A008589 -id:A008589 - cp's OEIS frontend

A127545 Multiples of 7, k, such that k +/- 1 are twin primes.

42, 420, 462, 882, 1050, 1092, 1302, 1428, 1722, 1932, 2142, 2268, 2310, 2688, 2730, 3360, 3528, 3822, 4158, 4242, 4788, 5418, 5502, 5880, 6090, 6132, 6300, 6552, 6762, 7308, 7350, 7560, 8232, 8820, 8862, 9240, 9282, 10038, 10332, 10458, 10500, 10710

Offset: 1

Zak Seidov, Apr 01 2007

nonn

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

Intersection of A008589 and A014574.

Select[Range[7,10710,7],PrimeQ[#-1]&&PrimeQ[#+1]&] (* James C. McMahon, Jan 01 2025 *)
Select[7 Range[2000],AllTrue[#+{1,-1},PrimeQ]&] (* Harvey P. Dale, Aug 16 2025 *)

a(n) = !(n %7) && isprime(n+1) && isprime(n-1); \\ Michel Marcus, Oct 11 2013

Name edited by James C. McMahon, Jan 01 2025

A217009 Multiples of 7 in base 8.

Original entry on oeis.org

7, 16, 25, 34, 43, 52, 61, 70, 77, 106, 115, 124, 133, 142, 151, 160, 167, 176, 205, 214, 223, 232, 241, 250, 257, 266, 275, 304, 313, 322, 331, 340, 347, 356, 365, 374, 403, 412, 421, 430, 437, 446, 455, 464, 473, 502, 511, 520, 527, 536, 545, 554, 563

Offset: 1

Jon Perry, Sep 23 2012

Digit sum is always divisible by 7.

Reinterpreting this sequence in base 10, these are numbers of the form 9n + 7 but with all numbers containing 8s and/or 9s removed. - Alonso del Arte, Sep 23 2012

			a(10) = 106 because 7 * 10 = 70, or 1 * 8^2 + 0 * 8^1 + 6 * 8^0 = 64 + 6 = 106_8.

Cf. A216994, A216995, A216996, A216997, A216998.

Cf. A008589.

k = 7;
for (i = 1; i <= 200; i++) {
x = i * k;
document.write(x.toString(k + 1) + ", ");
}

Table[BaseForm[7*n, 8], {n, 100}] (* Alonso del Arte, Sep 23 2012 *)
Select[9*Range[0, 99] + 7, DigitCount[#, 10, 8] == 0 && DigitCount[#, 10, 9] == 0 &] (* Alonso del Arte, Sep 23 2012 *)
Table[FromDigits[IntegerDigits[7*n, 8]], {n, 100}] (* T. D. Noe, Sep 24 2012 *)

a(n) = A007094(A008589(n)). -

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).

A085708 Arithmetic derivative of 10^n.

Original entry on oeis.org

0, 7, 140, 2100, 28000, 350000, 4200000, 49000000, 560000000, 6300000000, 70000000000, 770000000000, 8400000000000, 91000000000000, 980000000000000, 10500000000000000, 112000000000000000, 1190000000000000000, 12600000000000000000, 133000000000000000000

Offset: 0

Reinhard Zumkeller, Jul 19 2003

a(n) = A003415(A011557(n)) = A008589(n)*A011557(n-1).

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (20,-100).

Cf. A001787, A003415, A008589, A011557, A027471.

[7*n*10^(n-1): n in [0..20]]; // Vincenzo Librandi, Jun 09 2011

Table[7*n*10^(n-1),{n,0,20}] (* Harvey P. Dale, Jul 09 2021 *)

a(n) = 7*n*10^(n-1).

G.f.: 7*x/(10*x-1)^2.

A154684 Triangle read by rows where T(m,n)=2mn + m + n - 3, 1<=n<=m.

Original entry on oeis.org

1, 4, 9, 7, 14, 21, 10, 19, 28, 37, 13, 24, 35, 46, 57, 16, 29, 42, 55, 68, 81, 19, 34, 49, 64, 79, 94, 109, 22, 39, 56, 73, 90, 107, 124, 141, 25, 44, 63, 82, 101, 120, 139, 158, 177, 28, 49, 70, 91, 112, 133, 154, 175, 196, 217, 31, 54, 77, 100, 123, 146, 169

Offset: 1

Vincenzo Librandi, Jan 18 2009

2*T(m,n)+7 = (2n+1)*(2m+1) is not prime.

First column: A016777; second column: A016897; third column: A008589; fourth column: A017173. - Vincenzo Librandi, Nov 19 2012

			Triangle begins:
1;
4,  9;
7,  14, 21;
10, 19, 28, 37;
13, 24, 35, 46, 57;
16, 29, 42, 55, 68,  81;
19, 34, 49, 64, 79,  94,  109;
22, 39, 56, 73, 90,  107, 124, 141;
25, 44, 63, 82, 101, 120, 139, 158, 177;
28, 49, 70, 91, 112, 133, 154, 175, 196, 217; etc.

Vincenzo Librandi, Rows n = 1..100, flattened

Cf. A153053, A105760, A162265 (row sums), A016777, A016897, A008589, A017173.

[(2*n*k + n + k - 3): k in [1..n], n in [1..11]]; // Vincenzo Librandi, Nov 19 2012

t[n_,k_]:=2 n*k + n + k - 3; Table[t[n, k], {n, 20}, {k, n}]//Flatten (* Vincenzo Librandi, Nov 19 2012 *)

A168374 a(n) = 7 * floor(n/2).

Original entry on oeis.org

0, 7, 7, 14, 14, 21, 21, 28, 28, 35, 35, 42, 42, 49, 49, 56, 56, 63, 63, 70, 70, 77, 77, 84, 84, 91, 91, 98, 98, 105, 105, 112, 112, 119, 119, 126, 126, 133, 133, 140, 140, 147, 147, 154, 154, 161, 161, 168, 168, 175, 175, 182, 182, 189, 189, 196, 196, 203, 203, 210

Offset: 1

Vincenzo Librandi, Nov 24 2009

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

Cf. A004526, A008589.

[n eq 1 select 0 else 7*n-Self(n-1)-7: n in [1..70]]; // Vincenzo Librandi, Sep 17 2013

CoefficientList[Series[7 x/((1 + x) (1 - x)^2), {x, 0, 70}], x] (* Vincenzo Librandi, Sep 17 2013 *)
Table[7 Floor[n/2], {n, 60}] (* Bruno Berselli, Sep 18 2013 *)

a(n) = 7*n - a(n-1) - 7, with n>1, a(1)=0.
G.f.: 7*x^2/((1+x)*(1-x)^2). - Bruno Berselli, Sep 17 2013
a(n) = 7*A004526(n). - Bruno Berselli, Sep 17 2013
E.g.f.: (7/2)*(x*cosh(x) + (x - 1)*sinh(x)). - G. C. Greubel, Jul 19 2016

New definition by Bruno Berselli, Sep 18 2013

A176972 a(n) = 7^n + 7*n + 1.

Original entry on oeis.org

2, 15, 64, 365, 2430, 16843, 117692, 823593, 5764858, 40353671, 282475320, 1977326821, 13841287286, 96889010499, 678223072948, 4747561510049, 33232930569714, 232630513987327, 1628413597910576, 11398895185373277, 79792266297612142, 558545864083284155, 3909821048582988204

Offset: 0

Jonathan Vos Post, Apr 29 2010

			a(5) = 7^5 + 7*5 + 1 = 16843 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for linear recurrences with constant coefficients, signature (9,-15,7).

Cf. A000420, A008589, A016993, A176691, A176805, A176916.

Cf. A370664.

[7^n + 7*n + 1: n in [0..25]]; // Vincenzo Librandi, May 06 2011

Table[7^n+7n+1,{n,0,20}] (* or *) LinearRecurrence[{9,-15,7},{2,15,64},20] (* Harvey P. Dale, Apr 17 2014 *)

a(n) = A000420(n) + A008589(n) + 1 = A000420(n) + A016993(n).
a(n) = 7*a(n-1) - 42*(n-1) + 1, with n > 0. For n=5, a(5) = 7*2430 - 42*4 + 1 = 16843. - Bruno Berselli, May 18 2010
From R. J. Mathar, May 22 2010: (Start)
a(n) = 9*a(n-1) - 15*a(n-2) + 7*a(n-3).
G.f.: (-2 + 3*x + 41*x^2) / ((7*x-1)*(x-1)^2). (End)
E.g.f.: exp(x)*(1 + exp(6*x) + 7*x). - Stefano Spezia, Aug 19 2024

A220528 a(n) = n^7 + 7*n + 7^n.

Original entry on oeis.org

1, 15, 191, 2551, 18813, 94967, 397627, 1647135, 7862009, 45136639, 292475319, 1996813991, 13877119093, 96951759015, 678328486451, 4747732369423, 33233199005169, 232630924325999, 1628414210130607, 11398896079245015, 79792267577612141, 558545865884372695

Offset: 0

Jonathan Vos Post, Dec 15 2012

			a(1) = 1^7 + 7*1 + 7^1 = 15.
a(2) = 2^7 + 7*2 + 7^2 = 191.

Index entries for linear recurrences with constant coefficients, signature (15,-84,252,-462,546,-420,204,-57,7).

Cf. A000420, A001015, A008589, A220425, A220509, A220511, A220703.

Table[n^7 + 7*n + 7^n, {n, 0, 30}] (* T. D. Noe, Dec 17 2012 *)

makelist(n^7 + 7*n + 7^n,n,0,20); /* Martin Ettl, Jan 15 2013 */

a(n) = A001015(n) + A008589(n) + A000420(n).

G.f.: (55*x^8+546*x^7+8966*x^6+14692*x^5+6726*x^4-694*x^3-50*x^2-1) / ((x-1)^8*(7*x-1)). - Colin Barker, May 09 2013

A258188 Smallest multiple of 7 not appearing earlier that ends with n.

Original entry on oeis.org

21, 42, 63, 14, 35, 56, 7, 28, 49, 210, 511, 112, 413, 714, 315, 616, 217, 518, 119, 420, 721, 322, 623, 224, 525, 126, 427, 728, 329, 630, 231, 532, 133, 434, 735, 336, 637, 238, 539, 140, 441, 742, 343, 644, 245, 546, 147, 448, 749, 350, 651, 252, 553, 154

Offset: 1

Eric Angelini and Reinhard Zumkeller, May 23 2015

a(10*n) = 10*a(n).
The sequence is a permutation of the positive multiples of 7. - Vladimir Shevelev, May 24 2015
A258329(n) = a(n) / 7 is a permutation of the positive integers. - Reinhard Zumkeller, May 27 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..9999

Cf. A008589, A258083, A258217, A258329.

import Data.List (isPrefixOf, delete)
a258188 n = a258188_list !! (n-1)
a258188_list = f 1 $ tail $ zip
   a008589_list $ map (reverse . show) a008589_list where
   f x ws = g ws where
     g ((u, vs) : uvs) = if isPrefixOf xs vs
                         then u : f (x + 1) (delete (u, vs) ws) else g uvs
     xs = reverse $ show x

a[n_] := a[n] = For[k = 7, True, k = k + 7, If[Divisible[k - n, 10^IntegerLength[n]] && FreeQ[Array[a, n-1], k], Return[k]]]; Array[a, 54] (* Jean-François Alcover, Feb 07 2018 *)

A258217 Smallest multiple of 7 starting with n, that did not appear earlier.

Original entry on oeis.org

14, 21, 35, 42, 56, 63, 7, 84, 91, 105, 112, 126, 133, 140, 154, 161, 175, 182, 196, 203, 210, 224, 231, 245, 252, 266, 273, 28, 294, 301, 315, 322, 336, 343, 350, 364, 371, 385, 392, 406, 413, 420, 434, 441, 455, 462, 476, 483, 49, 504, 511, 525, 532, 546

Offset: 1

Eric Angelini and Reinhard Zumkeller, May 23 2015

The sequence is a permutation of the positive multiples of 7. - Vladimir Shevelev, May 24 2015

A258334(n) = a(n) / 7 is a permutation of the positive integers. - Reinhard Zumkeller, May 27 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Éric Angelini, Multiples of 3 ending with n, SeqFan list, May 23 2015.

Cf. A008589, A258188, A258083, A258334.

import Data.List (isPrefixOf, delete)
a258217 n = a258217_list !! (n-1)
a258217_list = f 1 $ tail $ zip a008589_list $ map show a008589_list where
   f x ws = g ws where
     g ((u, vs) : uvs) = if isPrefixOf (show x) vs
                         then u : f (x + 1) (delete (u, vs) ws) else g uvs

a[1] = 14; a[n_] := a[n] = For[dn = IntegerDigits[n]; k = 7, True, k = k+7, dk = IntegerDigits[k]; lg = Min[Length[dn], Length[dk]]; If[Union[ Take[dk, lg] - Take[dn, lg]] == {0} && FreeQ[Array[a, n-1], k], Return[k]]]; Array[a, 54] (* Jean-François Alcover, Feb 09 2018 *)

cp's OEIS Frontend

A127545 Multiples of 7, k, such that k +/- 1 are twin primes.

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A217009 Multiples of 7 in base 8.

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

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A085708 Arithmetic derivative of 10^n.

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Magma

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A154684 Triangle read by rows where T(m,n)=2mn + m + n - 3, 1<=n<=m.

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Magma

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A168374 a(n) = 7 * floor(n/2).

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Magma

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A176972 a(n) = 7^n + 7*n + 1.

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Magma

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A220528 a(n) = n^7 + 7*n + 7^n.

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