Naschie, Statistical geometry of a Cantor discretum and semiconductors, Computers Math. Applic., 29 (No, 12, 1995), 103-110.
C. J. Glasby, S. P. Glasby and F. Pleijel, Worms by number, Proc. Roy. Soc. B, Proc. Biol. Sci. 275 (1647) (2008) 2071-2076.
H. W. Gould, J. B. Kim and V. E. Hoggatt, Jr., Sequences associated with t-ary coding of Fibonacci's rabbits, Fib. Quart., 15 (1977), 311-318.

Cf. A005614, A003849.

Column k=10 of A144287.

a036299 n = a036299_list !! n
a036299_list = map read rabbits :: [Integer] where
   rabbits = "1" : "10" : zipWith (++) (tail rabbits) rabbits
-- Reinhard Zumkeller, Jul 06 2014

nxt[{a_,b_}]:=FromDigits[Join[IntegerDigits[b],IntegerDigits[a]]]; Transpose[NestList[{Last[#],nxt[#]}&,{1,10},10]][[1]] (* Harvey P. Dale, Oct 16 2011 *)

def aupton(terms):
  alst = [1, 10]
  while len(alst) < terms: alst.append(int(str(alst[-1]) + str(alst[-2])))
  return alst[:terms]
print(aupton(9)) # Michael S. Branicky, Jan 10 2021

a(n+1) = concatenation of a(n) and a(n-1).

cp's OEIS Frontend

A270083 Near-miss circular primes: Primes p where all but one of the numbers obtained by cyclically permuting the digits of p are prime.

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A293663 Circular primes that are not repunits.

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A001637 Numbers with an even number of digits.

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A046758 Equidigital numbers.

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A091762 Last n digits of concatenation of first n primes.

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A171797 A modified Sisyphus function: a(n) = concatenation of (number of digits in n) (number of even digits) (number of odd digits).

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A227362 Distinct digits of n arranged in decreasing order.

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