A020338 -id:A020338 - cp's OEIS frontend

A068899 Triangular numbers containing 2n digits obtained by duplicating the first n digits; i.e., triangular numbers in A020338.

55, 66, 5050, 5151, 203203, 255255, 426426, 500500, 501501, 581581, 828828, 930930, 39653965, 50005000, 50015001, 61566156, 3347133471, 5000050000, 5000150001, 6983669836, 220028220028, 500000500000, 500001500001

Offset: 1

Amarnath Murthy, Mar 21 2002

The sequence is infinite: the 10^n-th and the (10^n + 1)-th triangular numbers are members. It is a subsequence of A068898.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A020338, A068898.

N:= 10: # to get all terms of up to 2N digits
Res:= NULL:
for n from 1 to N do
   Divs:= select(t -> igcd(t,(10^n+1)/t)=1, numtheory:-divisors(10^n+1));
   for d in Divs do
     for e in [1,3] do
      u:= chrem([1,-1,e],[d,(10^n+1)/d,4]);
      y:= (u^2-1)/8/(10^n+1);
      if y >= 10^(n-1) and y < 10^n then Res:= Res, y*(10^n+1) fi;
od od od:
sort([Res]); # Robert Israel, Feb 27 2017

Select[Accumulate[Range[5*10^6]],EvenQ[IntegerLength[#]]&&Take[ IntegerDigits[ #],IntegerLength[ #]/2]== Take[IntegerDigits[#],-IntegerLength[#]/2]&] (* Harvey P. Dale, Aug 20 2022 *)

Corrected and extended by Larry Reeves (larryr(AT)acm.org), Jan 10 2003

A121826 a(n) = Product_{k=1..n} D(k), where D() are the doublets, A020338.

Original entry on oeis.org

11, 242, 7986, 351384, 19326120, 1275523920, 98215341840, 8642950081920, 855652058110080, 864208578691180800, 960135730925901868800, 1163684505882193064985600, 1527917756223319494326092800, 2160475707299773764977095219200, 3273120696559157253940299257088000

Offset: 1

Jason Earls, Aug 27 2006

a(n) is divisible by n! because D(k) is divisible by k. - Michel Marcus, Jan 07 2021

			a(4)=351384 because 11*22*33*44 = 351384.

Cf. A000142, A020338.

D(n) = eval(Str(n, n)); \\ A020338
a(n) = prod(k=1, n, D(k)); \\ Michel Marcus, Jan 07 2021

A121837 Least positive j such that Product_{k=1..n} D(k) + j is prime, where D() are the doublets, A020338.

Original entry on oeis.org

2, 9, 7, 7, 17, 7, 29, 17, 19, 23, 23, 13, 29, 79, 19, 89, 97, 53, 43, 347, 127, 127, 149, 29, 167, 331, 379, 61, 59, 167, 199, 557, 107, 113, 43, 191, 439, 41, 263, 227, 109, 71, 227, 137, 149, 409, 271, 53, 157, 79, 503, 103, 461, 137, 587, 233, 491, 73, 367, 233, 449

Offset: 1

Jason Earls, Aug 28 2006

Is every term for n > 2 always prime?
a(159) = 1. - Michel Marcus, Jan 07 2021
a(n) = 1 for n = 245 and 702 (using ispseudoprime() in PARI). - Michel Marcus, Jan 08 2021

Michel Marcus, Table of n, a(n) for n = 1..200

Cf. A013632, A020338, A121826, A151800.

D(n) = eval(Str(n, n)); \\ A020338
f(n) = prod(k=1, n, D(k)); \\ A121826
a(n) = my(q=f(n)); nextprime(q+1) - q; \\ Michel Marcus, Jan 07 2021

a(n) = A013632(A121826(n)). - Michel Marcus, Jan 07 2021

A121882 Numbers k such that k + D(k) + 1 is prime, where D() are the doublets, A020338.

Original entry on oeis.org

1, 3, 5, 6, 8, 9, 10, 11, 13, 14, 15, 21, 23, 25, 28, 30, 31, 34, 35, 36, 38, 45, 49, 50, 53, 60, 63, 64, 66, 69, 71, 74, 76, 79, 80, 81, 83, 90, 91, 99, 101, 105, 106, 108, 110, 113, 114, 124, 128, 130, 134, 135, 136, 140, 141, 143, 144, 150, 151, 159, 161, 163, 165

Offset: 1

Jason Earls, Aug 31 2006

			9 is a term because 9 + 99 + 1 = 109 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A020338.

filter:= n -> isprime(1+n*(2+10^(1+ilog10(n)))):
select(filter, [$1..1000]);# Robert Israel, Feb 23 2022

isok(k) = isprime(k + eval(Str(k, k)) + 1); \\ Michel Marcus, Feb 25 2022

A380233 Odd abundant numbers not divisible by 5 that are also doublets (cf. A020338).

Original entry on oeis.org

153153, 171171, 189189, 207207, 243243, 261261, 279279, 297297, 351351, 459459, 513513, 567567, 621621, 729729, 783783, 837837, 891891, 999999, 1392313923, 1556115561, 1719917199, 1883718837, 2034920349, 2211322113, 2375123751, 2538925389, 2702727027, 3194131941, 4176941769, 4668346683

Offset: 1

Omar E. Pol, Jan 17 2025

There are 26 odd abundant numbers not divisible by 5 less than 10^6. The surprising fact is that 18 of them are doublets.

Another interesting fact is that here there are no terms with 8 digits.

Intersection of A064001 and A020338.

Cf. A005101, A005231, A047201.

doublet:= n -> n * (10^(1+ilog10(n))+1):
select(t -> numtheory:-sigma(t) > 2*t, [seq(seq(doublet(10*x+i),i=[1,3,7,9]),x=1..10000); # Robert Israel, Jan 17 2025

Select[Table[FromDigits[Join[#, #] &@ IntegerDigits[n]], {n, Select[Range[50000], CoprimeQ[#, 10] &]}], DivisorSigma[-1, #] > 2 &] (* Amiram Eldar, Jan 17 2025 *)

select(x->((x%5) && (sigma(x)>2*x)), vector(50000, n, eval(Str(2*n-1, 2*n-1)))) \\ Michel Marcus, Jan 17 2025

More terms from Michel Marcus, Jan 17 2025

A348166 a(n) = abs(A020338(n)-A338754(n)).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 90, 0, 90, 180, 270, 360, 450, 540, 630, 720, 180, 90, 0, 90, 180, 270, 360, 450, 540, 630, 270, 180, 90, 0, 90, 180, 270, 360, 450, 540, 360, 270, 180, 90, 0, 90, 180, 270, 360, 450, 450, 360, 270, 180, 90, 0, 90, 180, 270, 360, 540, 450, 360, 270, 180, 90, 0, 90, 180, 270, 630, 540, 450, 360, 270, 180, 90

Offset: 1

Nicolas Bělohoubek, Oct 04 2021

All terms are multiples of 90.

			a(1) = abs(11-11) = 0
a(15) = abs(1515-1155) = 360
a(1965) = abs(19651965-11996655) = 7655310

Cf. A020338, A338754, A010785.

a(n)=abs(eval(Str(n, n))-fromdigits(digits(n), 100)*11) \\ Charles R Greathouse IV, Oct 04 2021

a(A010785(n)) = 0

Corrected by Charles R Greathouse IV, Oct 04 2021

A380232 Odd abundant numbers that are also doublets (cf. A020338).

Original entry on oeis.org

105105, 135135, 153153, 165165, 171171, 189189, 195195, 207207, 225225, 243243, 255255, 261261, 279279, 285285, 297297, 315315, 345345, 351351, 375375, 405405, 435435, 459459, 465465, 495495, 513513, 525525, 555555, 567567, 585585, 615615, 621621, 645645, 675675, 705705, 729729, 735735, 765765, 783783, 795795, 825825, 837837

Offset: 1

Omar E. Pol, Jan 17 2025

			a(1) = 105105 is an odd abundant number (cf. A005231) because it is and odd number and the sum of its divisors is equal to 229824 which exceeds 2*105105 = 210210. Also 105105 is a doublet (cf. A020338) because it is 105||105, that is the concatenation of 105 and 105, so 105105 is in the sequence.

Intersection of A005231 and A020338.
A subsequence is A380233.
Cf. A005101, A064001.

Select[Table[FromDigits[Join[#, #] &@ IntegerDigits[n]], {n, 1, 850, 2}], DivisorSigma[-1, #] > 2 &] (* Amiram Eldar, Jan 18 2025 *)

select(x->(sigma(x)>2*x), vector(1000, n, eval(Str(2*n-1, 2*n-1)))) \\ Michel Marcus, Jan 18 2025

A007732 Period of decimal representation of 1/n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 6, 6, 1, 1, 16, 1, 18, 1, 6, 2, 22, 1, 1, 6, 3, 6, 28, 1, 15, 1, 2, 16, 6, 1, 3, 18, 6, 1, 5, 6, 21, 2, 1, 22, 46, 1, 42, 1, 16, 6, 13, 3, 2, 6, 18, 28, 58, 1, 60, 15, 6, 1, 6, 2, 33, 16, 22, 6, 35, 1, 8, 3, 1, 18, 6, 6, 13, 1, 9, 5, 41, 6, 16, 21, 28, 2, 44, 1

Offset: 1

N. J. A. Sloane, Hal Sampson [ hals(AT)easynet.com ]

Appears to be a divisor of A007733*A007736. - Henry Bottomley, Dec 20 2001
Primes p such that a(p) = p-1 are in A001913. - Dmitry Kamenetsky, Nov 13 2008
When 1/n has a finite decimal expansion (namely, when n = 2^a*5^b), a(n) = 1 while A051626(n) = 0. - M. F. Hasler, Dec 14 2015
a(n.n) >= a(n) where n.n is A020338(n). - Davide Rotondo, Jun 13 2024

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, pp. 159 etc.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Project Euler, Reciprocal cycles: Problem 26
Index entries for sequences related to decimal expansion of 1/n

Cf. A121341, A066799, A121090, A001913, A084680.

A007732 := proc(n)
    a132740 := 1 ;
    for pe in ifactors(n)[2] do
        if not op(1,pe) in {2,5} then
            a132740 := a132740*op(1,pe)^op(2,pe) ;
        end if;
    end do:
    if a132740 = 1 then
        1 ;
    else
        numtheory[order](10,a132740) ;
    end if;
end proc:
seq(A007732(n),n=1..50) ; # R. J. Mathar, May 05 2023

Table[r = n/2^IntegerExponent[n, 2]/5^IntegerExponent[n, 5]; MultiplicativeOrder[10, r], {n, 100}] (* T. D. Noe, Oct 17 2012 *)

a(n)=znorder(Mod(10,n/2^valuation(n,2)/5^valuation(n,5))) \\ Charles R Greathouse IV, Jan 14 2013

from sympy import n_order, multiplicity
def A007732(n): return n_order(10,n//2**multiplicity(2,n)//5**multiplicity(5,n)) # Chai Wah Wu, Feb 07 2022

def a(n):
    n = ZZ(n)
    rad = 2**n.valuation(2) * 5**n.valuation(5)
    return Zmod(n // rad)(10).multiplicative_order()
[a(n) for n in range(1, 20)]
# F. Chapoton, May 03 2020

Note that if n=r*s where r is a power of 2 and s is odd then a(n)=a(s). Also if n=r*s where r is a power of 5 and s is not divisible by 5 then a(n) = a(s). So we just need a(n) for n not divisible by 2 or 5. This is the smallest number m such that n divides 10^m - 1; m is a divisor of phi(n), where phi = A000010.
phi(n) = n-1 only if n is prime and since a(n) divides phi(n), a(n) can only equal n-1 if n is prime. - Scott Hemphill (hemphill(AT)alumni.caltech.edu), Nov 23 2006
a(n)=a(A132740(n)); a(A132741(n))=a(A003592(n))=1. - Reinhard Zumkeller, Aug 27 2007

More terms from James Sellers, Feb 05 2000

A056524 Palindromes with even number of digits.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 3003, 3113, 3223, 3333, 3443, 3553, 3663, 3773, 3883, 3993, 4004, 4114, 4224, 4334, 4444, 4554

Offset: 1

Henry Bottomley, Jun 16 2000

Concatenation of n with reverse of n (keeping leading zeros in the reverse).
A178788(a(n)) = 0 for n > 1. - Reinhard Zumkeller, Jun 30 2010
All of the terms are divisible by eleven. - James Burling, Aug 08 2014

Reinhard Zumkeller, Table of n, a(n) for n = 1..9999

Cf. A002113, A004086, A056525, A066492.
Cf. A020338, A055642, A178788.
Cf. A110745 (permutation).

a056524 n = a056524_list !! (n-1)
a056524_list = [read (ns ++ reverse ns) :: Integer |
                n <- [0..], let ns = show n]
-- Reinhard Zumkeller, Dec 28 2011

d[n_]:=IntegerDigits[n]; Table[FromDigits[Join[x=d[n],Reverse[x]]],{n,45}] (* Jayanta Basu, May 29 2013 *)
Select[Flatten[Table[Range[10^n,10^(n+1)-1],{n,1,3,2}]],PalindromeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 22 2018 *)

def a(n): s = str(n); return int(s + s[::-1])
print([a(n) for n in range(1, 46)]) # Michael S. Branicky, Nov 02 2021

a(n) = n*10^A055642(n) + A004086(n).

a(n) = 11 * A066492(n).

A019550 a(n) is the concatenation of n and 2n.

Original entry on oeis.org

12, 24, 36, 48, 510, 612, 714, 816, 918, 1020, 1122, 1224, 1326, 1428, 1530, 1632, 1734, 1836, 1938, 2040, 2142, 2244, 2346, 2448, 2550, 2652, 2754, 2856, 2958, 3060, 3162, 3264, 3366, 3468, 3570, 3672, 3774, 3876, 3978, 4080, 4182, 4284, 4386, 4488, 4590

Offset: 1

R. Muller

Concatenation of digits of n and 2*n. - Harvey P. Dale, Sep 13 2011

All terms are divisible by 6. - Robert Israel, Sep 21 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Sylvester Smith, A Set of Conjectures on Smarandache Sequences, Bulletin of Pure and Applied Sciences, (Bombay, India), Vol. 15 E (No. 1), 1996, pp. 101-107.

Cf. concatenation of n and k*n: A020338 (k=1), this sequence (k=2), A019551 (k=3), A019552 (k=4), A019553 (k=5), A009440 (k=6), A009441 (k=7), A009470 (k=8), A009474 (k=9).
Cf. A235497.
Supersequence of A117304.

[Seqint(Intseq(2*n) cat Intseq(n)): n in [1..50]]; // Vincenzo Librandi, Feb 04 2014

seq(n*(10^(1+ilog10(2*n))+2),n=1..100); # Robert Israel, Sep 21 2015

nxt[n_]:=Module[{idn=IntegerDigits[n],idn2=IntegerDigits[2n]}, FromDigits[ Join[ idn,idn2]]]; Array[nxt,40] (* Harvey P. Dale, Sep 13 2011 *)

a(n) = eval(Str(n, 2*n)); \\ Michel Marcus, Sep 21 2015

def a(n): return int(str(n) + str(2*n))
print([a(n) for n in range(1, 46)]) # Michael S. Branicky, Dec 24 2021

From Robert Israel, Sep 21 2015 (Start)
G.f.: (6*(2*x+75*x^5-60*x^6) + 90*Sum_{k>=1} 10^k*x^(5*10^k)*(5*10^k - (5*10^k-1)*x))/(1-x)^2.
a(n+2) - 2*a(n+1) + a(n) = 45*10^(2*k+1) if n = 5*10^k-2, 90*10^k-450*10^(2*k) if n = 5*10^k-1, 0 otherwise. (End)

Offset changed from 0 to 1 by Vincenzo Librandi, Feb 04 2014

cp's OEIS Frontend

A068899 Triangular numbers containing 2n digits obtained by duplicating the first n digits; i.e., triangular numbers in A020338.

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A121826 a(n) = Product_{k=1..n} D(k), where D() are the doublets, A020338.

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A121837 Least positive j such that Product_{k=1..n} D(k) + j is prime, where D() are the doublets, A020338.

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A121882 Numbers k such that k + D(k) + 1 is prime, where D() are the doublets, A020338.

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A380233 Odd abundant numbers not divisible by 5 that are also doublets (cf. A020338).

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PARI

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A348166 a(n) = abs(A020338(n)-A338754(n)).

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A380232 Odd abundant numbers that are also doublets (cf. A020338).

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Mathematica

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A007732 Period of decimal representation of 1/n.

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