A008593 -id:A008593 - cp's OEIS frontend

A212773 Amounts (in cents) of coins in denominations 1, 5, 10, 25, and 50 (cents) which can consist of equal numbers of coins of all denominations present when two or more denominations are used (or none are used: term 0).

0, 6, 11, 12, 15, 16, 18, 22, 24, 26, 30, 31, 32, 33, 35, 36, 40, 41, 42, 44, 45, 48, 51, 52, 54, 55, 56, 60, 61, 62, 64, 65, 66, 70, 72, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 91, 93, 96, 99, 102, 104, 105, 108, 110, 112, 114, 120, 121, 122, 123, 124

Offset: 1

Rick L. Shepherd, May 26 2012

nonn

Nonnegative multiples of each of 6, 11, 15, 16, 26, 31, 35, 40, 41, 51, 56, 61, 65, 76, 81, 85, 86, and 91.

All products of terms are terms.

			4 is not a term because it is not an appropriate multiple. Also 4 = 4*1 cannot be represented with more than one denomination of coin. Similarly 7 is not a term; although 7 = 7*1 = 2*1 + 1*5 does have a representation in terms of two denominations, 1 and 5, there are unequal numbers of each.
a(11) = 30 is a term because it is a multiple of 6. 30 = 5*1 + 5*5 = 2*5 + 2*10 = 1*5 + 1*25, so five coins each of denominations 1 and 5, two each of 5 and 10, or one each of 5 and 25 totals 30.
The term 34924118340711600 (5 times the LCM of the numbers in the first comment, so also divisible by 75) is the smallest which can be expressed in 26 such ways, one for each possible combination of two or more of these five coin denominations. (It also can be expressed as a multiple of each of these five alone of course.)

Rick L. Shepherd, Table of n, a(n) for n = 1..10000
Index entries for sequences related to making change.

Cf. A212774, A008588, A008593, A008597, A008598, A135631.

{c = 0; n = -1; until(c==10000, n++;
if(n%6==0 || n%11==0 || n%15==0 || n%16==0 || n%26==0 ||
  n%31==0 || n%35==0 || n%40==0 || n%41==0 || n%51==0 ||
  n%56==0 || n%61==0 || n%65==0 || n%76==0 || n%81==0 ||
  n%85==0 || n%86==0 || n%91==0,
  c++; write("b212773.txt", c, " ", n)))}

A220653 a(n) = n^11 + 11*n + 11^n.

Original entry on oeis.org

1, 23, 2191, 178511, 4208989, 48989231, 364568683, 1996813991, 8804293561, 33739007399, 125937424711, 570623341343, 3881436747541, 36314872538111, 383799398753059, 4185897925275191, 45967322049616753, 505481300395601591, 5559981581902310911, 61159206938673444719

Offset: 0

Jonathan Vos Post, Dec 17 2012

This is to A220425 as 11 is to 2, to A220509 as 11 is to 3, to A220511 as 11 is to 5, and to A220528 as 11 is to 7.

The subsequence of primes begins: 23, 4185897925275191, see A220787 for the associated n.

			a(1) = 1^11 + 11*1 + 11^1 = 23.

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Cf. A008455, A008593, A001020, A220425, A220509, A220511, A220528, A220787.

Table[n^11 + 11*n + 11^n, {n, 0, 30}] (* T. D. Noe, Dec 17 2012 *)

A220653(n):=n^11+11*n+11^n$ makelist(A220653(n),n,0,20); /* Martin Ettl, Dec 17 2012 */

a(n) = A008455(n) + A008593(n) + A001020(n).

G.f.: (131*x^12 +21186*x^11 +1682460*x^10 +24070936*x^9 +104942001*x^8 +163196604*x^7 +91422264*x^6 +14484216*x^5 -518211*x^4 -131726*x^3 -1860*x^2 -1) / ((x -1)^12*(11*x -1)). - Colin Barker, May 09 2013

a(19) from Stefano Spezia, May 03 2025

A265187 Nonnegative m for which 2floor(m^2/11) = floor(2m^2/11).

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, 16, 17, 18, 20, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 64, 65, 66, 67, 68, 70, 71, 72, 73, 75, 76, 77, 78, 79, 81, 82, 83, 84

Offset: 1

Bruno Berselli, Dec 04 2015

Also, nonnegative m not congruent to 3 or 8 (mod 11).
Integers x >= 0 satisfying k*floor(x^2/11) = floor(k*x^2/11) with k >= 0:
k = 0, 1:  x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...   (A001477);
k = 2:     x = 0, 1, 2, 4, 5, 6, 7, 9, 10, 11, 12, 13, ... (this sequence);
k = 3:     x = 0, 1, 5, 6, 10, 11, 12, 16, 17, 21, 22, ... (A265188);
k = 4..10: x = 0, 1, 10, 11, 12, 21, 22, 23, 32, 33, ...   (A112654);
k > 10:    x = 0, 11, 22, 33, 44, 55, 66, 77, 88, 99, ...  (A008593).
Primes in sequence: 2, 5, 7, 11, 13, 17, 23, 29, 31, 37, 43, 53, 59, ...

Bruno Berselli, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,1,-1).

Cf. A008593, A112654, A265188.

Cf. similar sequences provided by 2*floor(m^2/h) = floor(2*m^2/h): A005843 (h=2), A001477 (h=3,4), A008854 (h=5), A047266 (h=6), A047299 (h=7), A042965 (h=8), A060464 (h=9), A237415 (h=10), this sequence (h=11), A047263 (h=12).

[n: n in [0..100] | 2*Floor(n^2/11) eq Floor(2*n^2/11)];

Select[Range[0, 100], 2 Floor[#^2/11] == Floor[2 #^2/11] &]
Select[Range[0, 100], ! MemberQ[{3, 8}, Mod[#, 11]] &]
LinearRecurrence[{1, 0, 0, 0, 0, 0, 0, 0, 1, -1}, {0, 1, 2, 4, 5, 6, 7, 9, 10, 11}, 80]

is(n)=2*(n^2\11) == (2*n^2)\11 \\ Anders Hellström, Dec 05 2015

[n for n in (0..100) if 2*floor(n^2/11) == floor(2*n^2/11)]

G.f.: x^2*(1 + x + 2*x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + x^8)/((1 - x)^2*(1 + x + x^2)*(1 + x^3 + x^6)).

a(n) = a(n-1) + a(n-9) - a(n-10) for n>10.

A337940 Triangle read by rows: T(n, k) = T(n+2) - T(n-k), with the triangular numbers T = A000217, for n >= 1, k = 1, 2, ..., n.

Original entry on oeis.org

6, 9, 10, 12, 14, 15, 15, 18, 20, 21, 18, 22, 25, 27, 28, 21, 26, 30, 33, 35, 36, 24, 30, 35, 39, 42, 44, 45, 27, 34, 40, 45, 49, 52, 54, 55, 30, 38, 45, 51, 56, 60, 63, 65, 66, 33, 42, 50, 57, 63, 68, 72, 75, 77, 78, 36, 46, 55, 63, 70, 76, 81, 85, 88, 90, 91

Offset: 1

Wolfdieter Lang, Nov 23 2020

This number triangle results from the array A(n, m) = T(n+m+1) - T(n-1), with T = A000217, for n, m >= 1. For this array see the example by Bob Selcoe, in A111774 (but with rows continued). The present triangle is obtained by reading the array by upwards antidiagonals: T(n, k) = A(n+1-k, k). See also the Jul 09 2019 comment by Ralf Steiner with the formula c_k(n) (rows k >= 1, columns n >= 3), rewritten for A(n, m) = (m+2)*(2*n+m+1)/2, leading to T(n, k) = (k+2)*(2*n-k+3)/2.
Therefore this triangle is related to the problem of giving the numbers which are sums of at least three consecutive positive integers given as sequence A111774. It allows us to find the multiplicities for the numbers of A111774. They are given in A338428(n).
To obtain the multiplicity for number N (>= 6) from A111774 one has to consider only the triangle rows n = 1, 2, ..., floor((N-3)/3).
The row reversed triangle, considered by Bob Selcoe in A111774, is T(n, n-k+1) = T(n+2) - T(k-1), for n >= 1, and k=1, 2, ..., n.
This triangle contains no odd prime numbers and no exact powers 2^m, for m >= 0. This can be seen by considering the diagonal sequences D(d, k), for d >= 1, k >= 1 or the row sequences of the array A(n, m), for n >= 1 and m >= 1. The result is A(r+1, s-2) = s*(s + 2*r + 1)/2, for r >= 0 and s >= 3 (from the g.f. of the diagonals of T given below). This is also given in the Jul 09 2019 comment by Ralf Steiner in A111774. Therefore A(r+1, s-2) is a product of two numbers >= 2, hence not a prime. And in both cases (i) s/2 integer or (ii) (s + 2*r + 1)/2 integer not both numbers can be powers of 2 by simple parity arguments.
The previous comment means that each T(n, k) has at least one odd prime as a proper divisor.
A number N appears in this triangle, or in A111774, if and only if floor(N/2) - delta(N) >= 1, where delta(N) = A055034(N). For the sequence b(n) := floor(n/2) - delta(n), for n >= 2, see A219839(n), b(1) = -1. See a W. Lang comment in A111774 for the proof.

			The triangle T(n, k) begins:
n \ k  1  2  3  4  5   6   7   8   9  10  11  12  13  14  15 ...
1:     6
2:     9 10
3:    12 14 15
4:    15 18 20 21
5:    18 22 25 27 28
6:    21 26 30 33 35  36
7:    24 30 35 39 42  44  45
8:    27 34 40 45 49  52  54  55
9:    30 38 45 51 56  60  63  65  66
10:   33 42 50 57 63  68  72  75  77  78
11:   36 46 55 63 70  76  81  85  88  90  91
12:   39 50 60 69 77  84  90  95  99 102 104 105
13:   42 54 65 75 84  92  99 105 110 114 117 119 120
14:   45 58 70 81 91 100 108 115 121 126 130 133 135 136
15:   48 62 75 87 98 108 117 125 132 138 143 147 150 152 153
...
N = 15 appears precisely twice from the sums 4+5+6 = A(4, 1) = T(4, 1), and (1+2+3)+4+5 = A(1, 3) = T(3, 3), i.e., with a sum of 3 and 5 consecutive positive integers.
N = 42 appears three times from the sums 13+14+15 = A(13, 1) = T(13, 1), 9+10+11 +12 = A(9, 2) = T(10, 2), 3+4+5+6+7+8+9 = A(3, 5) = T(7, 5); i.e., 42 can be written as a sum of 3, 4 and 7 consecutive positive integers.

Cf. A055034, A111774, A338428 (multiplicities), A219839.
For columns k = 1, 2, ..., 10 see A008585, A016825, A008587, A016945, A008589, A017113, A008591, A017329, A008593, A017593.
For diagonals d = 1, 2, ..., 10 see A000217, A000096, A055998, A055999, A056000, A056115, A056119 , A056121, A056126, A051942.

Flatten[Table[((n+2)*(n+3)-(n-k)*(n-k+1))/2,{n,11},{k,n}]] (* Stefano Spezia, Nov 24 2020 *)

T(n, k) = ((n+2)*(n+3) - (n-k)*(n-k+1))/2, for n >= 1 and k = 1, 2, ..., n (see the name).
T(n, k) = (k+2)*(2*n-k+3)/2 (factorized).
G.f. columns k = 2*j+1, for j >= 0: Go(j, x) = x^(2*j+1)*(2*j+3)*(j+2 - (j+1)*x)/(1-x)^2,
G.f. columns k = 2*j, for j >= 1: Ge(j, x) = x^(2*j)*(j+1)*(2*j+3 - (2*j+1)*x)/(1-x)^2.
G.f. row polynomials: G(z,x) = z*x*(1 + z*x)^3*{3*(2-z) - (8-3*z)*(z*x) + (3-z)*(z*x)^2}/((1 - z)^2*(1 - (z*x)^2)^3).
G.f. diagonals d >= 1: GD(d, x) = ((d+1)*3 - (5*d+3)*x + (2*d+1)*x^2)/(1-x)^3.
G.f. of GD(d, x): GGD(z,x) = (6-8*x+3*x^2 - (3-3*x+x^2)*z)/((1-x)^3*(1-z)^2).

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

A060979 |First digit - second digit + third digit - fourth digit ...| = 11.

Original entry on oeis.org

209, 308, 319, 407, 418, 429, 506, 517, 528, 539, 605, 616, 627, 638, 649, 704, 715, 726, 737, 748, 759, 803, 814, 825, 836, 847, 858, 869, 902, 913, 924, 935, 946, 957, 968, 979, 1309, 1408, 1419, 1507, 1518, 1529, 1606, 1617, 1628, 1639, 1705, 1716

Offset: 1

Robert G. Wilson v, May 10 2001

Note that all terms are divisible by eleven.

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

Cf. A008593, A060978-A060980, A060982, A061470-A061479, A061870-A061882.

Cf. A135499.

a060979 n = a060979_list !! (n-1)
a060979_list = filter (\x -> let digs = map (read . return) $ show x in
                             evens digs /= odds digs) [11, 22 ..]
   where evens [] = 0; evens [x] = x; evens (x:_:xs) = x + evens xs
         odds [] = 0; odds [x] = 0; odds (_:x:xs) = x + odds xs
-- Reinhard Zumkeller, Jul 05 2014

filter:= proc(n) local L,i;
  L:= convert(n,base,10);
  abs(add(L[i]*(-1)^i,i=1..nops(L))) = 11
end proc:
select(filter, [$1..1000] *~ 11); # Robert Israel, Jun 02 2023

Do[ a = IntegerDigits[ n ]; l = Length[ a ]; e = o = {}; Do[ o = Append[ o, a[ [ 2k - 1 ] ] ], {k, 1, l/2 + .5} ]; Do[ e = Append[ e, a[ [ 2k ] ] ], {k, 1, l/2} ]; If[ Abs[ Apply[ Plus, o ] - Apply[ Plus, e ] ] == 11, Print[ n ] ], {n, 1, 2000} ]
d11Q[n_]:=Module[{idn=IntegerDigits[n]},Abs[Total[Table[(-1)^(i+1) idn[[i]],{i,Length[idn]}]]]==11]; Select[Range[1800],d11Q] (* Harvey P. Dale, Aug 26 2012 *)

Erroneous comment deleted by Robert Israel, Jun 02 2023

A061478 First (leftmost) digit - second digit + third digit - fourth digit .... = 9.

Original entry on oeis.org

9, 90, 108, 119, 207, 218, 229, 306, 317, 328, 339, 405, 416, 427, 438, 449, 504, 515, 526, 537, 548, 559, 603, 614, 625, 636, 647, 658, 669, 702, 713, 724, 735, 746, 757, 768, 779, 801, 812, 823, 834, 845, 856, 867, 878, 889, 900, 911, 922, 933, 944, 955

Offset: 1

Amarnath Murthy, May 05 2001

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

Cf. A008593, A060978-A060980, A060982, A061470-A061479, A061870-A061882.

Select[Range[1000],Total[Times@@@Partition[Riffle[IntegerDigits[#],{1,-1},{2,-1,2}],2]]==9&] (* Harvey P. Dale, Jul 16 2017 *)

More terms from Robert G. Wilson v, May 10 2001 and from Larry Reeves (larryr(AT)acm.org), May 15 2001

A067042 Numbers in which the product of digits in even positions = product of digits in odd positions.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 100, 111, 122, 133, 144, 155, 166, 177, 188, 199, 200, 221, 242, 263, 284, 300, 331, 362, 393, 400, 441, 482, 500, 551, 600, 661, 700, 771, 800, 881, 900, 991, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009

Offset: 1

Amarnath Murthy, Dec 29 2001

			2364 is a member as 2*6 = 3*4.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A008593 (similar for sums).

Select[Range[1010], Product[Part[(digits=IntegerDigits[#]), 2i], {i, Floor[(len=IntegerLength[#])/2]}] == Product[Part[digits, 2i-1], {i,Ceiling[len/2]}] &] (* Stefano Spezia, Jan 05 2025 *)

from math import prod
def ok(n):
    s = str(n)
    return s != '1' and prod(map(int, s[::2])) == prod(map(int, s[1::2]))
print([k for k in range(1010) if ok(k)]) # Michael S. Branicky, Nov 22 2021

Asymptotics: For any n, let f(n) be the number of entries in this sequence that are less than n. Then f(n)/n approaches 1 as n goes to infinity. This is because among numbers with a large number of digits, almost all have 0's in both odd positions and even positions. - David Wasserman, Jan 16 2002

Corrected by David Wasserman, Jan 16 2002

More terms from Sascha Kurz, Mar 23 2002

A080466 Multiples of 11 in which the even positioned digits from left are even and the odd positioned ones are odd.

Original entry on oeis.org

121, 143, 165, 187, 341, 363, 385, 561, 583, 781, 1012, 1034, 1056, 1078, 1210, 1232, 1254, 1276, 1298, 1430, 1452, 1474, 1496, 1650, 1672, 1694, 1870, 1892, 3014, 3036, 3058, 3212, 3234, 3256, 3278, 3410, 3432, 3454, 3476, 3498, 3630, 3652, 3674, 3696

Offset: 1

Amarnath Murthy, Mar 02 2003

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

Intersection of A008593 and A376692.

Cf. A080467.

filter:= proc(n) local L,L1,L2;
L:= convert(n,base,10);
L1:= {seq(L[-i],i=2..nops(L),2)};
L2:= {seq(L[-i],i=1..nops(L),2)};
andmap(type,L1,even) and andmap(type,L2,odd)
end proc:
select(filter, [seq(i,i=11..10000,11)]); # Robert Israel, Mar 06 2018

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003

A083851 Decimal palindromes that are not multiples of 11.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 101, 111, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 494, 505, 515, 525, 535, 545, 555, 565, 575, 585, 595, 606

Offset: 1

Reinhard Zumkeller, May 06 2003

A083850(a(n)) = 0; all members have odd length.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A083852, A002113, A008593.

max = 600; Complement[Select[Range[max], IntegerDigits[#] == Reverse[IntegerDigits[#]] &], 11Range[Ceiling[max/11]]] (* Alonso del Arte, May 11 2014 *)

is(n)=n%11 && subst(Polrev(digits(n)), 'x, 10)==n \\ Charles R Greathouse IV, May 14 2014

from itertools import chain, count, islice
def A083851_gen(): # generator of terms
    return filter(lambda n: n % 11,chain.from_iterable((int((s:=str(d))+s[-2::-1]) for d in range(10**l,10**(l+1))) for l in count(0)))
A083851_list = list(islice(A083851_gen(),20)) # Chai Wah Wu, Jun 23 2022

cp's OEIS Frontend

A212773 Amounts (in cents) of coins in denominations 1, 5, 10, 25, and 50 (cents) which can consist of equal numbers of coins of all denominations present when two or more denominations are used (or none are used: term 0).

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PARI

A220653 a(n) = n^11 + 11*n + 11^n.

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Maxima

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A265187 Nonnegative m for which 2*floor(m^2/11) = floor(2*m^2/11).

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A337940 Triangle read by rows: T(n, k) = T(n+2) - T(n-k), with the triangular numbers T = A000217, for n >= 1, k = 1, 2, ..., n.

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A059632 Carryless product 11 X n base 10.

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A060979 |First digit - second digit + third digit - fourth digit ...| = 11.

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Maple

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A061478 First (leftmost) digit - second digit + third digit - fourth digit .... = 9.

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A067042 Numbers in which the product of digits in even positions = product of digits in odd positions.

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A265187 Nonnegative m for which 2floor(m^2/11) = floor(2m^2/11).