A297127 -id:A297127 - cp's OEIS frontend

A297124 Numbers having an up-first zigzag pattern in base 3; see Comments.

5, 15, 16, 46, 47, 48, 50, 138, 140, 141, 142, 145, 146, 150, 151, 415, 416, 420, 421, 424, 425, 426, 428, 435, 437, 438, 439, 451, 452, 453, 455, 1245, 1247, 1248, 1249, 1261, 1262, 1263, 1265, 1272, 1274, 1275, 1276, 1279, 1280, 1284, 1285, 1306, 1307

Offset: 1

Clark Kimberling, Jan 13 2018

A number n having base-b digits d(m), d(m-1), ..., d(0) such that d(i) != d(i+1) for 0 <= i < m shows a zigzag pattern of one or more segments, in the following sense. Writing U for up and D for down, there are two kinds of patterns: U, UD, UDU, UDUD, ... and D, DU, DUD, DUDU, ... . In the former case, we say n has an "up-first zigzag pattern in base b"; in the latter, a "down-first zigzag pattern in base b". Example: 2,4,5,3,0,1,4,2 has segments 2,4,5; 5,3,0; 0,1,4; and 4,2, so that 24530142, with pattern UDUD, has an up-first zigzag pattern in base 10, whereas 4,2,5,3,0,1,4,2 has a down-first pattern. The sequences A297124..A297127 partition the natural numbers. See the guide at A297146.

			Base-3 digits of 1307: 1,2,1,0,1,0,1, with pattern UDUDU, so that 1307 is in the sequence.

Cf. A297125, A297126.

a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
b = 3; t = Table[a[n, b], {n, 1, 10*z}];
u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297124 *)
v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297125 *)
Complement[Range[z], Union[u, v]]  (* A297126 *)

A297125 Numbers having a down-first zigzag pattern in base 3; see Comments.

Original entry on oeis.org

3, 6, 7, 10, 11, 19, 20, 21, 23, 30, 32, 33, 34, 57, 59, 60, 61, 64, 65, 69, 70, 91, 92, 96, 97, 100, 101, 102, 104, 172, 173, 177, 178, 181, 182, 183, 185, 192, 194, 195, 196, 208, 209, 210, 212, 273, 275, 276, 277, 289, 290, 291, 293, 300, 302, 303, 304

Offset: 1

Clark Kimberling, Jan 13 2018

A number n having base-b digits d(m), d(m-1), ..., d(0) such that d(i) != d(i+1) for 0 <= i < m shows a zigzag pattern of one or more segments, in the following sense. Writing U for up and D for down, there are two kinds of patterns: U, UD, UDU, UDUD, ... and D, DU, DUD, DUDU, ... . In the former case, we say n has an "up-first zigzag pattern in base b"; in the latter, a "down-first zigzag pattern in base b". Example: 2,4,5,3,0,1,4,2 has segments 2,4,5; 5,3,0; 0,1,4; and 4,2, so that 24530142, with pattern UDUD, has an up-first zigzag pattern in base 10, whereas 4,2,5,3,0,1,4,2 has a down-first pattern. The sequences A297124..A297127 partition the natural numbers. See the guide at A297146.

			Base-3 digits of 307: 1,0,2,1,0,1, with pattern DUDU, so that 307 is in the sequence.

Cf. A297124, A297126.

a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
b = 3; t = Table[a[n, b], {n, 1, 10*z}];
u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297124 *)
v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297125 *)
Complement[Range[z], Union[u, v]]  (* A297126 *)

A297494 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^10H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

Original entry on oeis.org

513, 20708, 584874, 4714408, 72449100, 200562418, 1012788198, 1953009460, 6172747128, 24788658690, 37242612640, 107770200778, 198936710910, 265200653548, 449592659568, 931777815258, 1775665528380, 2155635964450, 3812897562148, 5368106367720, 6351988507678

Offset: 1

Seiichi Manyama, Dec 31 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1000
N. Lygeros, O. Rozier, A new solution to the equation tau(p) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.
Eric Weisstein's World of Mathematics, Tau Function.

(1/2) * Sum_{|k|<=2*sqrt(p)} k^m*H(4*p-k^2): A000040 (m=0), A084920 (m=2), A297491 (m=4), A297492 (m=6), A297493 (m=8), this sequence (m=10).

Cf. A000594, A259825, A295645, A297127.

Let b(n) = 42*n^6 - 90*n^4 - 75*n^3 - 35*n^2 - 9*n - 1.
a(n) = b(prime(n)) - tau(prime(n)) where tau(n)=A000594(n) is Ramanujan's tau function.
So tau(prime(n)) + 1 == -a(n) (mod prime(n)).

A297122 a(n) = n^5*H(n) where H() is the Hurwitz class number.

Original entry on oeis.org

0, 0, 0, 81, 512, 0, 0, 16807, 32768, 0, 0, 161051, 331776, 0, 0, 1518750, 1572864, 0, 0, 2476099, 6400000, 0, 0, 19309029, 15925248, 0, 0, 19131876, 34420736, 0, 0, 85887453, 100663296, 0, 0, 105043750, 151165440, 0, 0, 360896796, 204800000, 0, 0, 147008443

Offset: 0

Seiichi Manyama, Dec 26 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..5000

Cf. A000594, A259825, A297127.

PARI
```
{a(n) = n^5*qfbhclassno(n)}
```

a(n) = n^5*A259825(n)/12.

a(4*n+1) = a(4*n+2) = 0.

A297123 a(n) = 462n^6 + 330n^4 - 165n^3 + 55n^2 - 11*n + 1.

Original entry on oeis.org

1, 672, 33727, 359536, 1967109, 7405696, 21949027, 55092192, 122381161, 247574944, 465140391, 823079632, 1386090157, 2239057536, 3490880779, 5278630336, 7772038737, 11178323872, 15747344911, 21777090864, 29619501781, 39686622592, 52457089587

Offset: 0

Seiichi Manyama, Dec 26 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Olivier Rozier, Ramanujan's tau function
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000594, A297127.

{a(n) = 462*n^6+330*n^4-165*n^3+55*n^2-11*n+1}

Vec((1 + 665*x + 29044*x^2 + 137524*x^3 + 135139*x^4 + 29243*x^5 + 1024*x^6) / (1 - x)^7 + O(x^40)) \\ Colin Barker, Dec 26 2017

From Colin Barker, Dec 26 2017: (Start)
G.f.: (1 + 665*x + 29044*x^2 + 137524*x^3 + 135139*x^4 + 29243*x^5 + 1024*x^6) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>6.
(End)

A297424 a(n) = tau((10^n)-th prime) where tau(n)=A000594(n) is Ramanujan's tau function.

Original entry on oeis.org

-24, 128406630, -1695266465052058, -4467161474022023509680, 2643202687128887204371152330, 5631587063815097155948902224731910, -6022712388820421179671063354886818730888, 3519878895631571325515625172934456394359869922, 452493700789420934344344052367209646628330983653992

Offset: 0

Seiichi Manyama, Dec 30 2017

sign

Olivier Rozier, Ramanujan's tau function

Cf. A000594, A006988, A076847, A297127.

PARI
```
{a(n) = ramanujantau(prime(10^n))}
```

a(n) = A076847(10^n) = A000594(A006988(n)).

cp's OEIS Frontend

A297124 Numbers having an up-first zigzag pattern in base 3; see Comments.

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Mathematica

A297125 Numbers having a down-first zigzag pattern in base 3; see Comments.

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Mathematica

A297494 a(n) = (1/2) * Sum_{|k|<=2*sqrt(p)} k^10*H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

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Formula

A297122 a(n) = n^5*H(n) where H() is the Hurwitz class number.

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Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A297123 a(n) = 462*n^6 + 330*n^4 - 165*n^3 + 55*n^2 - 11*n + 1.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A297424 a(n) = tau((10^n)-th prime) where tau(n)=A000594(n) is Ramanujan's tau function.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A297494 a(n) = (1/2) * Sum_{|k|<=2sqrt(p)} k^10H(4*p-k^2) where H() is the Hurwitz class number and p is n-th prime.

A297123 a(n) = 462n^6 + 330n^4 - 165n^3 + 55n^2 - 11*n + 1.