A008597 -id:A008597 - 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)))}

A214855 Fibonacci numbers divisible by 10.

Original entry on oeis.org

0, 610, 832040, 1134903170, 1548008755920, 2111485077978050, 2880067194370816120, 3928413764606871165730, 5358359254990966640871840, 7308805952221443105020355490, 9969216677189303386214405760200, 13598018856492162040239554477268290

Offset: 1

Reinhard Zumkeller, Mar 08 2013

nonn

Fibonacci numbers having a trailing zero in decimal representation.

A122840(a(n)) > 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..250
Index entries for linear recurrences with constant coefficients, signature (1364, 1).

Cf. A008597.

a214855 = a000045 . a008597 . subtract 1

a(n) = A000045(15*(n-1)).

G.f.: -610*x^2 / (x^2+1364*x-1). - Colin Barker, Mar 12 2013

New name from T. D. Noe, Mar 09 2013

A274824 Triangle read by rows: T(n,k) = (n-k+1)*sigma(k), n>=1, 1<=k<=n.

Original entry on oeis.org

1, 2, 3, 3, 6, 4, 4, 9, 8, 7, 5, 12, 12, 14, 6, 6, 15, 16, 21, 12, 12, 7, 18, 20, 28, 18, 24, 8, 8, 21, 24, 35, 24, 36, 16, 15, 9, 24, 28, 42, 30, 48, 24, 30, 13, 10, 27, 32, 49, 36, 60, 32, 45, 26, 18, 11, 30, 36, 56, 42, 72, 40, 60, 39, 36, 12, 12, 33, 40, 63, 48, 84, 48, 75, 52, 54, 24, 28, 13, 36, 44, 70, 54, 96, 56, 90, 65, 72, 36, 56, 14

Offset: 1

Omar E. Pol, Oct 02 2016

Theorem: for any sequence S the partial sums of the partial sums are also the antidiagonal sums of the square array in which the n-th row gives n times the sequence S. Therefore they are also the row sums of the triangular array in which the n-th diagonal gives n times the sequence S.
In this case the sequence S is A000203.
The n-th diagonal of this triangle gives n times A000203.
The row sums give A175254 which gives the partial sums of A024916 which gives the partial sums of A000203.
T(n,k) is also the total number of unit cubes that are exactly below the terraces of the k-th level (starting from the top) up the base of the stepped pyramid with n levels described in A245092. This fact is because the mentioned terraces have the same shape as the symmetric representation of sigma(k). For more information see A237593 and A237270.
In the definition of this sequence the value n-k+1 is also the height of the terraces associated to sigma(k) in the mentioned pyramid with n levels, or in other words, the distance between the mentioned terraces and the base of the pyramid.
The sum of the n-th row of triangle equals the volume (also the number of cubes) of the mentioned pyramid with n levels.
For an illustration of the pyramid, see the Links section.
The sum of the n-th row is also 1/4 of the volume of the stepped pyramid described in A244050 with n levels.
Column k lists the positive multiples of sigma(k).
The k-th term in the n-th diagonal is equal to n*sigma(k).
Note that this is also a square array read by antidiagonals upwards: T(i,j) = i*sigma(j), i>=1, j>=1. The first row of the array is A000203. So consider that the pyramid is upside down. The value of "i" is the distance between the base of the pyramid and the terraces associated to sigma(j). The antidiagonal sums give the partial sums of the partial sums of A000203.

			Triangle begins:
1;
2,  3;
3,  6,  4;
4,  9,  8,  7;
5,  12, 12, 14, 6;
6,  15, 16, 21, 12, 12;
7,  18, 20, 28, 18, 24,  8;
8,  21, 24, 35, 24, 36,  16, 15;
9,  24, 28, 42, 30, 48,  24, 30,  13;
10, 27, 32, 49, 36, 60,  32, 45,  26,  18;
11, 30, 36, 56, 42, 72,  40, 60,  39,  36,  12;
12, 33, 40, 63, 48, 84,  48, 75,  52,  54,  24, 28;
13, 36, 44, 70, 54, 96,  56, 90,  65,  72,  36, 56,  14;
14, 39, 48, 77, 60, 108, 64, 105, 78,  90,  48, 84,  28, 24;
15, 42, 52, 84, 66, 120, 72, 120, 91,  108, 60, 112, 42, 48, 24;
16, 45, 56, 91, 72, 132, 80, 135, 104, 126, 72, 140, 56, 72, 48, 31;
...
For n = 16 and k = 10 the sum of the divisors of 10 is 1 + 2 + 5 + 10 = 18, and 16 - 10 + 1 = 7, and 7*18 = 126, so T(16,10) = 126.
On the other hand, the symmetric representation of sigma(10) has two parts of 9 cells, giving a total of 18 cells. In the stepped pyramid described in A245092, with 16 levels, there are 16 - 10 + 1 = 7 cubes exactly below every cell of the symmetric representation of sigma(10) up the base of pyramid; hence the total numbers of cubes exactly below the terraces of the 10th level (starting from the top) up the base of the pyramid is equal to 7*18 = 126. So T(16,10) = 126.
The sum of the 16th row of the triangle is 16 + 45 + 56 + 91 + 72 + 132 + 80 + 135 + 104 + 126 + 72 + 140 + 56 + 72 + 48 + 31 = A175254(16) = 1276, equaling the volume (also the number of cubes) of the stepped pyramid with 16 levels described in A245092 (see Links section).

Indranil Ghosh, Rows 1..100 of triangle, flattened
Omar E. Pol, Illustration of the stepped pyramid with 16 levels

Row sums of triangle give A175254.
Diagonals 1-6: A000203, A074400, A272027, A239050, A274535, A274536.
Column 1 is A000027.
Initial zeros should be omitted in the following sequences related to the columns of triangle:
Columns 2-5: A008585, A008586, A008589, A008588.
Columns 6 and 11: A008594.
Columns 7-9: A008590, A008597, A008595.
Columns 10 and 17: A008600.
Columns 12-13: A135628, A008596.
Columns 14, 15 and 23: A008606.
Columns 16 and 25: A135631.
(There are many other OEIS sequences that are also columns of this triangle.)
Cf. A004736, A024916, A054973, A196020, A236104, A235791, A237591, A237593, A237270, A237271, A244050, A245092, A245093, A262612.

T(n,k) = (n-k+1) * A000203(k).

T(n,k) = A004736(n,k) * A245093(n,k).

A129512 Numbers with at least two pairs of distinct divisors having equal differences.

Original entry on oeis.org

6, 12, 15, 18, 20, 24, 28, 30, 36, 40, 42, 45, 48, 54, 56, 60, 63, 66, 70, 72, 75, 78, 80, 84, 88, 90, 91, 96, 99, 100, 102, 105, 108, 110, 112, 114, 120, 126, 130, 132, 135, 138, 140, 144, 150, 153, 156, 160, 162, 165, 168, 174, 176, 180, 182, 186, 189, 190, 192, 195

Offset: 1

Reinhard Zumkeller, Apr 19 2007

nonn

			See example for a(12) = 45 in A129510.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A129510, A066446, A129511 (complement).

Subsequences: A008588, A008597, A129521, A259366.

import Data.List.Ordered (minus)
a129512 n = a129512_list !! (n-1)
a129512_list = minus [1..] a129511_list
-- Reinhard Zumkeller, Aug 10 2015

q[k_] := Count[Tally[Differences /@ Subsets[Divisors[k], {2}] // Flatten][[;; , 2]], ?(# > 1 &)] > 0; Select[Range[200], q] (* _Amiram Eldar, Jan 27 2025 *)

is(n)=my(d=divisors(n)); for(i=1,#d-2, for(j=i+1,#d-1, for(k=1,#d, if(i!=k && setsearch(d, d[j]-d[i]+d[k]), return(1))))); 0 \\ Charles R Greathouse IV, Aug 26 2015

A129510(a(n)) < A066446(a(n)).

A317315 Multiples of 15 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 15, 3, 30, 5, 45, 7, 60, 9, 75, 11, 90, 13, 105, 15, 120, 17, 135, 19, 150, 21, 165, 23, 180, 25, 195, 27, 210, 29, 225, 31, 240, 33, 255, 35, 270, 37, 285, 39, 300, 41, 315, 43, 330, 45, 345, 47, 360, 49, 375, 51, 390, 53, 405, 55, 420, 57, 435, 59, 450, 61, 465, 63, 480, 65, 495, 67, 510, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 19-gonal numbers (A303813).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 19-gonal numbers.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008597 and A005408 interleaved.
Column 15 of A195151.
Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 (k=13), A195817 (k=14).
Cf. A303813.

a[n_] := If[OddQ[n], n, 15*n/2]; Array[a, 70, 0] (* Amiram Eldar, Oct 14 2023 *)

concat(0, Vec(x*(1 + 15*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

a(2n) = 15*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 15*x + x^2) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
Multiplicative with a(2^e) = 15*2^(e-1), and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 13/2^s). - Amiram Eldar, Oct 25 2023

A319073 Square array read by antidiagonals upwards: T(n,k) = k*sigma(n), n >= 1, k >= 1.

Original entry on oeis.org

1, 3, 2, 4, 6, 3, 7, 8, 9, 4, 6, 14, 12, 12, 5, 12, 12, 21, 16, 15, 6, 8, 24, 18, 28, 20, 18, 7, 15, 16, 36, 24, 35, 24, 21, 8, 13, 30, 24, 48, 30, 42, 28, 24, 9, 18, 26, 45, 32, 60, 36, 49, 32, 27, 10, 12, 36, 39, 60, 40, 72, 42, 56, 36, 30, 11, 28, 24, 54, 52, 75, 48, 84, 48, 63, 40, 33, 12

Offset: 1

Omar E. Pol, Sep 22 2018

			The corner of the square array begins:
         A000203 A074400 A272027 A239050 A274535 A274536 A319527 A319528
A000027:       1,      2,      3,      4,      5,      6,      7,      8, ...
A008585:       3,      6,      9,     12,     15,     18,     21,     24, ...
A008586:       4,      8,     12,     16,     20,     24,     28,     32, ...
A008589:       7,     14,     21,     28,     35,     42,     49,     56, ...
A008588:       6,     12,     18,     24,     30,     36,     42,     48, ...
A008594:      12,     24,     36,     48,     60,     72,     84,     96, ...
A008590:       8,     16,     24,     32,     40,     48,     56,     64, ...
A008597:      15,     30,     45,     60,     75,     90,    105,    120, ...
A008595:      13,     26,     39,     52,     65,     78,     91,    104, ...
A008600:      18,     36,     54,     72,     90,    108,    126,    144, ...
...

Index entries for sequences related to sigma(n)

Another version of A274824.
Antidiagonal sums give A175254.
Main diagonal gives A064987.
Row n lists the multiples of A000203(n).
Row 1 is A000027.
Initial zeros should be omitted in the following sequences related to the rows of the array:
Row 2-5: A008585, A008586, A008589, A008588.
Rows 6 and 11: A008594.
Rows 7-9: A008590, A008597, A008595.
Rows 10 and 17: A008600.
Rows 12-13: A135628, A008596.
Rows 14, 15 and 23: A008606.
Rows 16 and 25: A135631.
(Note that in the OEIS there are many other sequences that are also rows of this square array.)
Cf. A000203, A237593, A319526.

T:=Flat(List([1..12],n->List([1..n],k->k*Sigma(n-k+1))));; Print(T); # Muniru A Asiru, Jan 01 2019

with(numtheory): T:=(n,k)->k*sigma(n-k+1): seq(seq(T(n,k),k=1..n),n=1..12); # Muniru A Asiru, Jan 01 2019

Table[k DivisorSigma[1, #] &[m - k + 1], {m, 12}, {k, m}] // Flatten (* Michael De Vlieger, Dec 31 2018 *)

A139616 a(n) = 120*n + 16.

Original entry on oeis.org

16, 136, 256, 376, 496, 616, 736, 856, 976, 1096, 1216, 1336, 1456, 1576, 1696, 1816, 1936, 2056, 2176, 2296, 2416, 2536, 2656, 2776, 2896, 3016, 3136, 3256, 3376, 3496, 3616, 3736, 3856, 3976, 4096, 4216, 4336, 4456, 4576, 4696, 4816

Offset: 0

Omar E. Pol, Apr 27 2008

Numbers of the 16th column of positive numbers in the square array of nonnegative and polygonal numbers A139600. Also, numbers of the 16th column in the square array A057145.

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008597, A057145, A139600.

[8*(15*n + 2): n in [0..60]]; // Vincenzo Librandi, Jul 23 2011

a(n)=120*n+16 \\ Charles R Greathouse IV, Oct 05 2011

a(n) = 2*a(n-1)-a(n-2). - Wesley Ivan Hurt, Jun 17 2021
From Elmo R. Oliveira, Apr 04 2024: (Start)
G.f.: 8*(2+13*x)/(x-1)^2.
E.g.f.: 8*exp(x)*(2 + 15*x).
a(n) = 8*(A008597(n) + 2). (End)

A164284 a(n) = 15*n-7.

Original entry on oeis.org

8, 23, 38, 53, 68, 83, 98, 113, 128, 143, 158, 173, 188, 203, 218, 233, 248, 263, 278, 293, 308, 323, 338, 353, 368, 383, 398, 413, 428, 443, 458, 473, 488, 503, 518, 533, 548, 563, 578, 593, 608, 623, 638, 653, 668, 683, 698, 713, 728, 743, 758, 773, 788

Offset: 1

Vincenzo Librandi, Aug 12 2009

Alternative definition: numbers m such that (13*m+1)/15 is an integer.

The previous comment is the case t=0 of ((13-t*(t+1))*m + t*(t+1)/2 + 1)/15, where t = 0, 1, 2 or 3. - Bruno Berselli, Feb 22 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008597, A139606.

Cf. similar sequences with closed form (2*k-1)*n+k listed in A269044.

Range[8, 1000, 15] (* Vladimir Joseph Stephan Orlovsky, May 31 2011 *)
LinearRecurrence[{2,-1},{8,23},60] (* Harvey P. Dale, Dec 25 2016 *)

x='x+O('x^50); Vec(x*(8+7*x)/(x-1)^2) \\ G. C. Greubel, Sep 12 2017

def a(n): return 15*n - 7
print([a(n) for n in range(1, 54)]) # Michael S. Branicky, Aug 23 2021

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

E.g.f.: 7 + (15*x - 7)*exp(x). - G. C. Greubel, Sep 12 2017

Definition simplified by R. J. Mathar, Aug 21 2009

A169825 Multiples of 420.

Original entry on oeis.org

0, 420, 840, 1260, 1680, 2100, 2520, 2940, 3360, 3780, 4200, 4620, 5040, 5460, 5880, 6300, 6720, 7140, 7560, 7980, 8400, 8820, 9240, 9660, 10080, 10500, 10920, 11340, 11760, 12180, 12600, 13020, 13440, 13860, 14280, 14700, 15120, 15540, 15960, 16380, 16800

Offset: 0

N. J. A. Sloane, May 30 2010

Numbers that are divisible by all of 1,2,3,4,5,6,7.

Erich Friedman, What's Special About This Number? (See the entry for "420")
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A001477, A008589, A008596, A008597, A008602, A008603.

Cf. A135628, A169823, A169827, A249674.

Range[0, 100]*420 (* Paolo Xausa, Apr 17 2024 *)

a(n)=420*n \\ Charles R Greathouse IV, Apr 16 2024

a(n) = 420*n. - Wesley Ivan Hurt, Apr 11 2021
From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 420*x/(x-1)^2.
E.g.f.: 420*x*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 7*A169823(n) = 14*A249674(n) = 15*A135628(n) = 20*A008603(n) = 21*A008602(n) = 28*A008597(n) = 30*A008596(n) = 60*A008589(n) = 420*A001477(n) = A169827(n)/2. (End)

A380820 a(0) = 0, a(1) = 1, and for n >= 2, a(n) = a(n-1) + a(n-2) if a(n-1) < n, otherwise a(n-1) - n.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 1, 9, 0, 9, 9, 18, 5, 23, 8, 31, 14, 45, 26, 6, 32, 10, 42, 18, 60, 34, 7, 41, 12, 53, 22, 75, 42, 8, 50, 14, 64, 26, 90, 50, 9, 59, 16, 75, 30, 105, 58, 10, 68, 18, 86, 34, 120, 66, 11, 77, 20, 97, 38, 135, 74, 12, 86, 22, 108, 42, 150

Offset: 0

Ya-Ping Lu, Feb 04 2025

Sequence starts with the first 7 Fibonacci numbers. For n >= 12, a(n) takes the values of (8*n+30)/7, (n+22)/7, (9*n+35)/7, (2*n+26)/7, (11*n+41)/7, (4*n+30)/7, and (15*n+45)/7 sequentially for n = 5, 6, 0, 1, 2, 3, 4 mod 7 (see plot in Links), which correspond to A017089 (n>=2), A000027 (n>=5), A017221 (n>=2), A005843 (n>=4), A017497 (n>=2), A016825 (n>=3), and A008597 (n>=3), respectively.

Terms for n >= 16 are the same as A322558(n) for n >= 17.

Ya-Ping Lu, Plot of A380820
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,2,0,0,0,0,0,0,-1).

Cf. A000027, A000045, A005843, A008597, A016825, A017089, A017221, A017497, A322558.

s={0,1};Do[AppendTo[s,If[s[[-1]]James C. McMahon, Feb 14 2025 *)

def A380820(n): R = [0, 1, 1, 2, 3, 5, 8, 1, 9, 0, 9, 9]; X = [9, 2, 11, 4, 15, 8, 1]; Y = [35, 26, 41, 30, 45, 30, 22]; return R[n] if n < 12 else (X[n%7]*n + Y[n%7])//7

a(n) = A322558(n+1) for n >= 16.

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

A214855 Fibonacci numbers divisible by 10.

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Extensions

A274824 Triangle read by rows: T(n,k) = (n-k+1)*sigma(k), n>=1, 1<=k<=n.

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A129512 Numbers with at least two pairs of distinct divisors having equal differences.

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Haskell

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A317315 Multiples of 15 and odd numbers interleaved.

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Mathematica

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A319073 Square array read by antidiagonals upwards: T(n,k) = k*sigma(n), n >= 1, k >= 1.

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Maple

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A139616 a(n) = 120*n + 16.

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Magma

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A164284 a(n) = 15*n-7.

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