A002279 -id:A002279 - cp's OEIS frontend

A285094 Corresponding values of geometric means of digits of numbers from A061430.

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

Offset: 0

Jaroslav Krizek, Apr 14 2017

Jaroslav Krizek, Table of n, a(n) for n = 0..3000

Cf. A061430 (numbers with integer geometric mean of digits in base 10).

Sequences of numbers n such that a(n) = k for k = 0 - 9: A011540 (k = 0), A002275 (k = 1), A061426 (k = 2), A061427 (k = 3), A061428 (k = 4), A002279 (k = 5), A061429 (k = 6), A002281 (k = 7), A002282 (k = 8), A002283 (k = 9).

[0] cat [Floor(&*Intseq(n) ^ (1/#Intseq(n))): n in [1..100000] | IsIntegral(&*Intseq(n) ^ (1/#Intseq(n)))];

A322927 Expansion of x(1 + 5x + 40x^2)/((1 - x^2)(1 - 10*x^2)).

Original entry on oeis.org

0, 1, 5, 51, 55, 551, 555, 5551, 5555, 55551, 55555, 555551, 555555, 5555551, 5555555, 55555551, 55555555, 555555551, 555555555, 5555555551, 5555555555, 55555555551, 55555555555, 555555555551, 555555555555, 5555555555551, 5555555555555, 55555555555551

Offset: 0

Vincenzo Librandi, Mar 17 2019

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,11,0,-10).

Bisections give: A002279 (even part), A173804 (odd part).

I:=[0, 1, 5, 51]; [n le 4 select I[n] else 11*Self(n-2)-10*Self(n-4): n in [1..30]];

seq(coeff(series(x*(1+5*x+40*x^2)/((1-x^2)*(1-10*x^2)),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Mar 17 2019

CoefficientList[Series[x (1 + 5 x + 40 x^2) / (10 x^4 - 11 x^2 + 1), {x, 0, 25}], x]

G.f.: x*(1 + 5*x + 40*x^2)/((1 - x^2)*(1 - 10*x^2)).
a(n) = 11*a(n-2) - 10*a(n-4).
a(n) = 5*(10^n - 1)/9 for n even; a(n) = (5*10^n - 41)/9 otherwise.

A332152 a(n) = 5(10^(2n+1)-1)/9 - 3*10^n.

Original entry on oeis.org

2, 525, 55255, 5552555, 555525555, 55555255555, 5555552555555, 555555525555555, 55555555255555555, 5555555552555555555, 555555555525555555555, 55555555555255555555555, 5555555555552555555555555, 555555555555525555555555555, 55555555555555255555555555555, 5555555555555552555555555555555

Offset: 0

M. F. Hasler, Feb 09 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002279 (5*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332112 .. A332192 (variants with different repeated digit 1, ..., 9).
Cf. A332150 .. A332159 (variants with different middle digit 0, ..., 9).

A332152 := n -> 5*(10^(2*n+1)-1)/9-3*10^n;

Array[5 (10^(2 # + 1)-1)/9 - 3*10^# &, 15, 0]

apply( {A332152(n)=10^(n*2+1)\9*5-3*10^n}, [0..15])

def A332152(n): return 10**(n*2+1)//9*5-3*10**n

a(n) = 5*A138148(n) + 2*10^n = A002279(2n+1) - 3*10^n.
G.f.: (2 + 303*x - 800*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332153 a(n) = 5(10^(2n+1)-1)/9 - 2*10^n.

Original entry on oeis.org

3, 535, 55355, 5553555, 555535555, 55555355555, 5555553555555, 555555535555555, 55555555355555555, 5555555553555555555, 555555555535555555555, 55555555555355555555555, 5555555555553555555555555, 555555555555535555555555555, 55555555555555355555555555555, 5555555555555553555555555555555

Offset: 0

M. F. Hasler, Feb 09 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002279 (5*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332113 .. A332193 (variants with different repeated digit 1, ..., 9).
Cf. A332150 .. A332159 (variants with different middle digit 0, ..., 9).

A332153 := n -> 5*(10^(2*n+1)-1)/9-2*10^n;

Array[5 (10^(2 # + 1)-1)/9 - 2*10^# &, 15, 0]

apply( {A332153(n)=10^(n*2+1)\9*5-2*10^n}, [0..15])

def A332153(n): return 10**(n*2+1)//9*5-2*10**n

a(n) = 5*A138148(n) + 3*10^n = A002279(2n+1) - 2*10^n.
G.f.: (3 + 202*x - 700*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332154 a(n) = 5(10^(2n+1)-1)/9 - 10^n.

Original entry on oeis.org

4, 545, 55455, 5554555, 555545555, 55555455555, 5555554555555, 555555545555555, 55555555455555555, 5555555554555555555, 555555555545555555555, 55555555555455555555555, 5555555555554555555555555, 555555555555545555555555555, 55555555555555455555555555555, 5555555555555554555555555555555

Offset: 0

M. F. Hasler, Feb 09 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002279 (5*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332114 .. A332194 (variants with different repeated digit 1, ..., 9).
Cf. A332150 .. A332159 (variants with different middle digit 0, ..., 9).

A332154 := n -> 5*(10^(2*n+1)-1)/9-10^n;

Array[5 (10^(2 # + 1)-1)/9 - 10^# &, 15, 0]
LinearRecurrence[{111,-1110,1000},{4,545,55455},20] (* or *) Table[FromDigits[Join[PadRight[{},n,5],{4},PadRight[{},n,5]]],{n,0,20}] (* Harvey P. Dale, Mar 09 2025 *)

apply( {A332154(n)=10^(n*2+1)\9*5-10^n}, [0..15])

def A332154(n): return 10**(n*2+1)//9*5-10**n

a(n) = 5*A138148(n) + 4*10^n = A002279(2n+1) - 10^n.
G.f.: (4 + 101*x - 600*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332156 a(n) = 5(10^(2n+1)-1)/9 + 10^n.

Original entry on oeis.org

6, 565, 55655, 5556555, 555565555, 55555655555, 5555556555555, 555555565555555, 55555555655555555, 5555555556555555555, 555555555565555555555, 55555555555655555555555, 5555555555556555555555555, 555555555555565555555555555, 55555555555555655555555555555, 5555555555555556555555555555555

Offset: 0

M. F. Hasler, Feb 09 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002279 (5*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332116 .. A332196 (variants with different repeated digit 1, ..., 9).
Cf. A332150 .. A332159 (variants with different middle digit 0, ..., 9).

A332156 := n -> 5*(10^(2*n+1)-1)/9+10^n;

Array[5 (10^(2 # + 1)-1)/9 + 10^# &, 15, 0]

apply( {A332156(n)=10^(n*2+1)\9*5+10^n}, [0..15])

def A332156(n): return 10**(n*2+1)//9*5+10**n

a(n) = 5*A138148(n) + 6*10^n = A002279(2n+1) + 10^n.
G.f.: (6 - 101*x - 400*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
E.g.f.: exp(x)*(50*exp(99*x) + 9*exp(9*x) - 5)/9. - Stefano Spezia, Jul 13 2024

A332157 a(n) = 5(10^(2n+1)-1)/9 + 2*10^n.

Original entry on oeis.org

7, 575, 55755, 5557555, 555575555, 55555755555, 5555557555555, 555555575555555, 55555555755555555, 5555555557555555555, 555555555575555555555, 55555555555755555555555, 5555555555557555555555555, 555555555555575555555555555, 55555555555555755555555555555, 5555555555555557555555555555555

Offset: 0

M. F. Hasler, Feb 09 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002279 (5*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332117 .. A332197 (variants with different repeated digit 1, ..., 9).
Cf. A332150 .. A332159 (variants with different middle digit 0, ..., 9).

A332157 := n -> 5*(10^(2*n+1)-1)/9+2*10^n;

Array[5 (10^(2 # + 1)-1)/9 + 2*10^# &, 15, 0]

apply( {A332157(n)=10^(n*2+1)\9*5+2*10^n}, [0..15])

def A332157(n): return 10**(n*2+1)//9*5+2*10**n

a(n) = 5*A138148(n) + 7*10^n = A002279(2n+1) + 2*10^n.
G.f.: (7 - 202*x - 300*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A332158 a(n) = 5(10^(2n+1)-1)/9 + 3*10^n.

Original entry on oeis.org

8, 585, 55855, 5558555, 555585555, 55555855555, 5555558555555, 555555585555555, 55555555855555555, 5555555558555555555, 555555555585555555555, 55555555555855555555555, 5555555555558555555555555, 555555555555585555555555555, 55555555555555855555555555555, 5555555555555558555555555555555

Offset: 0

M. F. Hasler, Feb 09 2020

Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002279 (5*R_n), A011557 (10^n).
Cf. A138148 (cyclops numbers with binary digits), A002113 (palindromes).
Cf. A332118 .. A332178, A181965 (variants with different repeated digit 1, ..., 9).
Cf. A332150 .. A332159 (variants with different middle digit 0, ..., 9).

A332158 := n -> 5*(10^(2*n+1)-1)/9+3*10^n;

Array[5 (10^(2 # + 1)-1)/9 + 3*10^# &, 15, 0]

apply( {A332158(n)=10^(n*2+1)\9*5+3*10^n}, [0..15])

def A332158(n): return 10**(n*2+1)//9*5+3*10**n

a(n) = 5*A138148(n) + 8*10^n = A002279(2n+1) + 3*10^n.
G.f.: (8 - 303*x - 200*x^2)/((1 - x)(1 - 10*x)(1 - 100*x)).
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.

A365933 a(n) is the period of the remainders when repdigits are divided by n.

Original entry on oeis.org

1, 9, 27, 9, 9, 27, 54, 9, 81, 9, 18, 27, 54, 54, 27, 9, 144, 81, 162, 9, 54, 18, 198, 27, 9, 54, 243, 54, 252, 27, 135, 9, 54, 144, 54, 81, 27, 162, 54, 9, 45, 54, 189, 18, 81, 198, 414, 27, 378, 9, 432, 54, 117, 243, 18, 54, 162, 252, 522, 27, 540, 135, 162, 9, 54

Offset: 1

Karl-Heinz Hofmann, Nov 07 2023

For n>1: Periods are divisible by 9 (= a full cycle in the sequence of repdigits). a(n)/9 is the period of the remainders when repunits are divided by n. So the digit part of the repdigits has no effect on periods generally. For most n the beginning of the periodic part is always A010785(1). If n is a term of A083118 the periodic part starts later after some initial remainders that do not repeat.

			For n = 6:                Remainders of A010785(1..54) mod n.
A010785( 1...9) mod n:      [1, 2, 3, 4, 5, 0, 1, 2, 3]
A010785(10..18) mod n:      [5, 4, 3, 2, 1, 0, 5, 4, 3]
A010785(19..27) mod n:      [3, 0, 3, 0, 3, 0, 3, 0, 3]
So the period is 3*9 = 27. Thus a(n) = 27. And the pattern seen above starts again:
A010785(28..36) mod n:      [1, 2, 3, 4, 5, 0, 1, 2, 3]
A010785(37..45) mod n:      [5, 4, 3, 2, 1, 0, 5, 4, 3]
A010785(46..54) mod n:      [3, 0, 3, 0, 3, 0, 3, 0, 3]

Karl-Heinz Hofmann, Table with additional information.

Cf. A010785, A002275.
Cf. A305322 (divisor 3), A002279 (divisor 5), A366596 (divisor 7).
Cf. A083118 (the impossible divisors).

def A365933(n):
    if n == 1: return 1
    remainders, exponent = [], 1
    while (rem:=(10**exponent // 9 % n)) not in remainders:
        remainders.append(rem); exponent += 1
    return (exponent - remainders.index(rem) - 1) * 9

def A365933(n):
    if n==1: return 1
    a,b,x,y=1,1,1%n,11%n
    while x!=y:
        if a==b:
            a<<=1
            x,b=y,0
        y = (10*y+1)%n
        b+=1
    return 9*b # Chai Wah Wu, Jan 23 2024

A361820 Palindromes in A329150.

Original entry on oeis.org

0, 2, 3, 5, 7, 11, 22, 33, 55, 77, 202, 222, 232, 252, 272, 303, 313, 323, 333, 353, 373, 505, 525, 535, 555, 575, 707, 717, 727, 737, 757, 777, 1111, 2002, 2112, 2222, 2332, 2552, 2772, 3003, 3113, 3223, 3333, 3553, 3773, 5005, 5115, 5225, 5335, 5555, 5775, 7007, 7117

Offset: 1

Bernard Schott, Mar 25 2023

If m is a palindrome with no digit greater than 5 in A118597, then A329147(m) is a term, but there exist terms that are not of this form as 313, 717, ...

			232 is a term which has two preimages since A329147(91) = A329147(121) = 232.
313 = A329147(26) is a term whose preimage is not in A118597.
2002 is a term since A329147(1001) = 2002.
2112 is a term since A329147(151) = 2112.
27172 = A329147(1471) is a term whose preimage is not in A118597.

Intersection of A002113 and A329150.
Cf. A118597, A329147, A361750, A361821.
Subsequences: A002276, A002277, A002279, A002281, A099814.

p[n_] := If[n > 0, Prime[n], 0]; seq[ndigmax_] := Module[{t = Table[FromDigits[ Flatten@ IntegerDigits@ (p /@ IntegerDigits[n])], {n, 0, 10^ndigmax - 1}]}, Union@ Select[t, # < 10^ndigmax && PalindromeQ[#] &]]; seq[4] (* Amiram Eldar, Mar 26 2023 *)

ispal(n) = my(d=digits(n)); d==Vecrev(d);
f(n) = if (n, fromdigits(concat(apply(d -> if (d, digits(prime(d)), [0]), digits(n)))), 0); \\ A329147
lista(nn) = my(list = List(), m); for (n=0, nn, m = f(n); if ((m <= nn) && ispal(m), listput(list, m));); vecsort(Set(list)); \\ Michel Marcus, Mar 26 2023

cp's OEIS Frontend

A285094 Corresponding values of geometric means of digits of numbers from A061430.

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A322927 Expansion of x*(1 + 5*x + 40*x^2)/((1 - x^2)*(1 - 10*x^2)).

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Maple

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A332152 a(n) = 5*(10^(2*n+1)-1)/9 - 3*10^n.

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A332153 a(n) = 5*(10^(2*n+1)-1)/9 - 2*10^n.

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Maple

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Formula

A332154 a(n) = 5*(10^(2*n+1)-1)/9 - 10^n.

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Maple

Mathematica

PARI

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Formula

A332156 a(n) = 5*(10^(2*n+1)-1)/9 + 10^n.

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Maple

Mathematica

PARI

Python

Formula

A332157 a(n) = 5*(10^(2*n+1)-1)/9 + 2*10^n.

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Maple

Mathematica

PARI

Python

Formula

A322927 Expansion of x(1 + 5x + 40x^2)/((1 - x^2)(1 - 10*x^2)).

A332152 a(n) = 5(10^(2n+1)-1)/9 - 3*10^n.

A332153 a(n) = 5(10^(2n+1)-1)/9 - 2*10^n.

A332154 a(n) = 5(10^(2n+1)-1)/9 - 10^n.

A332156 a(n) = 5(10^(2n+1)-1)/9 + 10^n.

A332157 a(n) = 5(10^(2n+1)-1)/9 + 2*10^n.

A332158 a(n) = 5(10^(2n+1)-1)/9 + 3*10^n.