A008602 -id:A008602 - cp's OEIS frontend

A215145 a(n) = 20*n + 1.

1, 21, 41, 61, 81, 101, 121, 141, 161, 181, 201, 221, 241, 261, 281, 301, 321, 341, 361, 381, 401, 421, 441, 461, 481, 501, 521, 541, 561, 581, 601, 621, 641, 661, 681, 701, 721, 741, 761, 781, 801, 821, 841, 861, 881, 901, 921, 941, 961, 981

Offset: 0

Jeremy Gardiner, Aug 04 2012

Jeremy Gardiner, Table of n, a(n) for n = 0..999
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A124204, A008602.

I:=[1,21]; [n le 2 select I[n] else 2*Self(n-1) - Self(n-2): n in [1..30]]; // G. C. Greubel, Apr 19 2018

Range[1, 1001, 20]
LinearRecurrence[{2, -1}, {1, 21}, 50] (* G. C. Greubel, Apr 19 2018 *)

for(n=0, 50, print1(20*n + 1, ", ")) \\ G. C. Greubel, Apr 19 2018

(1 to 1001 by 20).toList // Alonso del Arte, Feb 20 2020

From G. C. Greubel, Apr 19 2018: (Start)
a(n) = 2*a(n - 1) - a(n - 2).
G.f.: (1 + 19*x)/(1 - x)^2.
E.g.f.: (20*x + 1)*exp(x). (End)

A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.

Original entry on oeis.org

1, 3, 4, 5, 8, 9, 7, 12, 15, 16, 9, 16, 21, 24, 25, 11, 20, 27, 32, 35, 36, 13, 24, 33, 40, 45, 48, 49, 15, 28, 39, 48, 55, 60, 63, 64, 17, 32, 45, 56, 65, 72, 77, 80, 81, 19, 36, 51, 64, 75, 84, 91, 96, 99, 100, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 121

Offset: 0

Andrew S. Plewe and Franklin T. Adams-Watters, Jul 11 2005

A "Goldbach Conjecture" for this sequence: when there are n terms between consecutive odd integers (2n+1) and (2n+3) for n > 0, at least one will be the product of 2 primes (not necessarily distinct). Example: n=3 for consecutive odd integers a(7)=7 and a(11)=9 and of the 3 sequence entries a(8)=12, a(9)=15 and a(10)=16 between them, one is the product of 2 primes a(9)=15=3*5. - Michael Hiebl, Jul 15 2007
A024352 gives distinct values in the array, minus the first row (1, 4, 9, 16, etc.). a(n) gives all solutions to the equation x^2 + xy = n, with y mod 2 = 0, x > 0, y >= 0. - Andrew S. Plewe, Oct 19 2007
Alternatively, triangular sequence of coefficients of Dynkin diagram weights for the Cartan groups C_n: t(n,m) = m*(2*n - m). Row sums are A002412. - Roger L. Bagula, Aug 05 2008

			Array begins:
  1  4  9 16 25 36  49  64  81 100 ...
  3  8 15 24 35 48  63  80  99 120 ...
  5 12 21 32 45 60  77  96 117 140 ...
  7 16 27 40 55 72  91 112 135 160 ...
  9 20 33 48 65 84 105 128 153 180 ...
  ...
Triangle begins:
   1;
   3,  4;
   5,  8,  9;
   7, 12, 15, 16;
   9, 16, 21, 24, 25;
  11, 20, 27, 32, 35, 36;
  13, 24, 33, 40, 45, 48, 49;
  15, 28, 39, 48, 55, 60, 63, 64;
  17, 32, 45, 56, 65, 72, 77, 80, 81;
  19, 36, 51, 64, 75, 84, 91, 96, 99, 100;

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

G. C. Greubel, Antidiagonals n = 0..50, flattened

Rows give A000290, A005563, A028347, A028560, A028566, A098603, A098847, A098848, A098849, A098850.
Columns give A005408, A008586, A016945, A008590, A017329, A008594, A008598, A008602, A008606, A000567.
Diagonals give A033428, A045944, A067725.
Cf. A000384, A002412, A024352, A105020, A128076.

[(k+1)*(2*n-k+1): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 15 2023

t[n_, m_]:= (n^2 - m^2); Flatten[Table[t[i, j], {i,12}, {j,i-1,0,-1}]]
(* to view table *) Table[t[i, j], {j,0,6}, {i,j+1,10}]//TableForm (* Robert G. Wilson v, Jul 11 2005 *)
Table[(k+1)*(2*n-k+1), {n,0,15}, {k,0,n}]//Flatten (* Roger L. Bagula, Aug 05 2008 *)

def A105020(n,k): return (k+1)*(2*n-k+1)
flatten([[A105020(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Mar 15 2023

a(n) = r^2 - (r^2 + r - m)^2/4, where r = round(sqrt(m)) and m = 2*n+2. - Wesley Ivan Hurt, Sep 04 2021
a(n) = A128076(n+1) * A105020(n+1). - Wesley Ivan Hurt, Jan 07 2022
From G. C. Greubel, Mar 15 2023: (Start)
Sum_{k=0..n} T(n, k) = A002412(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*((1+(-1)^n)*A000384((n+2)/2) - (1- (-1)^n)*A000384((n+1)/2)). (End)

More terms from Robert G. Wilson v, Jul 11 2005

A317320 Multiples of 20 and odd numbers interleaved.

Original entry on oeis.org

0, 1, 20, 3, 40, 5, 60, 7, 80, 9, 100, 11, 120, 13, 140, 15, 160, 17, 180, 19, 200, 21, 220, 23, 240, 25, 260, 27, 280, 29, 300, 31, 320, 33, 340, 35, 360, 37, 380, 39, 400, 41, 420, 43, 440, 45, 460, 47, 480, 49, 500, 51, 520, 53, 540, 55, 560, 57, 580, 59, 600, 61, 620, 63, 640, 65, 660, 67, 680, 69

Offset: 0

Omar E. Pol, Jul 25 2018

Partial sums give the generalized 24-gonal numbers (A303814).

a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 24-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. A008602 and A005408 interleaved.
Column 20 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. A303814.

With[{nn=40},Riffle[20*Range[0,nn],Range[1,2*nn+1,2]]] (* Harvey P. Dale, Feb 16 2020 *)

concat(0, Vec(x*(1 + 20*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ Colin Barker, Jul 29 2018

a(n) = n, if n is odd.
a(n) = 10*n, if n is even.
a(2n) = 20*n, a(2n+1) = 2*n + 1.
From Colin Barker, Jul 29 2018: (Start)
G.f.: x*(1 + 20*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) = 5*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 + 9*2^(1-s)). - Amiram Eldar, Oct 26 2023

A257391 Numbers of the form 4^p(4^p+1)(2^p-1) with p an odd prime.

Original entry on oeis.org

29120, 32537600, 34093383680, 36011213418659840, 36888985097480437760, 38685331082014736871587840, 39614005699412557795646504960, 41538369916519054182462860998737920, 44601490313984496701256699111250939955118080, 45671926145323068271210017365594287580527984640

Offset: 1

Danny Rorabaugh, Apr 21 2015

5 divides (4^m+1) for odd m, so every term in this sequence is a multiple of 5 (A008587).
A064487(k) = 4^(2k+1)*(4^(2k+1)+1)*(2^(2k+1)-1), so this sequence is a subsequence of A064487.
Every non-solvable number (A056866) is divisible by 12 or 20. All non-solvable numbers not divisible by 12 (A008594) are divisible by a member of this sequence. In particular, every primitive non-solvable number (A257146) not divisible by 12 is in this sequence.
All terms are divisible by 320 and have at least 4 distinct prime factors. - Jianing Song, Apr 04 2022

See A056866 and A064487 for references.

Danny Rorabaugh, Table of n, a(n) for n = 1..100

Subsequence of A008587, A008602, A056866, and A064487.

Table[4^p (4^p+1)(2^p-1),{p,Prime[Range[2,20]]}] (* Harvey P. Dale, Jul 17 2024 *)

a(n)=my(p=prime(n+1)); 4^p*(4^p+1)*(2^p-1) \\ Charles R Greathouse IV, Apr 21 2015

[4^nth_prime(n)*(4^nth_prime(n)+1)*(2^nth_prime(n)-1) for n in range(2,12)]

a(n) = 4^p*(4^p+1)*(2^p-1) where p = A065091(n) = A000040(n+1).

A117798 Icosagonal numbers divisible by 20.

Original entry on oeis.org

0, 20, 820, 1200, 3440, 4180, 7860, 8960, 14080, 15540, 22100, 23920, 31920, 34100, 43540, 46080, 56960, 59860, 72180, 75440, 89200, 92820, 108020, 112000, 128640, 132980, 151060, 155760, 175280, 180340, 201300, 206720, 229120, 234900, 258740

Offset: 1

Luc Stevens (lms022(AT)yahoo.com), Apr 29 2006

Intersection of A008602 and A051872. - Michel Marcus, Feb 27 2014

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A008602, A051872.

Select[Table[PolygonalNumber[20, n], {n, 0, 200}], Divisible[#, 20] &] (* Amiram Eldar, Mar 22 2021 *)

isok(n) = ispolygonal(n, 20) && !(n % 20); \\ Michel Marcus, Feb 27 2014

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)

A317095 a(n) = 40*n.

Original entry on oeis.org

0, 40, 80, 120, 160, 200, 240, 280, 320, 360, 400, 440, 480, 520, 560, 600, 640, 680, 720, 760, 800, 840, 880, 920, 960, 1000, 1040, 1080, 1120, 1160, 1200, 1240, 1280, 1320, 1360, 1400, 1440, 1480, 1520, 1560, 1600, 1640, 1680, 1720, 1760, 1800, 1840, 1880

Offset: 0

Felix Fröhlich, Sep 07 2018

a(n) is equal to the freshwater zone below sea level for a water table of elevation n above sea level in a simplified freshwater-saltwater interface in a coastal water-table aquifer (cf. Barlow, 2003, p. 14, eq. (2) and p. 15, Fig. B-1 and B-2).
From Bruno Berselli, Sep 10 2018: (Start)
After 0, subsequence of A065607: 1/a(n)^2 + 1/(30*n)^2 = 1/(24*n)^2, with n > 0 and a(n) > 30*n.
Also, all positive terms belong to A049094: 2^(40*n)-1 = 1024^(4*n)-1 and (25*41-1)^(4*n)-1 is divisible by 25. (End)

P. M. Barlow, Ground Water in Freshwater-Saltwater Environments of the Atlantic Coast, Circular 1262 (2003).
Wikipedia, Aquifer
Wikipedia, Saltwater intrusion
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A005843, A007395, A008586, A008592, A008602, A010692, A049094, A065607.
Row n = 40 of A004247. Intersection of A008587 and A008590.
After 0, subsequence of A005101.

Table[40 n, {n, 0, 50}] (* or *)
LinearRecurrence[{2, -1}, {0, 40}, 50] (* or *)
CoefficientList[Series[40*x/(1 - x)^2, {x, 0, 50}], x] (* Stefano Spezia, Sep 07 2018 *)

PARI
```
a(n) = 40*n
```

a(n) = if(n==0, 0, if(n==1, 40, 2*a(n-1)-a(n-2)))

PARI
```
concat(0, Vec(40*x/(1-x)^2 + O(x^60)))
```

O.g.f.: 40*x/(1 - x)^2.
E.g.f.: 40*x*exp(x). - Bruno Berselli, Sep 10 2018
a(n) = 2*a(n - 1) - a(n - 2) for n > 1. - Stefano Spezia, Sep 07 2018
a(n) = A008586(A008592(n)) = 4*A008592(n).
a(n) = A010692(n)*A008586(n) = 10*A008586(n).
a(n) = A008602(A005843(n)) = 20*A005843(n).
a(n) = A007395(n)*A008602(n) = 2*A008602(n).

A332410 a(n) = 2a(n-1) - a(n-2) + a(n-5) - 2a(n-6) + a(n-7) with a(0)=0, a(1)=1, a(2)=3, a(3)=6, a(4)=11, a(5)=17, a(6)=24.

Original entry on oeis.org

0, 1, 3, 6, 11, 17, 24, 32, 41, 52, 64, 77, 91, 106, 123, 141, 160, 180, 201, 224, 248, 273, 299, 326, 355, 385, 416, 448, 481, 516, 552, 589, 627, 666, 707, 749, 792, 836, 881, 928, 976, 1025, 1075, 1126, 1179

Offset: 0

Paul Curtz, Feb 11 2020

This sequence occurs twice as a linear spoke in the hexagonal spiral constructed from A002266:
                       17  17  17  17  17  18  18
                     16  11  11  11  11  12  12  18
                   16  11   6   6   7   7   7  12  18
                 16  10   6   3   3   3   3   7  12  18
               16  10   6   3   1   1   1   4   7  12  19
             16  10   6   2   0   0   0   1   4   8  13  19
               15  10   5   2   0   0   1   4   8  13  19
                 15  10   5   2   2   2   4   8  13  19
                   15   9   5   5   5   4   8  13  19
                     15   9   9   9   9   8  13  20
                       15  14  14  14  14  14  20
a(-1-n) = 0, 1, 4, 8, 13, 19, 26, 35, 45, ... also occurs twice in the same spiral.
Difference table:
  0, 1, 3, 6, 11, 17, 24, 32, 41, 52, ... = a(n)
  1, 2, 3, 5,  6,  7,  8,  9, 11, 12, ... = A047256(n+1)
  1, 1, 2, 1,  1,  1,  1,  2,  1,  1, ... = A130782.
There is no linear spoke with three copies in this spiral. Compare with the spiral illustrated in sequence A330707 and constructed from A002265 where the same spokes occur three times: A006578, A001859 and A077043, essentially. Strictly, three times from 1, 1, 1 for A006578, from 2, 2, 2 for A001859 and from 7, 7, 7 for A077043.

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

Cf. A002266, A008602, A047256, A130782, A330707.

Cf. A001859, A006578, A077043.

LinearRecurrence[{2, -1, 0, 0, 1, -2, 1}, {0, 1, 3, 6, 11, 17, 24}, 45] (* Amiram Eldar, Feb 12 2020 *)

concat(0, Vec(x*(1 + x)*(1 + x^2 + x^3) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)) + O(x^50))) \\ Colin Barker, Feb 11 2020, Apr 24 2020

a(8+n) - a(8-n) = 20*n.

G.f.: x*(1 + x)*(1 + x^2 + x^3) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Feb 11 2020

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);

A105179 Numbers having in decimal representation the same final digit as their largest proper divisor has; a(1) = 1.

Original entry on oeis.org

1, 11, 15, 20, 25, 31, 40, 41, 45, 60, 61, 71, 75, 80, 100, 101, 105, 120, 121, 125, 131, 135, 140, 143, 151, 160, 165, 175, 180, 181, 187, 191, 195, 200, 209, 211, 220, 225, 240, 241, 251, 253, 255, 260, 271, 275, 280, 281, 285, 300, 311, 315, 319, 320, 325

Offset: 1

Reinhard Zumkeller, Apr 29 2005

A010879(A032742(a(n))) = A010879(a(n));
A008602 is a subsequence apart from the initial term;
A030430 is a subsequence.

			n=105=3*35 and 105 == 35 modulo 10, therefore 105 is a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to final digits of numbers

a105179 n = a105179_list !! (n-1)
a105179_list = 1 : filter (\x -> a010879 (a032742 x) == a010879 x) [2..]
-- Reinhard Zumkeller, Jan 10 2013

Join[{1},Select[Range[2,400],Last[IntegerDigits[Divisors[#][[-2]]]] == Last[IntegerDigits[#]]&]]  (* Harvey P. Dale, Apr 21 2011 *)

lpf(n)=factor(n)[1,1]
is(n)=if(n%2, n%15==0||n%25==0||n==1||lpf(n)%5==1, n%20==0) \\ Charles R Greathouse IV, Jan 02 2013

cp's OEIS Frontend

A215145 a(n) = 20*n + 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Scala

Formula

A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A317320 Multiples of 20 and odd numbers interleaved.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A257391 Numbers of the form 4^p*(4^p+1)*(2^p-1) with p an odd prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

A117798 Icosagonal numbers divisible by 20.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A169825 Multiples of 420.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A317095 a(n) = 40*n.

Views

Author

Keywords

Comments

Links

Crossrefs

A257391 Numbers of the form 4^p(4^p+1)(2^p-1) with p an odd prime.

A332410 a(n) = 2a(n-1) - a(n-2) + a(n-5) - 2a(n-6) + a(n-7) with a(0)=0, a(1)=1, a(2)=3, a(3)=6, a(4)=11, a(5)=17, a(6)=24.