A008592 -id:A008592 - cp's OEIS frontend

A178359 Rounded up arithmetic mean of digits of n appended to n, cf. A004427.

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 122, 132, 143, 153, 164, 174, 185, 195, 201, 212, 222, 233, 243, 254, 264, 275, 285, 296, 302, 312, 323, 333, 344, 354, 365, 375, 386, 396, 402, 413, 423, 434, 444, 455, 465, 476, 486, 497, 503, 513, 524, 534

Offset: 0

Reinhard Zumkeller, May 27 2010

A010879(a(n)) = A004427(a(n)) = A004427(n);
a(A178358(n)) = A178358(a(n));
subsequence of A178403.

			For n=8379: A004427(n) = ceiling((8+3+7+9)/4) = 7; so a(8379) = 10*8379 + 7 = 83797.

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

amd[n_]:=Module[{m=Ceiling[Mean[IntegerDigits[n]]]},n*10^IntegerLength[ m]+ m]; Array[amd,60,0] (* Harvey P. Dale, Aug 12 2015 *)

a(n) = A008592(n) + A004427(n).

A217562 Even numbers not divisible by 5.

Original entry on oeis.org

2, 4, 6, 8, 12, 14, 16, 18, 22, 24, 26, 28, 32, 34, 36, 38, 42, 44, 46, 48, 52, 54, 56, 58, 62, 64, 66, 68, 72, 74, 76, 78, 82, 84, 86, 88, 92, 94, 96, 98, 102, 104, 106, 108, 112, 114, 116, 118, 122, 124, 126, 128, 132

Offset: 1

Jeremy Gardiner, Oct 06 2012

Numbers ending with 2,4,6,8 in base 10.
No term is divisible by 10 therefore a subsequence of A067251 (Numbers with no trailing zeros in decimal representation).
Union of this sequence with A005408 (The odd numbers) gives A067251.
Union of this sequence with A045572 (Numbers that are odd but not divisible by 5) gives A047201.
The even numbers divisible by 5 are A008592 (Multiples of 10).

Jeremy Gardiner, Table of n, a(n) for n = 1..4000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A005408, A005843, A045572, A047201, A067251.

for n=1 to 199
if n mod 5 <> 0 and n mod 2 <> 1 then print str$(n)+", ";
next n
print

I:=[2, 4, 6, 8, 12]; [n le 5 select I[n] else Self(n-1) + Self(n-4) - Self(n-5): n in [1..60]]; // Vincenzo Librandi, Dec 28 2012

CoefficientList[Series[2*(1 + x + x^2 + x^3 + x^4)/((1 + x)*(1 + x^2)*(x - 1)^2), {x, 0, 100}], x] (* Vincenzo Librandi, Dec 28 2012 *)

A217562(n)=(n-1)*5\2+2 \\ M. F. Hasler, Oct 07 2012

def A217562(n): return (5*n-1>>1)&-2 # Chai Wah Wu, Apr 21 2025

a(n) = 2*A047201(n).
G.f.: 2*x*(1+x+x^2+x^3+x^4) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 06 2012
a(n) = 2*(n+floor((n-1)/4)). - Aaron J Grech, Sep 28 2024
E.g.f.: (4 - cos(x) + (5*x - 3)*cosh(x) + sin(x) + (5*x - 2)*sinh(x))/2. - Stefano Spezia, Sep 28 2024

A017270 a(n) = (10*n)^2.

Original entry on oeis.org

0, 100, 400, 900, 1600, 2500, 3600, 4900, 6400, 8100, 10000, 12100, 14400, 16900, 19600, 22500, 25600, 28900, 32400, 36100, 40000, 44100, 48400, 52900, 57600, 62500, 67600, 72900, 78400, 84100, 90000, 96100, 102400, 108900, 115600, 122500, 129600, 136900, 144400

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A008592, A016850.

[(10*n)^2: n in [0..40]]; // Vincenzo Librandi, Jul 28 2011

LinearRecurrence[{3,-3,1},{0,100,400},40] (* Harvey P. Dale, Oct 02 2017 *)

a(n)=(10*n)^2 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = a(n-1) + 200*n - 100, n > 0 ; a(0)=0. - Miquel Cerda, Oct 30 2016
G.f.: 100*x*(1 + x)/(1 - x)^3. - Ilya Gutkovskiy, Oct 30 2016
a(n) = 100*A000290(n). - Michel Marcus, Oct 30 2016
From Amiram Eldar, Jan 25 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/600.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/1200.
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/10)/(Pi/10).
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/10)/(Pi/10) = 5*(sqrt(5)-1)/(2*Pi). (End)
From Elmo R. Oliveira, Nov 30 2024: (Start)
E.g.f.: 100*x*(1 + x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
a(n) = A008592(n)^2 = A000290(A008592(n)) = A016850(2*n). (End)

A169669 (first digit of n) * (last digit of n) in decimal representation.

Original entry on oeis.org

0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 0

Offset: 0

Reinhard Zumkeller, Apr 05 2010

a(n) = A000030(n)*A010879(n);
a(n) = A115300(n) for n<=100, A115300(101) = 0;
a(n) = A111707(n) for n<=109, A111707(110) = 1;
0 <= a(n) <= 81, range = A174995;
a(10*n + n mod 10) = a(n);
a(A008592(n)) = 0;
A000030(a(A017281(n))) = A000030(A017281(n));
a(n) = a(A004086(n))*A168184(n);

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

Cf. A088133.

Cf. A000030, A008592, A010879, A111707, A115300, A174995.

a169669 n = a000030 n * mod n 10
-- Reinhard Zumkeller, Apr 29 2015

def a(n): return int(str(n)[0])*(n%10)
print([a(n) for n in range(81)]) # Michael S. Branicky, Jul 13 2022

A190997 Product of digits of all the divisors of n.

Original entry on oeis.org

1, 2, 3, 8, 5, 36, 7, 64, 27, 0, 1, 288, 3, 56, 75, 384, 7, 2592, 9, 0, 42, 8, 6, 18432, 50, 72, 378, 3584, 18, 0, 3, 2304, 27, 168, 525, 373248, 21, 432, 243, 0, 4, 16128, 12, 512, 13500, 288, 28, 3538944, 252, 0, 105, 2880, 15, 725760, 125, 860160, 945

Offset: 1

Jaroslav Krizek, Jun 15 2011

Product of digits of concatenation of all divisors of n (A037278).

			For n = 12: a(12) = 1 * 2 * 3 * 4 * 6 * 1 * 2 = 288.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A134681, A034690.

A190997:=proc(n) local d, i, p: d:=numtheory[divisors](n): p:=1: for i from 1 to nops(d) do p:=p*mul(d, d=convert(d[i], base, 10)): od: return p: end: seq(A190997(n),n=1..57); # Nathaniel Johnston, Jun 15 2011

Table[Times@@Flatten[IntegerDigits/@Divisors[n]],{n,100}] (* Harvey P. Dale, Nov 27 2022 *)

a007954(n) = my(d=digits(n)); prod(i=1, #d, d[i]);
a(n) = my(div=divisors(n), pdt=1); for(k=1, #div, pdt=pdt*a007954(div[k])); pdt \\ Felix Fröhlich, Sep 22 2016

a(n) = 0 for n = multiples of 10; a(A008592(n)) = 0 for n >=1.

A355223 The k-th rightmost digit of a(n) is the least of the k rightmost digits of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0, 11, 22, 23, 24, 25, 26, 27, 28, 29, 0, 11, 22, 33, 34, 35, 36, 37, 38, 39, 0, 11, 22, 33, 44, 45, 46, 47, 48, 49, 0, 11, 22, 33, 44, 55, 56, 57, 58, 59, 0, 11, 22, 33, 44, 55, 66, 67, 68

Offset: 0

Rémy Sigrist, Jun 24 2022

Leading zeros are ignored.

			For n = 1402:
- min({1, 4, 0, 2}) = 0,
- min({4, 0, 2}) = 0,
- min({0, 2}) = 0,
- min({2}) = 2,
- so a(1402) = 2.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

See A355221, A355222 and A355224 for similar sequences.

Cf. A008592, A009994 (fixed points), A135481 (binary analog).

a(n, base=10) = { my (d=digits(n, base), m=oo); forstep (k=#d, 1, -1, d[k]=m=min(m, d[k])); fromdigits(d, base) }

def a(n):
    s, m = str(n), "9"
    return int("".join((m:=min(m, s[-1-k])) for k in range(len(s)))[::-1])
print([a(n) for n in range(69)]) # Michael S. Branicky, Jun 24 2022

from itertools import accumulate
def A355223(n): return int(''.join(accumulate(str(n)[::-1],func=min))[::-1]) # Chai Wah Wu, Jun 25 2022

a(n) <= n with equality iff n belongs to A009994.
a(a(n)) = a(n).
a(n) = 0 iff n is a multiple of 10.

A059632 Carryless product 11 X n base 10.

Original entry on oeis.org

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176, 187, 198, 109, 220, 231, 242, 253, 264, 275, 286, 297, 208, 219, 330, 341, 352, 363, 374, 385, 396, 307, 318, 329, 440, 451, 462, 473, 484, 495, 406, 417, 428, 439, 550, 561, 572, 583

Offset: 0

Henry Bottomley, Feb 19 2001

a(n) <= 11*n; a(m) = 11*m iff m is a term of A039691. - Reinhard Zumkeller, Jul 05 2014

			a(19)=109 since we have 11 X 19 = carryless sum of 100, 90, 10 and 9 =109

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version.

Cf. A001477 for carryless 1 X n, A004520 for carryless 2 X 10 base 10, A055120 for carryless 9 X n, A008592 for carryless 10 X n.
Cf. A048724 carryless 3Xn in base 2, A242399 carryless 4Xn in base 3.
Cf. A008593.

a059632 n = foldl (\v d -> 10 * v + d) 0 $
                  map (flip mod 10) $ zipWith (+) ([0] ++ ds) (ds ++ [0])
            where ds = map (read . return) $ show n
-- Reinhard Zumkeller, Jul 05 2014

A069498 Triangular numbers of the form 10*k.

Original entry on oeis.org

0, 10, 120, 190, 210, 300, 630, 780, 820, 990, 1540, 1770, 1830, 2080, 2850, 3160, 3240, 3570, 4560, 4950, 5050, 5460, 6670, 7140, 7260, 7750, 9180, 9730, 9870, 10440, 12090, 12720, 12880, 13530, 15400, 16110, 16290, 17020, 19110, 19900, 20100, 20910, 23220

Offset: 1

Amarnath Murthy, Mar 30 2002

Intersection of A000217 and A008592. - Michel Marcus, Sep 17 2013

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000217, A008592.

a[0] := 0:a[1] := 4:a[2] := 15:a[3] := 19:seq((20*(floor(i/4))+a[i mod 4])*(20*(floor(i/4))+a[i mod 4]+1)/2,i=0..100);

Select[Accumulate[Range[0,250]],Divisible[#,10]&] (* Harvey P. Dale, Aug 28 2016 *)

G.f.: -10*x^2*(x^6+11*x^5+7*x^4+2*x^3+7*x^2+11*x+1) / ((x-1)^3*(x+1)^2*(x^2+1)^2). - Colin Barker, Sep 23 2013

Corrected and extended by Harvey P. Dale and Sascha Kurz, Mar 31 2002

A085315 Numbers such that first reversing digits and after forming its cube equals the result of first-form-cube and after-reverse operation with exclusion of cases divisible by 10.

Original entry on oeis.org

1, 2, 7, 11, 101, 111, 1001, 1011, 1101, 10001, 10011, 10101, 11001, 11011, 100001, 100011, 100101, 100111, 101001, 101011, 101101, 110001, 110011, 110101, 111001, 1000001, 1000011, 1000101, 1000111, 1001001, 1001011, 1001101, 1010001, 1010011, 1011001, 1100001, 1100011, 1100101, 1101001, 1110001

Offset: 1

Labos Elemer, Jul 01 2003

			n=100111,rev[n]=111001, n^3=1003333697667631.
rev[n^3]=111001^3=1367667963333001=rev[n]^3.

Cf. A000578, A004086, A008592, A069494, A085305.

r:= n-> (s-> parse(cat(seq(s[-i], i=1..length(s)))))(""||n):
q:= n-> irem(n, 10)>0 and r(n^3)=r(n)^3:
select(q, [$1..2000000])[];  # Alois P. Heinz, Oct 22 2021

nd[x_, y_] := 10*x+y; tn[x_] := Fold[nd, 0, x] rt[x_] := tn[Reverse[IntegerDigits[x]]] Do[s=rt[n^3]; s1=rt[n]^3; If[Equal[s, s1]&& !Equal[Mod[n, 10], 0], k=k+1; Print[n]], {n, 1, 10000000}]; k

Solutions to rev[x^3]=rev[x]^3 without numbers divisible by 10.

{ A069494 } minus { A008592 }. - Alois P. Heinz, Oct 22 2021

A158814 Multiples of 10 in the EKG sequence A064413.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 310, 320, 330, 350, 340, 360, 370, 380, 390, 400, 410, 420, 430, 440, 450, 460, 470, 480, 490, 500, 510, 520, 530, 540, 550, 560, 570

Offset: 1

Paul Curtz, Mar 27 2009

nonn

The first 33 terms are regular, as in A008592.
The multiples of 1 is A064413 itself. The multiples of 2 is A064469. The multiples of 3 are 3*A155963(n).
The first 15 multiples of 4 are 4*A115510, but then become different. The multiples of 5 are A158486,
also represented by A158504. The first 85 multiples of 6 coincide with A008588(1..85), then deviate.
The first 56 multiples of 9 are A008591(1..56), and then deviate. Multiples of 13 are in A158605.

Edited and extended by R. J. Mathar, Apr 04 2009

Corrected by Paul Curtz, Apr 17 2009

cp's OEIS Frontend

A178359 Rounded up arithmetic mean of digits of n appended to n, cf. A004427.

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A217562 Even numbers not divisible by 5.

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A017270 a(n) = (10*n)^2.

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A169669 (first digit of n) * (last digit of n) in decimal representation.

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A190997 Product of digits of all the divisors of n.

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A355223 The k-th rightmost digit of a(n) is the least of the k rightmost digits of n.

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A059632 Carryless product 11 X n base 10.

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