A067079 -id:A067079 - cp's OEIS frontend

A067080 If n = ab...def in decimal notation then the left digitorial function Ld(n) = ab...defab...deab...d...ab*a.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 160, 164, 168, 172, 176, 180, 184, 188, 192, 196, 250, 255, 260, 265, 270, 275, 280, 285, 290, 295, 360, 366, 372

Offset: 1

Amarnath Murthy, Jan 05 2002

This entry should probably start at n=0, just as A067079 does. But that would require a number of changes, so it can wait until the editors have more free time. - N. J. A. Sloane, Nov 29 2014

			Ld(256) = 256*25*2 =12800.
a(31)=floor(31/10^0)*floor(31/10^1)=31*3=93;
a(42)=168 since 42=42(base-10) and so a(42)=42*4(base-10)=42*4=168.

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A067079, A065039, A048651, A098844, A132019, A132026, A132038, A000217.
For formulas regarding a general parameter p (i.e. terms floor(n/p^k)) see A132264.
For the product of terms floor(n/p^k) for p=2 to p=12 see A098844(p=2), A132027(p=3)-A132033(p=9), A132263(p=11), A132264(p=12).
For the products of terms 1+floor(n/p^k) see A132269-A132272, A132327, A132328.

a067080 n = if n <= 9 then n else n * a067080 (n `div` 10)
-- Reinhard Zumkeller, Nov 29 2012

Table[d = IntegerDigits[n]; rd = 1; While[ Length[d] > 0, rd = rd*FromDigits[d]; d = Drop[d, -1]]; rd, {n, 1, 75} ]
Table[Times@@NestList[Quotient[#,10]&,n,IntegerLength[n]-1],{n,70}] (* Harvey P. Dale, Dec 16 2013 *)

a(n)=my(t=n);while(n\=10,t*=n); t \\ Charles R Greathouse IV, Nov 20 2012

a(n) = Product_{k=1..length(n)} floor(n/10^(k-1)). - Vladeta Jovovic, Jan 08 2002
From Hieronymus Fischer, Aug 13 2007: (Start)
a(n) = product{0<=k<=floor(log_10(n)), floor(n/10^k)}, n>=1.
Recurrence:
a(n) = n*a(floor(n/10));
a(n*10^m) = n^m*10^(m(m+1)/2)*a(n).
a(k*10^m) = k^(m+1)*10^(m(m+1)/2), for 0a(n) <= b(n), where b(n)=n^(1+floor(log_10(n)))/10^(1/2*(1+floor(log_10(n)))*floor(log_10(n))); equality holds for n=k*10^m, m>=0, 1<=k<10. Here b(n) can also be written n^(1+floor(log_10(n)))/10^A000217(floor(log_10(n))).
Also: a(n) <= 3^((1-log_10(3))/2)*n^((1+log_10(n))/2)=1.332718...*10^A000217(log_10(n)), equality for n=3*10^m, m>=0.
a(n) > c*b(n), where c=0.472362443816572... (see constant A132026).
Also: a(n) > c*2^((1-log_10(2))/2)*n^((1+log_10(n))/2) = 0.601839...*10^A000217(log_10(n)).
lim inf a(n)/b(n) = 0.472362443816572..., for n-->oo.
lim sup a(n)/b(n) = 1, for n-->oo.
lim inf a(n)/n^((1+log_10(n))/2) = 0.472362443816572...*sqrt(2)/2^log_10(sqrt(2)), for n-->oo.
lim sup a(n)/n^((1+log_10(n))/2) = sqrt(3)/3^log_10(sqrt(3)), for n-->oo.
lim inf a(n)/a(n+1) = 0.472362443816572... for n-->oo (see constant A132026).
a(n) = O(n^((1+log_10(n))/2)). (End)

More terms from Robert G. Wilson v, Jan 07 2002

A065039 If n in base 10 is d_1 d_2 ... d_k then a(n) = d_1 + d_1d_2 + d_1d_2d_3 + ... + d_1...d_k.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70

Offset: 0

Santi Spadaro, Nov 04 2001

a(n) = (D(n) - sod(n))/9, for n >= 1, with sod(n) the sum of digits of n, and with D(n) any of the 10 numbers given in base 10 representation by d_(nod(n)-1) d_(nod(n)-2) ... d_0 b_0, where nod(n) is the number of digits of n = d_(nod(n)-1) d_(nod(n)-2) ... d_0 in base 10, and b_0 from {0, 1, ..., 9}. E.g., D(1234) stands for any number from {12340, 12341, ..., 12349}. This corresponds the well known (and easy to prove) rule that any number after subtraction of its sum of digits is divisible by 9. In this subtraction any of the last digit b_0 leads to the same result. Some mathematical tricks are based on this rule. See the Gardner reference. - Wolfdieter Lang, May 04 2010

			a(1234)=1370 because 1+12+123+1234=1370.
With repunits: a(1234) = 4*1 + 3*11 + 2*111 + 1*1111 = 1370. - _Wolfdieter Lang_, May 04 2010

M. Gardner, Mathematische Zaubereien, Dumont, 2004, p. 39. German translation of: Mathematics, Magic and Mystery, Dover, 1956. [From Wolfdieter Lang, May 04 2010]

Harry J. Smith, Table of n, a(n) for n = 0..1000

Complement of A065438.

Cf. A067079, A067080, A067082, A054899, A054861, A067080, A098844, A132027, A005187.

import Data.List (inits)
a065039 n = sum $ map read $ tail $ inits $ show n
-- Reinhard Zumkeller, Mar 31 2011

A065039 := proc(n) local d,m: d:=convert(n,base,10): m:=nops(d): return add(op(convert(d[(m-k+1)..m], base, 10, 10^m)),k=1..m): end: seq(A065039(n),n=0..64); # Nathaniel Johnston, Jun 27 2011

a[n_] := Apply[Plus, Table[FromDigits[Take[IntegerDigits[n], k]], {k, 1, Length[IntegerDigits[n]]}]]
Table[d = IntegerDigits[n]; rd = 0; While[ Length[d] > 0, rd = rd + FromDigits[d]; d = Drop[d, -1]]; rd, {n, 0, 75} ]
f[n_] := Plus @@ NestList[ Quotient[ #, 10] &, n, Max[1, Floor@ Log[10, n]]]; Array[f, 70, 0] (* Robert G. Wilson v, Jun 29 2010 *)
Array[Total[Table[FromDigits[Take[IntegerDigits[#],x]],{x, IntegerLength[ #]}]]&,100,0](* Harvey P. Dale, Jan 02 2016 *)

{ for (n=0, 1000, a=0; k=n; until (k==0, a+=k; k\=10); write("b065039.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 04 2009

a(n) = sum( k>=0, floor(n/10^ k)) = n+A054899(n). - Benoit Cloitre, Aug 03 2002
From Hieronymus Fischer, Aug 14 2007: (Start)
a(10*n)=10*n+a(n); a(n*10^m)=10*n*(10^m-1)/9+a(n).
a(k*10^m)=k*(10^(m+1)-1)/2, 0<=k<10, m>=0.
a(n)=10/9*n+O(log(n)), a(n+1)-a(n)=O(log(n)); this follows from the inequalities below.
a(n)<=(10*n-1)/9; equality holds for powers of 10.
a(n)>=(10*n-9)/9-floor(log_10(n)); equality holds for n=10^m-1, m>0.
lim inf (10*n/9-a(n))=1/9, for n-->oo.
lim sup (10*n/9-log_10(n)-a(n))=0, for n-->oo.
lim sup (a(n+1)-a(n)-log_10(n))=1, for n-->oo.
G.f.: sum{k>=0, x^(10^k)/(1-x^(10^k))}/(1-x).
(End)
a(n) = sum(d_(k)*RU(k+1),k=0..nod(n)-1), with the notation nod(n)and d_k given in a comment above, and RU(k)is the repunit (10^k-1)/9 (k times 1). - Wolfdieter Lang, May 04 2010

A132772 a(n) = n*(n + 30).

Original entry on oeis.org

0, 31, 64, 99, 136, 175, 216, 259, 304, 351, 400, 451, 504, 559, 616, 675, 736, 799, 864, 931, 1000, 1071, 1144, 1219, 1296, 1375, 1456, 1539, 1624, 1711, 1800, 1891, 1984, 2079, 2176, 2275, 2376, 2479, 2584, 2691, 2800, 2911, 3024, 3139, 3256, 3375, 3496, 3619

Offset: 0

Omar E. Pol, Aug 28 2007

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566.
Cf. A028569, A067079, A098849, A098850, A098603, A098847, A098848, A102928, A120071.
Cf. A132759, A132760, A132761, A132762, A132763, A132764, A132765, A132766, A132767.
Cf. A132768, A132769, A132770, A132771.

Table[n(n+30),{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,31,64},50] (* Harvey P. Dale, Mar 06 2015 *)

a(n)=n*(n+30) \\ Charles R Greathouse IV, Jun 17 2017

[n*(n+30) for n in (0..50)] # G. C. Greubel, Mar 13 2022

G.f.: x*(31-29*x)/(1-x)^3. - R. J. Mathar, Nov 14 2007
a(n) = 2*n + a(n-1) + 29 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010
a(0)=0, a(1)=31, a(2)=64, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Mar 06 2015
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(30)/30 = A001008(30)/A102928(30) = 9304682830147/69872686884000, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 225175759291/9981812412000. (End)
E.g.f.: x*(31 + x)*exp(x). - G. C. Greubel, Mar 13 2022

A067082 If n = abc...def in decimal notation then the right digit sum function = abc...def + bc...def + c...def + ... + def + ef + f.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 70, 72

Offset: 0

Amarnath Murthy, Jan 05 2002

			a(256) = 256 + 56 + 6 = 318.

Cf. A067079, A067080, A065039.

Table[d = IntegerDigits[n]; rd = 0; While[ Length[d] > 0, rd = rd + FromDigits[d]; d = Drop[d, 1]]; rd, {n, 0, 75} ]

a(abcd) = abcd+(abcd-1000a)+(abcd-1000a-100b)+(abcd-1000a-100b-10c).

n*length(n)-Sum_{k=1..length(n)} 10^k*floor(n/10^k). - Vladeta Jovovic, Jan 08 2002

More terms from Robert G. Wilson v, Jan 07 2002

A321243 a(n) is the product of n and all its decimal digits individually except the leftmost digit.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 24, 39, 56, 75, 96, 119, 144, 171, 0, 21, 44, 69, 96, 125, 156, 189, 224, 261, 0, 31, 64, 99, 136, 175, 216, 259, 304, 351, 0, 41, 84, 129, 176, 225, 276, 329, 384, 441, 0, 51, 104, 159, 216, 275, 336, 399, 464, 531

Offset: 0

Jacob Swales, Nov 14 2018

First differs from A067079 at n=102: A067079(102) = 408 != a(102) = 0.

			a(33) = 33*3 = 99. a(234) = 234*3*4 = 2808.

Jacob Swales, Table of n, a(n) for n = 0..10000

Cf. A067079.

a[n_] := n * If[n<10, 1, Times@@Rest[IntegerDigits[n]]]; Array[a, 100, 0] (* Amiram Eldar, Nov 15 2018 *)
Table[n Times@@Rest[IntegerDigits[n]],{n,0,60}] (* Harvey P. Dale, Mar 06 2023 *)

def a(n):
    seq_num = n
    for letter in range(len(str(n))):
        if letter != 0:
            seq_num = seq_num * int(str(n)[letter])
    return seq_num

A071714 Numbers n such that Rd(n) + Ld(n) +/-1 is prime, where Rd and Ld are the right- and left-digital factorial functions.

Original entry on oeis.org

2, 3, 6, 9, 33, 90, 102, 195, 210, 276, 379, 380, 402, 414, 575, 588, 616, 618, 916, 939, 980, 984, 1110, 1112, 1210, 1314, 1614, 2132, 2136, 2166, 2190, 2280, 2372, 2394, 2438, 2468, 2730, 2780, 3360, 3436, 3510, 3816, 3842, 3940, 3950, 4222, 4236, 4344

Offset: 1

Jason Earls, Jun 03 2002

			195 is a term because (195*95*5)+(195*19*1)+1 = 96331 and (195*95*5)+(195*19*1)-1 = 96329 are both prime.

Cf. A067079, A067080.

Showing 1-8 of 8 results.

cp's OEIS Frontend

A324322 Numbers k such that Ld(k) == k (mod Rd(k)), where Ld(k) = A067080 and Rd(k) = A067079.

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A324321 Numbers k such that Rd(k) == k (mod Ld(k)), where Rd(k) = A067079 and Ld(k) = A067080.

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A067080 If n = ab...def in decimal notation then the left digitorial function Ld(n) = ab...def*ab...de*ab...d*...*ab*a.

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A065039 If n in base 10 is d_1 d_2 ... d_k then a(n) = d_1 + d_1d_2 + d_1d_2d_3 + ... + d_1...d_k.

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A132772 a(n) = n*(n + 30).

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A067082 If n = abc...def in decimal notation then the right digit sum function = abc...def + bc...def + c...def + ... + def + ef + f.

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A321243 a(n) is the product of n and all its decimal digits individually except the leftmost digit.

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A067080 If n = ab...def in decimal notation then the left digitorial function Ld(n) = ab...defab...deab...d...ab*a.