A008600 -id:A008600 - cp's OEIS frontend

A317318 Multiples of 18 and odd numbers interleaved.

0, 1, 18, 3, 36, 5, 54, 7, 72, 9, 90, 11, 108, 13, 126, 15, 144, 17, 162, 19, 180, 21, 198, 23, 216, 25, 234, 27, 252, 29, 270, 31, 288, 33, 306, 35, 324, 37, 342, 39, 360, 41, 378, 43, 396, 45, 414, 47, 432, 49, 450, 51, 468, 53, 486, 55, 504, 57, 522, 59, 540, 61, 558, 63, 576, 65, 594, 67, 612, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 22-gonal numbers (A303299).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 22-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. A008600 and A005408 interleaved.
Column 18 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. A303299.

a[n_] := If[OddQ[n], n, 9*n]; Array[a, 70, 0] (* Amiram Eldar, Oct 14 2023 *)

concat(0, Vec(x*(1 + 18*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

a(2n) = 18*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 18*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) = 9*2^e, and a(p^e) = p^e for an odd prime p. - Amiram Eldar, Oct 14 2023
Dirichlet g.f.: zeta(s-1) * (1 + 2^(4-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 *)

A373599 Numbers k such that k and A327860(k) are both multiples of 3, where A327860 is the arithmetic derivative of the primorial base exp-function.

Original entry on oeis.org

0, 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 222, 240, 258, 276, 294, 312, 330, 348, 366, 384, 402, 426, 444, 462, 480, 498, 516, 534, 552, 570, 588, 606, 624, 630, 648, 666, 684, 702, 720, 738, 756, 774, 792, 810, 828, 852, 870, 888, 906, 924, 942, 960, 978, 996, 1014, 1032, 1056, 1074, 1092, 1110, 1128

Offset: 1

Antti Karttunen, Jun 18 2024

nonn

If x and y are terms and if A329041(x,y) = 1 (i.e., when adding x and y together will not generate any carries in the primorial base), then x+y is also a term. This follows from the quasi-exponential nature of A276086 and because A373144 is a multiplicative semigroup.

			18 = 3*6 is included, because also A327860(18) = 75 is a multiple of 3.
222 = 3*74 is included, because also A327860(222) = 135 is a multiple of 3.
240 = 3*80 is included, because also A327860(240) = 18 is a multiple of 3.
258 = 3*86 is included, because also A327860(258) = 8025 is a multiple of 3. Note that A049345(18) = 300, A049345(240) = 11000, and A049345(240+18) = 11300, so the sum in this case is carry-free (cf. the comment).
2556 = 3*852 is included, because also A327860(2556) = 2556 is a multiple of 3 (see also A328110 and A373144).

Antti Karttunen, Table of n, a(n) for n = 1..20020
Index entries for sequences related to primorial base

Cf. A049345, A276086, A327860, A329041, A373598 (characteristic function).
Indices of multiples of 3 in A351083.
Intersection of A008585 and A369654.
Differs from A008600 (multiples of 18) for the first time at a(13) = 222, which is not a multiple of 18.
Cf. also A373144.

PARI
```
isA373599 = A373598;
```

A158910 First Differences of A061240.

Original entry on oeis.org

18, 18, 18, 54, 18, 18, 18, 72, 18, 36, 18, 36, 36, 18, 18, 72, 18, 54, 36, 18, 36, 90, 18, 36, 90, 54, 36, 36, 18, 18, 54, 90, 36, 54, 18, 18, 54, 36, 18, 54, 18, 54, 18, 36, 90, 36, 54, 36, 54, 36, 18, 18, 54, 36, 72, 18, 108, 36, 36, 72, 18, 18, 126, 36, 54, 54, 54, 36, 126

Offset: 1

Paul Curtz, Mar 30 2009

Since A061240 contains prime numbers of the form 9k+5, k even, all numbers here are multiples of 18.

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

Cf. A008600.

Differences[Select[Prime[Range[500]],Mod[#,9]==5&]] (* Harvey P. Dale, May 19 2018 *)

Edited and extended by R. J. Mathar, Apr 03 2009

A194272 Array T(n,k) with 6 columns read by rows in which row n lists 3n-2, 3n-1, 3n, 3n, 3n, 3n.

Original entry on oeis.org

1, 2, 3, 3, 3, 3, 4, 5, 6, 6, 6, 6, 7, 8, 9, 9, 9, 9, 10, 11, 12, 12, 12, 12, 13, 14, 15, 15, 15, 15, 16, 17, 18, 18, 18, 18, 19, 20, 21, 21, 21, 21, 22, 23, 24, 24, 24, 24, 25, 26, 27, 27, 27, 27, 28, 29, 30, 30, 30, 30, 31, 32, 33, 33, 33, 33, 34, 35, 36, 36, 36, 36

Offset: 1

Omar E. Pol, Aug 20 2011

Also first differences of A194273 which is also a sequence related to cellular automata.

			Array begins:
1,  2,  3,  3,  3,  3,
4,  5,  6,  6,  6,  6,
7,  8,  9,  9,  9,  9,
10, 11, 12, 12, 12, 12,
13, 14, 15, 15, 15, 15,
16, 17, 18, 18, 18, 18,
19, 20, 21, 21, 21, 21,
22, 23, 24, 24, 24, 24,
...
Sum of row n gives 18*n-3 = A008600(n) - 3.
G.f. = x + 2*x^2 + 3*x^3 + 3*x^4 + 3*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to cellular automata
Index entries for linear recurrences with constant coefficients, signature (2,-1,-1,2,-1).

Column 1: A016777. Column 2: A016789. Every column 3, 4, 5 and 6: positive integers of A008585.

Cf. A008600, A047926, A132868, A194273.

[Floor((n+3)/6) + Floor((n+4)/6) + Floor((n+5)/6) : n in [1..100]]; // Wesley Ivan Hurt, Apr 04 2015

A194272:=n->floor((n+3)/6) + floor((n+4)/6) + floor((n+5)/6): seq(A194272(n), n=1..100); # Wesley Ivan Hurt, Apr 04 2015

Table[Floor[(n + 3)/6] + Floor[(n + 4)/6] + Floor[(n + 5)/6], {n, 100}] (* Wesley Ivan Hurt, Apr 04 2015 *)

x='x+O('x^60); Vec(x*(1-x^3)/((1-x)^2*(1-x^6))) \\ G. C. Greubel, Aug 13 2018

From Michael Somos, May 12 2014: (Start)
Euler transform of length 6 sequence [2, 0, -1, 0, 0, 1].
G.f.: x * (1-x^3) / ( (1-x)^2 * (1-x^6) ).
a(n-1) = A047926(n) - A132868(n). (End)
From Wesley Ivan Hurt, Apr 04 2015, Sep 08 2015: (Start)
a(n) = 2*a(n-1)-a(n-2)-a(n-3)+2*a(n-4)-a(n-5), n>5.
a(n) = floor((n+3)/6) + floor((n+4)/6) + floor((n+5)/6).
a(n) = Sum_{i=0..n-1} floor(i/6) - floor((i-3)/6). (End)

A350522 a(n) = 18*n + 16.

Original entry on oeis.org

16, 34, 52, 70, 88, 106, 124, 142, 160, 178, 196, 214, 232, 250, 268, 286, 304, 322, 340, 358, 376, 394, 412, 430, 448, 466, 484, 502, 520, 538, 556, 574, 592, 610, 628, 646, 664, 682, 700, 718, 736, 754, 772, 790, 808, 826, 844, 862, 880, 898, 916, 934, 952, 970

Offset: 0

Omar E. Pol, Jan 03 2022

Sixth column of A006370 (the Collatz or 3x+1 map) when it is interpreted as a rectangular array with six columns read by rows.

Leo Tavares, Illustration: Triple Hexagonal Rings
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Bisection of A017245.

Cf. A006370, A008588, A008600, A016969, A017257, A062728, A239129.

GAP
```
List([0..53], n-> 18*n+16)
```
Magma
```
[18*n+16: n in [0..53]];
```
Maple
```
seq(18*n+16, n=0..53);
```
Mathematica
```
Table[18n+16, {n, 0, 53}]
```
Maxima
```
makelist(18*n+16, n, 0, 53);
```
PARI
```
a(n)=18*n+16
```
Python
```
[18*n+16 for n in range(53)]
```

a(n) = A239129(n+1) - 1.
From Stefano Spezia, Jan 04 2022: (Start)
O.g.f.: 2*(8 + x)/(1 - x)^2.
E.g.f.: 2*exp(x)*(8 + 9*x).
a(n) = 2*a(n-1) - a(n-2) for n > 1. (End)
a(n) = 3*A008588(n+1) - 2. - Leo Tavares, Sep 14 2022
From Elmo R. Oliveira, Apr 12 2024: (Start)
a(n) = 2*A017257(n) = A006370(A016969(n)).
a(n) = 2*(A062728(n+1) - A062728(n)). (End)

A383488 Numbers k that have at least one divisor d_i(k) for which a divisor d_j(k) exists such that d_i(k) < d_j(k) < sigma(d_i(k)).

Original entry on oeis.org

12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 130, 132, 138, 140, 144, 150, 154, 156, 160, 162, 168, 170, 174, 176, 180, 186, 189, 192, 196, 198, 200, 204, 208, 210, 216

Offset: 1

Felix Huber, May 03 2025

nonn

Numbers k (without multiplicity) that are multiples of lcm(c,i), where c is any composite and i is any integer from [c + 1, sigma(c) - 1].

			All multiples of 12 (A008594) are terms because 12 has the divisors 4 and 6 where sigma(4) = 7 > 6.
All multiples of 18 (A008600) are terms because 18 has the divisors 6 and 9 where sigma(6) = 12 > 9.
All multiples of 20 (A008602) are terms because 20 has the divisors 4 and 5 where sigma(4) = 7 > 5.

Felix Huber, Table of n, a(n) for n = 1..10000

Supersequence of A008594, A008600, A008602, A008606, A044102, A135628.

Cf. A000203, A002808, A027750, A109042, A383360, A383489.

with(NumberTheory):
A383488:=proc(n)
    option remember;
    local k,i,L;
    if n=1 then
        12
    else
        for k from procname(n-1)+1 do
            L:=Divisors(k);
            for i to nops(L)-1 do
                if sigma(L[i])>L[i+1] then
                    return k
                fi
            od
        od
    fi;
end proc;
seq(A383488(n),n=1..57);

A334403 Harshad numbers with sum of digits equal to 18.

Original entry on oeis.org

198, 288, 378, 396, 468, 486, 558, 576, 594, 648, 666, 684, 738, 756, 774, 792, 828, 846, 864, 882, 918, 936, 954, 972, 990, 1098, 1188, 1278, 1296, 1368, 1386, 1458, 1476, 1494, 1548, 1566, 1584, 1638, 1656, 1674, 1692, 1728, 1746, 1764, 1782, 1818, 1836, 1854

Offset: 1

Davide Rotondo, Sep 08 2020

Even numbers with sum of digits equal to 18 are Harshad numbers (A005349).

If k is a term, then so is 10*k. - Robert Israel, Mar 26 2023

			198/18 = 11.

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

Intersection of A005349 and A235228.
Subsequence of A008600.
Cf. A002998.

filter:= n -> convert(convert(n,base,10),`+`) = 18:
select(filter, [seq(i,i=18...4000, 18)]); # Robert Israel, Mar 26 2023

Select[18 * Range[100], Plus @@ IntegerDigits[#] == 18 &] (* Amiram Eldar, Sep 08 2020 *)

isok(m) = my(s=sumdigits(m)); (s==18) && !(m%s); \\ Michel Marcus, Sep 08 2020

first(n) = {my(res = vector(n), t = 0); forstep(i = 18, oo, 18, if(vecsum(digits(i)) == 18, t++; res[t] = i; if(t >= n, return(res) ) ) ) } \\ David A. Corneth, Sep 08 2020

More terms from Michel Marcus, Sep 08 2020

A350521 a(n) = 18*n + 4.

Original entry on oeis.org

4, 22, 40, 58, 76, 94, 112, 130, 148, 166, 184, 202, 220, 238, 256, 274, 292, 310, 328, 346, 364, 382, 400, 418, 436, 454, 472, 490, 508, 526, 544, 562, 580, 598, 616, 634, 652, 670, 688, 706, 724, 742, 760, 778, 796, 814, 832, 850, 868, 886, 904, 922, 940, 958

Offset: 0

Omar E. Pol, Jan 03 2022

Second column of A006370 (the Collatz or 3x+1 map) when it is interpreted as a rectangular array with six columns read by rows.

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

Bisection of A017209.

Cf. A006370, A008600, A016921, A017185, A242215, A298035.

GAP
```
List([0..53], n-> 18*n+4)
```
Magma
```
[18*n+4: n in [0..53]];
```
Maple
```
seq(18*n+4, n=0..53);
```
Mathematica
```
Table[18n+4, {n, 0, 53}]
```
Maxima
```
makelist(18*n+4, n, 0, 53);
```
PARI
```
a(n)=18*n+4
```
Python
```
[18*n+4 for n in range(53)]
```

a(n) = A242215(n) - 1.
a(n) = A298035(n+1) + 1.
From Elmo R. Oliveira, Apr 08 2024: (Start)
G.f.: 2*(2+7*x)/(1-x)^2.
E.g.f.: 2*exp(x)*(2 + 9*x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 2*A017185(n) = A006370(A016921(n)). (End)

A217591 Absolute differences between emirps (A006567) and their reversals.

Original entry on oeis.org

18, 54, 18, 36, 54, 36, 18, 18, 594, 198, 792, 594, 594, 792, 792, 198, 396, 396, 594, 594, 594, 198, 396, 198, 396, 594, 594, 198, 198, 198, 792, 594, 198, 792, 594, 792, 7992, 180, 270, 2268, 540, 8532, 810, 6804, 1908, 7902, 360, 2358, 630, 2718, 180, 1908, 5904, 1998, 7992, 90, 6084, 8172, 8262, 8442

Offset: 1

Jonathan Vos Post, Oct 07 2012

This is unsorted, and in order of appearance of emirps.

All values are multiples of 18 (A008600). - Charles R Greathouse IV, Oct 15 2012

			a(1) = absolute value of first emirp versus its reversal = |13 - 31| = |-18| = 18.
a(2) = |17 - 71| = |-54| = 54.
a(3) = |31 - 13| = |18| = 18.
a(4) = |37 - 73| = |-36| = 36.

Cf. A004086, A006567, A008600, A217286, A217386, A217387.

R:= n-> (s-> parse(cat(s[-i]$i=1..length(s))))(""||n):
q:= n-> isprime(n) and (p-> p<>n and isprime(p))(R(n)):
map(x-> abs(x-R(x)), select(q, [$2..1280]))[];  # Alois P. Heinz, Jul 12 2024

a(n) = | A006567(n) - R(A006567(n)) | = | A006567(n) - A004086(A006567(n)) |.

Corrected and more terms from Georg Fischer, Jul 12 2024

cp's OEIS Frontend

A317318 Multiples of 18 and odd numbers interleaved.

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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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A373599 Numbers k such that k and A327860(k) are both multiples of 3, where A327860 is the arithmetic derivative of the primorial base exp-function.

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A158910 First Differences of A061240.

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A194272 Array T(n,k) with 6 columns read by rows in which row n lists 3*n-2, 3*n-1, 3*n, 3*n, 3*n, 3*n.

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Formula

A350522 a(n) = 18*n + 16.

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GAP

Magma

Maple

Mathematica

Maxima

PARI

Python

Formula

A383488 Numbers k that have at least one divisor d_i(k) for which a divisor d_j(k) exists such that d_i(k) < d_j(k) < sigma(d_i(k)).

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A194272 Array T(n,k) with 6 columns read by rows in which row n lists 3n-2, 3n-1, 3n, 3n, 3n, 3n.