A008589 -id:A008589 - cp's OEIS frontend

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

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 *)

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).

A363922 a(n) = smallest number m > 0 such that n followed by m 7's yields a prime, or -1 if no such m exists.

Original entry on oeis.org

1, 2, 1, 1, 2, 1, -1, 2, 1, 1, 3, 1, 1, -1, 1, 1, 2, 2, 1, 6, -1, 1, 2, 2, 1, 2, 1, -1, 48, 1, 1, 5, 1, 1, -1, 1, 10, 2, 1, 12, 2, -1, 3, 3, 1, 1, 3, 1, -1, 2, 8, 7, 3, 1, 1, -1, 1, 1, 9, 1, 1, 2, -1, 1, 2, 5, 1, 3, 2, -1, 2, 1, 66, 2, 1, 3, -1, 1, 1, 3

Offset: 1

Toshitaka Suzuki, Jul 12 2023

a(n) = -1 when n = 7*k because no matter how many 7's are appended to n, the resulting number is always divisible by 7 and therefore cannot be prime.
a(n) = -1 when n = 15873*k + 891, 1261, 2889, 3263, 3300, 7810, 8917, 9812, 12617, 13024, 14615 or 15066, because n followed by any positive number, m say, of 7's is divisible by at least one of the primes {3,11,13,37}.
Similarly,
a(n) = -1 when n = 11111111*k + 964146, 1207525, 2342974, 3567630, 7525789, 8134540, 8591231 or 9641467 by primes {11,73,101,137};
a(n) = -1 when n = 429000429*k + 23928593, 27079312, 36492115, 41207969, 52285750, 80569929, 89920882, 93857078, 133928703, 217208145, 223492302, 236849444, 239285937, 247857232, 259793116, 270793127, 323985244, 332698824, 333570182, 334985255, 346849554, 364921157, 376698868 or 412079697 by primes {3,11,13,101,9901};
a(n) = -1 when n = 1221001221*k + 14569863, 28792885, 145698637, 167698659, 225079510, 235985156, 247079532, 287928857, 331921124, 399492478, 415286113, 421492500, 437286135, 455985376, 489857474, 529929099, 551921344, 635208563, 709857694, 877208805, 896850104, 993570842, 1029793886 or 1138850346 by primes {3,11,37,101,9901};
a(n) = -1 when n = 1443001443*k + 85928655, 167698659, 176928746, 218921011, 233985154, 247079532, 310492389, 326286024, 376857361, 585793442, 655208583, 700699192, 746208674, 780080065, 791570640, 805850013, 843492922, 859286557, 882570731, 896850104, 1027793884, 1219922012, 1234986155 or 1377858362 by primes {3,13,37,101,9901}.
a(4444) > 300000 or a(4444) = -1.

			a(11)=3 because 117 and 1177 are composite but 11777 is prime.

Toshitaka Suzuki, Table of n, a(n) for n = 1..4443

Cf. A008589, A069568, A090464, A113076.

a(n) = if ((n%7), my(m=1); while (!isprime(eval(concat(Str(n), Str(7*(10^m-1)/9)))), m++); m, -1); \\ Michel Marcus, Jul 17 2023

A084891 Multiples of 2, 3, 5, or 7, but not 7-smooth.

Original entry on oeis.org

22, 26, 33, 34, 38, 39, 44, 46, 51, 52, 55, 57, 58, 62, 65, 66, 68, 69, 74, 76, 77, 78, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 99, 102, 104, 106, 110, 111, 114, 115, 116, 117, 118, 119, 122, 123, 124, 129, 130, 132, 133, 134, 136, 138, 141, 142, 145, 146

Offset: 1

Reinhard Zumkeller, Jul 13 2003

nonn

Intersection of A068191 with (A005843, A008585, A008587 and A008589); union of (A005843, A008585, A008587 and A008589) without A002473.

A020639(a(n)) <= 7, A006530(a(n)) > 7.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Diagram showing numbers k in this sequence instead as k mod 210, in black, else white if k is coprime to 210, purple if k = 1, red if k | 210, and gold if rad(k) | 210, magnification 5X.
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A002473, A005843, A006530, A008585, A008587, A008589, A020639, A068191, A080672.

okQ[n_] := AnyTrue[{2, 3, 5, 7}, Divisible[n, #]&] && FactorInteger[n][[-1, 1]] > 7;
Select[Range[1000], okQ] (* Jean-François Alcover, Oct 15 2021 *)

mult2357(m,n) = \\ mult 2,3,5,7 not 7 smooth
{
local(x,a,j,f,ln);
for(x=m,n,
f=0;
if(gcd(x,210)>1,
a = ifactor(x);
for(j=1,length(a),
if(a[j]>7,f=1;break);
);
if(f,print1(x","));
);
);
}
ifactor(n) = \\ The vector of the prime factors of n with multiplicity.
{
local(f,j,k,flist);
flist=[];
f=Vec(factor(n));
for(j=1,length(f[1]),
for(k = 1,f[2][j],flist = concat(flist,f[1][j])
);
);
return(flist)
}
\\ Cino Hilliard, Jul 03 2009

from sympy import primefactors
def ok(n):
    pf = set(primefactors(n))
    return pf & {2, 3, 5, 7} and pf - {2, 3, 5, 7}
print(list(filter(ok, range(147)))) # Michael S. Branicky, Oct 15 2021

A177897 Triangle of octanomial coefficients read by rows: n-th row is obtained by expanding ((1+x)(1+x^2)(1+x^4))^n mod 2 and converting to decimal.

Original entry on oeis.org

1, 255, 21845, 3342387, 286331153, 64424509455, 5519032976645, 844437815230467, 72340172838076673, 18446744073709551615, 1567973246265311887445, 241781474574111093044019, 20552052528097949033496593, 4660480146812799619066433295, 396140812663555408357742346245, 61084913312720243968763869790979

Offset: 0

Vladimir Shevelev, Dec 15 2010

A generalization: Denote {a_k(n)}_(n>=0) the sequence of triangle of 2^k-nomial coefficients [read by rows: n-th row is obtained by expanding ((1+x)*(1+x^2)*...*(1+x^(2^(k-1)))^n ] mod 2 converted to decimal. Then a_k(n)=A001317((2^k-1)*n). [Proof is based on the fact (following from the Lucas theorem for the binomial coefficients) that the k-th row of Pascal triangle contains odd coefficients only iff k is Mersenne number (k=2^m-1)].

Cf. A001317, A008589, A177882.

a = Plus@@(x^Range[0, 7]); Table[FromDigits[Mod[CoefficientList[a^n, x], 2], 2], {n, 0, 15}]

a(n) = subst(lift(Mod(1+'x, 2)^(7*n)), 'x, 2); \\ Michel Marcus, Oct 14 2024

def A177897(n): return sum((bool(~(m:=7*n)&m-k)^1)<Chai Wah Wu, May 03 2023

a(n) = A001317(7*n).

More terms from Michel Marcus, Oct 14 2024

A302537 a(n) = (n^2 + 13*n + 2)/2.

Original entry on oeis.org

1, 8, 16, 25, 35, 46, 58, 71, 85, 100, 116, 133, 151, 170, 190, 211, 233, 256, 280, 305, 331, 358, 386, 415, 445, 476, 508, 541, 575, 610, 646, 683, 721, 760, 800, 841, 883, 926, 970, 1015, 1061, 1108, 1156, 1205, 1255, 1306, 1358, 1411, 1465, 1520, 1576

Offset: 0

Franck Maminirina Ramaharo, Apr 09 2018

Binomial transform of [1, 7, 1, 0, 0, 0, ...].

Numbers m > 0 such that 8*m + 161 is a square.

			Illustration of initial terms (by the formula a(n) = A052905(n) + 3*n):
.                                                                    o
.                                                                  o o
.                                                    o           o o o
.                                                  o o         o o o o
.                                      o         o o o       o o o o o
.                                    o o       o o o o     o o o o o o
.                          o       o o o     o o o o o   o . . . . . o
.                        o o     o o o o   o . . . . o   o . . . . . o
.                o     o o o   o . . . o   o . . . . o   o . . . . . o
.              o o   o . . o   o . . . o   o . . . . o   o . . . . . o
.        o   o . o   o . . o   o . . . o   o . . . . o   o . . . . . o
.      o o   o . o   o . . o   o . . . o   o . . . . o   o . . . . . o
.  o   o o   o o o   o o o o   o o o o o   o o o o o o   o o o o o o o
.        o     o o     o o o     o o o o     o o o o o     o o o o o o
.        o     o o     o o o     o o o o     o o o o o     o o o o o o
.        o     o o     o o o     o o o o     o o o o o     o o o o o o
----------------------------------------------------------------------
.  1     8      16        25          35            46              58

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, Addison-Wesley, 1994.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Sequences whose n-th terms are of the form binomial(n, 2) + n*k + 1:
A152947 (k = 0); A000124 (k = 1); A000217 (k = 2); A034856 (k = 3);
A052905 (k = 4); A051936 (k = 5); A246172 (k = 6).
Cf. A000578, A002939, A004120, A008589, A056119, A060544, A162261.

A302537:= func< n | ((n+1)^2 +12*n +1)/2 >;
[A302537(n): n in [0..50]]; // G. C. Greubel, Jan 21 2025

a := n -> (n^2 + 13*n + 2)/2;
seq(a(n), n = 0 .. 100);

Table[(n^2 + 13 n + 2)/2, {n, 0, 100}]
CoefficientList[ Series[(5x^2 - 5x - 1)/(x - 1)^3, {x, 0, 50}], x] (* or *)
LinearRecurrence[{3, -3, 1}, {1, 8, 16}, 51] (* Robert G. Wilson v, May 19 2018 *)

makelist((n^2 + 13*n + 2)/2, n, 0, 100);

a(n) = (n^2 + 13*n + 2)/2; \\ Altug Alkan, Apr 12 2018

def A302537(n): return (n**2 + 13*n + 2)//2
print([A302537(n) for n in range(51)]) # G. C. Greubel, Jan 21 2025

a(n) = binomial(n + 1, 2) + 6*n + 1 = binomial(n, 2) + 7*n + 1.
a(n) = a(n-1) + n + 6.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3, where a(0) = 1, a(1) = 8 and a(2) = 16.
a(n) = 2*a(n-1) - a(n-2) + 1.
a(n) = A004120(n+1) for n > 1.
a(n) = A056119(n) + 1.
a(n) = A152947(n+1) + A008589(n).
a(n) = A060544(n+1) - A002939(n).
a(n) = A000578(n+1) - A162261(n) for n > 0.
G.f.: (1 + 5*x - 5*x^2)/(1 - x)^3.
E.g.f.: (1/2)*(2 + 14*x + x^2)*exp(x).
Sum_{n>=0} 1/a(n) = 24097/45220 + 2*Pi*tan(sqrt(161)*Pi/2) / sqrt(161) = 1.4630922534498496... - Vaclav Kotesovec, Apr 11 2018

A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.

Original entry on oeis.org

2, 4, 3, 6, 6, 5, 8, 9, 10, 7, 10, 12, 15, 14, 11, 12, 15, 20, 21, 22, 13, 14, 18, 25, 28, 33, 26, 17, 16, 21, 30, 35, 44, 39, 34, 19, 18, 24, 35, 42, 55, 52, 51, 38, 23, 20, 27, 40, 49, 66, 65, 68, 57, 46, 29, 22, 30, 45, 56, 77, 78, 85, 76, 69, 58, 31

Offset: 1

Stefano Spezia, Jul 14 2019

T(n, k) is the k-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n).
The h-th subdiagonal of the triangle T gives the primes multiplied by (h + 1).
The k-th column of the triangle T gives the multiples of prime(1 + k).
Also array A(n, k) = n*prime(1 + k) read by ascending antidiagonals, with 0 <= k < n. - Michel Marcus, Jul 15 2019

			The triangle T(n, k) begins:
---+-----------------------------------------------------
n\k|    0     1     2     3     4     5     6     7     8
---+-----------------------------------------------------
1  |    2
2  |    4     3
3  |    6     6     5
4  |    8     9    10     7
5  |   10    12    15    14    11
6  |   12    15    20    21    22    13
7  |   14    18    25    28    33    26    17
8  |   16    21    30    35    44    39    34    19
9  |   18    24    35    42    55    52    51    38    23
...
For n = 3 the matrix M(3) is
          2,         3,         5
    M_{2,1},         2,         3
    M_{3,1},   M_{3,2},         2
and therefore T(3, 0) = 2 + 2 + 2 = 6, T(3, 1) = 3 + 3 = 6, and T(3, 2) = 5.

Wikipedia, Toeplitz matrix

Cf. A025581, A318173, A325516.

Cf. A000040: diagonal; A001747: 1st subdiagonal; A001748: 2nd subdiagonal; A001749: 3rd subdiagonal; A001750: 4th subdiagonal; A005843: 0th column; A008585: 1st column; A008587: 2nd column; A008589: 3rd column; A008593: 4th column; A008595: 5th column; A008599: 6th column; A008601: 7th column; A014148: row sums; A138636: 5th subdiagonal; A272470: 6th subdiagonal.

[[(n-k)*NthPrime(1+k): k in [0..n-1]]: n in [1..11]]; // triangle output

a:=(n, k)->(n-k)*ithprime(1+k): seq(seq(a(n, k), k=0..n-1), n=1..11);

Flatten[Table[(n-k)*Prime[1+k],{n,1,11},{k,0,n-1}]]

T(n, k) = (n - k)*prime(1 + k);
tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ triangle output

[[(n-k)*Primes().unrank(k) for k in (0..n-1)] for n in (1..11)] # triangle output

T(n, k) = A025581(n, k)*A000040(1 + k).

A366596 Repdigit numbers that are divisible by 7.

Original entry on oeis.org

0, 7, 77, 777, 7777, 77777, 111111, 222222, 333333, 444444, 555555, 666666, 777777, 888888, 999999, 7777777, 77777777, 777777777, 7777777777, 77777777777, 111111111111, 222222222222, 333333333333, 444444444444, 555555555555, 666666666666, 777777777777

Offset: 1

Kritsada Moomuang, Oct 14 2023

7 divides a repdigit iff it consists of only digit 7, or has length 6*k (for any digit).

Repdigit remainders A010785(k) mod 7 have period 54. - Karl-Heinz Hofmann, Dec 04 2023

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..2329
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,1000001,0,0,0,0,0,0,0,0,0,0,0,0,0,-1000000).

Intersection of A008589 and A010785.
Cf. A002281 (a subsequence).
Cf. A305322 (divisor 3), A002279 (divisor 5), A083118 (the impossible divisors).

r(n) = 10^((n+8)\9)\9*((n-1)%9+1); \\ A010785
lista(nn) = select(x->!(x%7), vector(nn, k, r(k-1))); \\ Michel Marcus, Oct 26 2023

def A366596(n):
    digitlen, digit = (n+12)//14*6, (n+12)%14-4
    if digit < 1: digitlen += digit - 1; digit = 7
    return 10**digitlen // 9 * digit # Karl-Heinz Hofmann, Dec 04 2023

From Karl-Heinz Hofmann, Dec 04 2023: (Start)
a(n) = A010785(floor((n-2)/14)*54 + ((n-2) mod 14) + 41),   for (n-2) mod 14 >  4.
a(n) = (10^(6*floor((n-2)/14) + 6)-1)/9*(((n-2) mod 14)-4), for (n-2) mod 14 >  4.
a(n) = A010785(floor((n-2)/14)*54 + ((n-2) mod 14)*9 + 7),  for (n-2) mod 14 <= 4.
a(n) = (10^(6*floor((n-2)/14) + 1 + ((n-2) mod 14))-1)/9*7, for (n-2) mod 14 <= 4.
(End)

A016982 a(n) = (7*n)^2.

Original entry on oeis.org

0, 49, 196, 441, 784, 1225, 1764, 2401, 3136, 3969, 4900, 5929, 7056, 8281, 9604, 11025, 12544, 14161, 15876, 17689, 19600, 21609, 23716, 25921, 28224, 30625, 33124, 35721, 38416, 41209, 44100, 47089, 50176, 53361, 56644, 60025, 63504, 67081, 70756, 74529, 78400

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A008589.

[(7*n)^2: n in [0..50]]; // Vincenzo Librandi, May 21 2011

a(n)=(7*n)^2 \\ Charles R Greathouse IV, Jun 17 2017

From Amiram Eldar, Jan 25 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/294.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/588.
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/7)/(Pi/7).
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/7)/(Pi/7). (End)
From Elmo R. Oliveira, Nov 29 2024: (Start)
G.f.: 49*x*(1 + x)/(1-x)^3.
E.g.f.: 49*x*(1 + x)*exp(x).
a(n) = 49*A000290(n) = A008589(n)^2 = A000290(A008589(n)).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A061825 Multiples of 7 containing only odd digits.

Original entry on oeis.org

7, 35, 77, 91, 119, 133, 175, 315, 357, 371, 399, 511, 539, 553, 595, 735, 777, 791, 917, 931, 959, 973, 1113, 1155, 1197, 1337, 1351, 1379, 1393, 1519, 1533, 1575, 1715, 1757, 1771, 1799, 1911, 1939, 1953, 1995, 3115, 3157, 3171, 3199, 3311, 3339

Offset: 1

Amarnath Murthy, May 28 2001

			315 = 7*45 is a term as it contains only odd digits.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for 10-automatic sequences.

Subsequence of A008589.

Cf. A061826.

R:= NULL:
for d from 1 to 4 do
  for n from 5^d to 2*5^d-1 do
    L:= map(t -> 2*t+1,convert(n,base,5)[1..d]);
    x:= add(L[i]*10^(i-1),i=1..nops(L));
    if x mod 7 = 0 then
      R:= R, x;
    fi;
od od:
R; # Robert Israel, Nov 02 2019

Select[7*Range[500], Count[IntegerDigits[#], ?EvenQ]==0&] (* _Harvey P. Dale, Jun 13 2011 *)

is(n)=n%7==0 && #setintersect(Set(digits(n)), [0,2,4,6,8])==0 \\ Charles R Greathouse IV, Feb 15 2017

More terms from Larry Reeves (larryr(AT)acm.org), May 30 2001

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A363922 a(n) = smallest number m > 0 such that n followed by m 7's yields a prime, or -1 if no such m exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A084891 Multiples of 2, 3, 5, or 7, but not 7-smooth.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

A177897 Triangle of octanomial coefficients read by rows: n-th row is obtained by expanding ((1+x)*(1+x^2)*(1+x^4))^n mod 2 and converting to decimal.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A302537 a(n) = (n^2 + 13*n + 2)/2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

Python

Formula

A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A177897 Triangle of octanomial coefficients read by rows: n-th row is obtained by expanding ((1+x)(1+x^2)(1+x^4))^n mod 2 and converting to decimal.