A008596 -id:A008596 - 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 *)

A158186 a(n) = 10n^2 - 7n + 1.

Original entry on oeis.org

1, 4, 27, 70, 133, 216, 319, 442, 585, 748, 931, 1134, 1357, 1600, 1863, 2146, 2449, 2772, 3115, 3478, 3861, 4264, 4687, 5130, 5593, 6076, 6579, 7102, 7645, 8208, 8791, 9394, 10017, 10660, 11323, 12006, 12709, 13432, 14175, 14938, 15721, 16524, 17347

Offset: 0

Reinhard Zumkeller, Mar 13 2009

Sequence found by reading the segment (1, 4) together with the line (one of the diagonal axes) from 4, in the direction 4, 27, ..., in the square spiral whose vertices are the generalized heptagonal numbers A085787. - Omar E. Pol, Sep 10 2011

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001622, A008596, A010010, A033571, A085787.

Table[10n^2-7n+1,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{1,4,27},50] (* Harvey P. Dale, Apr 06 2020 *)

a(n)=10*n^2-7*n+1 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = (2*n-1)*(5*n-1).
a(n) = A033571(n) - A008596(n) = A010010(n) - A033571(n).
G.f.: (1+x+18*x^2)/(1-x)^3. - Jaume Oliver Lafont, Mar 27 2009
a(n) = a(n-1) + 20*n - 17 (with a(0)=1). - Vincenzo Librandi, Dec 03 2010
Sum_{n>=0} 1/a(n) = 1 + (2*sqrt(1+2/sqrt(5))*Pi - 2*sqrt(5)*log(phi) - 5*log(5) + 8*log(2))/12, where phi is the golden ratio (A001622). - Amiram Eldar, Sep 22 2022

Typo in definition corrected by Reinhard Zumkeller, Dec 03 2009

A153432 Numbers m such that all ten numbers 10^k*m+1; k=0,1,...,9 are prime.

Original entry on oeis.org

15538734736, 28034367522, 41221375552, 89468493268, 92220460962, 95951017078, 105627706800, 108012908976, 147689513538, 161055865908, 177073756566, 181921823202, 185137153012, 192336309046, 212463686778

Offset: 1

Farideh Firoozbakht, Mar 31 2009

If m & n are in the sequence, k<10 and r=m*n*10^k -1 is prime then r has at least k+1 representations of the form p*q-(p+q)where p & q are prime.

Carlos Rivera, Puzzle 482. Two Bergot questions, The Prime Puzzles & Problems Connection.

Subsequence of A006093 and of A008596.

Cf. A153431, A153433.

a(n) >> n log^10 n. - Charles R Greathouse IV, Dec 29 2024

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)

A169827 Multiples of 840.

Original entry on oeis.org

0, 840, 1680, 2520, 3360, 4200, 5040, 5880, 6720, 7560, 8400, 9240, 10080, 10920, 11760, 12600, 13440, 14280, 15120, 15960, 16800, 17640, 18480, 19320, 20160, 21000, 21840, 22680, 23520, 24360, 25200, 26040, 26880, 27720, 28560, 29400, 30240, 31080, 31920

Offset: 0

N. J. A. Sloane, May 30 2010

Numbers that are divisible by all of 1,2,3,4,5,6,7,8.

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

Cf. A001477, A005843, A008596, A008603, A135628, A169823, A169825.

Cf. A249674, A317095.

840*Range[0,40] (* Harvey P. Dale, Nov 03 2015 *)

a(n)=840*n \\ Charles R Greathouse IV, Apr 16 2024

From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 840*x/(x-1)^2.
E.g.f.: 840*x*exp(x).
a(n) = 840*n = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 2*A169825(n) = 14*A169823(n) = 21*A317095(n) = 28*A249674(n) = 30*A135628(n) = 40*A008603(n) = 60*A008596(n) = 420*A005843(n) = 840*A001477(n). (End)

A179849 Sum of prime p and next prime after p is divisible by 7.

Original entry on oeis.org

19, 41, 53, 103, 151, 211, 229, 263, 313, 397, 419, 439, 461, 479, 523, 557, 571, 709, 859, 881, 919, 977, 983, 991, 1033, 1049, 1069, 1091, 1103, 1109, 1117, 1171, 1187, 1193, 1279, 1301, 1327, 1427, 1447, 1453, 1489, 1499, 1571, 1621, 1709, 1721, 1747

Offset: 1

Zak Seidov, Jan 10 2011

nonn

Also primes p such that the sum of p and next prime after p is a multiple of 14, since for p > 2 the sum of two consecutive primes is even. - Klaus Brockhaus, Jan 11 2011

			p=19, q=23, p+q=42=7*6=14*3; p=41, q=43, p+q=84=7*12=14*6.

Cf. A031932 (lower prime of a difference of 14 between consecutive primes), A008596 (multiples of 14).

IsA179849:=func< n | IsPrime(n) and (n+NextPrime(n)) mod 14 eq 0 >; [ p: p in PrimesUpTo(2000) | IsA179849(p) ]; // Klaus Brockhaus, Jan 11 2011

fQ[n_] := Block[{q = NextPrime@ n}, Mod[n + q, 7] == 0]; Select[ Prime@ Range@ 300, fQ]

{q=3;for(n=1,100,p=q;q=nextprime(p+1);(p+q)%7==0&print(p))}

A358053 a(n) = 14*n - 1.

Original entry on oeis.org

13, 27, 41, 55, 69, 83, 97, 111, 125, 139, 153, 167, 181, 195, 209, 223, 237, 251, 265, 279, 293, 307, 321, 335, 349, 363, 377, 391, 405, 419, 433, 447, 461, 475, 489, 503, 517, 531, 545, 559, 573, 587, 601, 615, 629, 643, 657, 671, 685, 699, 713, 727, 741, 755, 769, 783, 797

Offset: 1

Leo Tavares, Oct 27 2022

This sequence can be illustrated by a star outline with a central row of counters.

Subsequence of primes is A045473. - Bernard Schott, Jan 25 2023

Paolo Xausa, Table of n, a(n) for n = 1..10000
Leo Tavares, Illustration: Mid-line Stars.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008596, A003154, A017053, A045473, A045944, A202804.

14*Range[100] - 1 (* Paolo Xausa, Oct 04 2024 *)

a(n) = 14*n - 1.
a(n) = A003154(n+1) - A202804(n-1).
a(n) = A003154(n+1) - 2*A045944(n-1).
From Elmo R. Oliveira, Apr 03 2025: (Start)
G.f.: x*(13 + x)/(x - 1)^2.
E.g.f.: exp(x)*(14*x - 1) + 1.
a(n) = A017053(2*n-1).
a(n) = 2*a(n-1) - a(n-2) for n > 2. (End)

A233207 Triangle T(n,k), read by rows, given by T(n+k,k)=2k(2*n+1).

Original entry on oeis.org

0, 0, 2, 0, 6, 4, 0, 10, 12, 6, 0, 14, 20, 18, 8, 0, 18, 28, 30, 24, 10, 0, 22, 36, 42, 40, 30, 12, 0, 26, 44, 54, 56, 50, 36, 14, 0, 30, 52, 66, 72, 70, 60, 42, 16, 0, 34, 60, 78, 88, 90, 84, 70, 48, 18, 0, 38, 68, 90, 104, 110, 108, 98, 80, 54, 20, 0, 42, 76, 102

Offset: 0

Philippe Deléham, Dec 05 2013

Row sums are A006331(n).
Diagonal sums are A212964(n+1).
T(2n,n)=A002943(n).

			Triangle begins:
  0
  0, 2
  0, 6, 4
  0, 10, 12, 6
  0, 14, 20, 18, 8
  0, 18, 28, 30, 24, 10

Cf. columns: A000004, A016825, A017113, A017593, A051062, 10*A005408, A073762.

Cf. diagonals: A005843, A008588, A008592, A008596, A008600, A008604.

T(n+k,k) = A005843(k)*A005408(n).

Sum_{k=0..n} T(n,k) = n*(n+1)*(2*n+1)/3 = A006331(n).

A242570 a(n) = 252 * n.

Original entry on oeis.org

0, 252, 504, 756, 1008, 1260, 1512, 1764, 2016, 2268, 2520, 2772, 3024, 3276, 3528, 3780, 4032, 4284, 4536, 4788, 5040, 5292, 5544, 5796, 6048, 6300, 6552, 6804, 7056, 7308, 7560, 7812, 8064, 8316, 8568, 8820, 9072, 9324, 9576, 9828, 10080, 10332, 10584, 10836, 11088, 11340

Offset: 0

Derek Orr, May 17 2014

As lcm(1,2,3,...,9) = 2520, 10*a(n) + k is divisible by each k from 1 through 9.

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

Cf. A001477, A008600, A008603, A008589, A008591, A008594, A008596.

Cf. A044102, A135628.

252*Range[0, 49] (* Alonso del Arte, May 17 2014 *)
LinearRecurrence[{2,-1},{0,252},50] (* Harvey P. Dale, Mar 25 2025 *)

PARI
```
for(n=0,50,print(252*n))
```

From Elmo R. Oliveira, Apr 16 2024: (Start)
G.f.: 252*x/(x-1)^2.
E.g.f.: 252*x*exp(x).
a(n) = 2*a(n-1) - a(n-2) for n >= 2.
a(n) = 7*A044102(n) = 9*A135628(n) = 12*A008603(n) = 14*A008600(n) = 18*A008596(n) = 21*A008594(n) = 28*A008591(n) = 36*A008589(n) = 252*A001477(n). (End)

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

A158186 a(n) = 10*n^2 - 7*n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A153432 Numbers m such that all ten numbers 10^k*m+1; k=0,1,...,9 are prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A169825 Multiples of 420.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A169827 Multiples of 840.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A179849 Sum of prime p and next prime after p is divisible by 7.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma

Mathematica

PARI

A358053 a(n) = 14*n - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A233207 Triangle T(n,k), read by rows, given by T(n+k,k)=2*k*(2*n+1).

Views

Author

Keywords

Comments

A158186 a(n) = 10n^2 - 7n + 1.

A233207 Triangle T(n,k), read by rows, given by T(n+k,k)=2k(2*n+1).