A008594 -id:A008594 - cp's OEIS frontend

A168186 Positive numbers that are not multiples of 12.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78

Offset: 1

Reinhard Zumkeller, Nov 30 2009

121 (in decimal) is a member of this sequence but not a member of A023805. - Robert Munafo, Jan 26 2010

80 is a member of this sequence but is not a member of A160453. - Franklin T. Adams-Watters, Jan 26 2010

			156 is in A023805 but not in this sequence. - _Franklin T. Adams-Watters_, Jan 26 2010

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,1,-1).

Complement of A008594.
Cf. A109015, A168185.
All three of A023805, A160453, A168186 are different.

Select[Table[n,{n,200}],Mod[#,12]!=0&] (* Vladimir Joseph Stephan Orlovsky, Feb 18 2011*)

A168185(a(n)) = 1.
A109015(a(n)) < 12.
From Chai Wah Wu, Jan 16 2020: (Start)
a(n) = a(n-1) + a(n-11) - a(n-12) for n > 12.
G.f.: x*(x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)/(x^12 - x^11 - x + 1). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (12*sqrt(6) - 4*sqrt(3) + 6*sqrt(2) - 15)*Pi/72. - Amiram Eldar, May 12 2025

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 *)

A017522 a(n) = (12*n)^2.

Original entry on oeis.org

0, 144, 576, 1296, 2304, 3600, 5184, 7056, 9216, 11664, 14400, 17424, 20736, 24336, 28224, 32400, 36864, 41616, 46656, 51984, 57600, 63504, 69696, 76176, 82944, 90000, 97344, 104976, 112896, 121104, 129600, 138384, 147456, 156816, 166464, 176400, 186624, 197136

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 (n^2), A008594 (12*n).

[(12*n)^2: n in [0..35]]; // Vincenzo Librandi, Feb 10 2012

LinearRecurrence[{3, -3, 1}, {0, 144, 576}, 50] (* Vincenzo Librandi, Feb 10 2012 *)
(12 Range[0, 30])^2 (* Bruno Berselli, Feb 10 2012 *)

a(n)=(12*n)^2 \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 144*x*(1+x)/(1-x)^3. - Bruno Berselli, Feb 10 2012
From Amiram Eldar, Jan 25 2021: (Start)
Sum_{n>=1} 1/a(n) = Pi^2/864.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/1728.
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/12)/(Pi/12).
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/12)/(Pi/12) = 3*sqrt(2)*(sqrt(3)-1)/Pi. (End)
From Elmo R. Oliveira, Dec 02 2024: (Start)
E.g.f.: 144*x*(1 + x)*exp(x).
a(n) = 144*A000290(n) = A008594(n)^2 = A000290(A008594(n)).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A158547 a(n) = 24*n^2 + 1.

Original entry on oeis.org

1, 25, 97, 217, 385, 601, 865, 1177, 1537, 1945, 2401, 2905, 3457, 4057, 4705, 5401, 6145, 6937, 7777, 8665, 9601, 10585, 11617, 12697, 13825, 15001, 16225, 17497, 18817, 20185, 21601, 23065, 24577, 26137, 27745, 29401, 31105, 32857, 34657, 36505, 38401, 40345

Offset: 0

Vincenzo Librandi, Mar 21 2009

The identity (24*n^2 + 1)^2 - (144*n^2 + 12)*(2*n)^2 = 1 can be written as a(n)^2 - A158546(n)*A005843(n)^2 = 1.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Leo Tavares, Illustration: Stellar Stars.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A005843, A008594, A158546, A227776.

I:=[1, 25, 97]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Feb 14 2012

LinearRecurrence[{3, -3, 1}, {1, 25, 97}, 50] (* Vincenzo Librandi, Feb 14 2012 *)

for(n=0, 40, print1(24*n^2+1", ")); \\ Vincenzo Librandi, Feb 14 2012

G.f.: (1 + 22*x + 25*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 02 2023: (Start)
Sum_{n>=0} 1/a(n) = 1/2 + coth(Pi/(2*sqrt(6)))*Pi/(4*sqrt(6)).
Sum_{n>=0} (-1)^n/a(n) = 1/2 + cosech(Pi/(2*sqrt(6)))*Pi/(4*sqrt(6)). (End)
From Elmo R. Oliveira, Jan 16 2025: (Start)
E.g.f.: exp(x)*(1 + 24*x + 24*x^2).
a(n) = A227776(2*n). (End)

Comment rewritten, a(0) added by R. J. Mathar, Oct 16 2009

A207709 Floor((H(n) + exp(H(n))*log(H(n)))/sigma(n)), where H(n) is the harmonic number sum_{i=1..n} 1/i.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2

Offset: 1

Arkadiusz Wesolowski, Feb 19 2012

nonn

An assertion equivalent to the Riemann hypothesis is: a(n) > 0 for every n >= 1.
a(12*n) = 1 for all 1<=n<=43312.
For n >= 1, a(2^(10^n)) so far appears to equal floor(n*(exp(1)-2/3) - 1/3).
There exist integers n such that (H(n) + exp(H(n))*log(H(n)))/sigma(n) < 1.01 (i.e., n = 100630609505753353981293837481689271234222794240000*1087#). See A215640 for information on how to generate these numbers. - Arkadiusz Wesolowski, Aug 19 2012

			a(11) = 2 because (H(11) + exp(1)^H(11)*log(H(11)))/sigma(11) = 2.1387006307....

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543.
Eric W. Weisstein, MathWorld: Riemann Hypothesis
Wikipedia, Jeffrey Lagarias

Cf. A057641, A000203, A008594, A076633.

lst = {}; Do[h = NIntegrate[(1 - x^n)/(1 - x), {x, 0, 1}]; AppendTo[lst, Floor[(h + Exp@h*Log@h)/DivisorSigma[1, n]]], {n, 530}]; lst

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).

A166873 a(n) = a(n-1) + 12*n for n > 1; a(1) = 1.

Original entry on oeis.org

1, 25, 61, 109, 169, 241, 325, 421, 529, 649, 781, 925, 1081, 1249, 1429, 1621, 1825, 2041, 2269, 2509, 2761, 3025, 3301, 3589, 3889, 4201, 4525, 4861, 5209, 5569, 5941, 6325, 6721, 7129, 7549, 7981, 8425, 8881, 9349, 9829, 10321, 10825, 11341, 11869

Offset: 1

Klaus Brockhaus, Oct 22 2009

Binomial transform of 1,24,12,0,0,0,....

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

Cf. A008594 (multiples of 12).

A000217, A028387, A133694, A059993, A166137, A166143, A166146, A166147, A166148, A166150, A166144 have recurrence a(n-1)+k*n with a(1)=1 or a(0)=1 for k = 1..11 resp.

[ n eq 1 select 1 else Self(n-1)+12*n: n in [1..44] ];

LinearRecurrence[{3,-3,1},{1,25,61},50] (* G. C. Greubel, May 27 2016 *)

a(n)=6*n^2+6*n-11 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 6*n^2 + 6*n - 11.
a(n) = 2*a(n-1) - a(n-2) + 12.
G.f.: x*(1 + 22*x - 11*x^2)/(1-x)^3.
a(n) - a(n-1) = A008594(n) for n > 1.
From G. C. Greubel, May 27 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
E.g.f.: (-11 + 12*x + 6*x^2)*exp(x) + 11. (End)

A181475 a(n) = 3n^4 + 6n^3 - 3*n + 1.

Original entry on oeis.org

1, 7, 91, 397, 1141, 2611, 5167, 9241, 15337, 24031, 35971, 51877, 72541, 98827, 131671, 172081, 221137, 279991, 349867, 432061, 527941, 638947, 766591, 912457, 1078201, 1265551, 1476307, 1712341, 1975597, 2268091, 2591911, 2949217, 3342241, 3773287, 4244731

Offset: 0

Bruno Berselli, Oct 25 2010 - Oct 29 2010

If gcd(n,7) = gcd(n+1,7) = gcd(2*n+1,7) = 1 then a(n) == 0 (mod 7) (E. Picutti, see References).

Ettore Picutti, Sul numero e la sua storia, Feltrinelli Economica, 1977, p. 208.

Bruno Berselli, Table of n, a(n) for n = 0..10000.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Subsequence of A003215.

Magma
```
[3*n^4+6*n^3-3*n+1: n in [0..31]];
```

Table[3 n^4 + 6 n^3 - 3 n + 1, {n, 0, 40}] (* Vincenzo Librandi, Mar 26 2013 *)
LinearRecurrence[{5,-10,10,-5,1},{1,7,91,397,1141},40] (* Harvey P. Dale, Jul 12 2022 *)

G.f.: (1 + 2*x + 66*x^2 + 2*x^3 + x^4)/(1-x)^5.
a(n) = a(-n-1) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) + 6*12.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + 6*A008594(n-1).
a(n) = 2*a(n-1) - a(n-2) + 6*A003154(n).
a(n) = a(n-1) + 6*A007588(n).
a(n) = 1 + 6*A062392(n).
a(n) = 7*A000540(n)/A000330(n) = A154105(A000096(n-1)) for n > 0.
Sum_{i=0..n} a(i) = (3*n^5 + 15*n^4 + 20*n^3 - 3*n + 5)/5.
a(n) = 7*(3*n^2 + 3*n - 1)*(Sum_{k=1..n} k^6)/(5*Sum_{k=1..n} k^4), n > 0. - Gary Detlefs, Oct 18 2011

Formula, program and crossref added by Bruno Berselli, Aug 22 2011

A246930 a(n) = prime(12*n).

Original entry on oeis.org

37, 89, 151, 223, 281, 359, 433, 503, 593, 659, 743, 827, 911, 997, 1069, 1163, 1249, 1321, 1439, 1511, 1601, 1693, 1783, 1877, 1987, 2069, 2143, 2267, 2347, 2423, 2543, 2657, 2713, 2801, 2903, 3011, 3119, 3221, 3323, 3413, 3527, 3607, 3697, 3797, 3907

Offset: 1

Vincenzo Librandi, Sep 08 2014

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. similar sequences listed in A246929.

Cf. A000040, A008594.

Magma
```
[NthPrime(12*n): n in [1..50]];
```

A246930:=n->ithprime(12*n): seq(A246930(n), n=1..50); # Wesley Ivan Hurt, Sep 08 2014

Mathematica
```
Prime[12 Range[50]]
```

vector(60, i, prime(12*i)) \\ Michel Marcus, Sep 08 2014

[nth_prime(12*n) for n in (1..50)] # Bruno Berselli, Sep 08 2014

a(n) = A000040(A008594(n)). - Wesley Ivan Hurt, Sep 08 2014

A272975 Numbers that are congruent to {0,7} mod 12.

Original entry on oeis.org

0, 7, 12, 19, 24, 31, 36, 43, 48, 55, 60, 67, 72, 79, 84, 91, 96, 103, 108, 115, 120, 127, 132, 139, 144, 151, 156, 163, 168, 175, 180, 187, 192, 199, 204, 211, 216, 223, 228, 235, 240, 247, 252, 259, 264, 271, 276, 283, 288, 295, 300, 307, 312, 319, 324

Offset: 1

Wesley Ivan Hurt, May 30 2016

Numbers that are not congruent to {1, 2, 3, 4, 5, 6, 8, 9, 10, 11} mod 12.

Bisection of A083032.

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

Cf. A008594, A017605, A083032, A049453, A049598.

[n : n in [0..400] | n mod 12 in [0, 7]];

A272975:=n->(12*n-11+(-1)^n)/2: seq(A272975(n), n=1..100);

Mathematica
```
Table[(12n - 11 + (-1)^n)/2, {n, 80}]
```

concat(0, Vec(x^2*(7+5*x)/((x-1)^2*(x+1)) + O(x^99))) \\ Altug Alkan, May 31 2016

G.f.: x^2*(7+5*x) / ((x-1)^2*(x+1)).
a(n) = a(n-1) + a(n-2) - a(n-3) for n>3.
a(n) = (12*n - 11 + (-1)^n)/2.
a(2k) = A017605(k-1) k>0, a(2k-1) = A008594(k-1) k>0, a(2k)-a(2k-1) = 7.
a(n)-a(-n) = A008594(n) for n>0.
Sum_{i=1..n} a(2*i) = A049453(n) for n>0.
Sum_{i=1..n} a(2*i-1) = A049598(n-1) for n>0.
E.g.f.: 5 + ((12*x - 11)*exp(x) + exp(-x))/2. - David Lovler, Sep 04 2022
Sum_{n>=2} (-1)^n/a(n) = log(2)/4 + log(3)/8 - ((sqrt(3)-1)*Pi + 2*(sqrt(3)+3)*log(sqrt(3)+2))/(24*(sqrt(3)+1)). - Amiram Eldar, Sep 17 2023

cp's OEIS Frontend

A168186 Positive numbers that are not multiples of 12.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

A017522 a(n) = (12*n)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A158547 a(n) = 24*n^2 + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A207709 Floor((H(n) + exp(H(n))*log(H(n)))/sigma(n)), where H(n) is the harmonic number sum_{i=1..n} 1/i.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

A166873 a(n) = a(n-1) + 12*n for n > 1; a(1) = 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

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

A181475 a(n) = 3n^4 + 6n^3 - 3*n + 1.