A063938 -id:A063938 - cp's OEIS frontend

A299205 Numbers k such that k-1 divides tau(k), where tau(k) = A000594(k) is Ramanujan's tau function.

2, 3, 10, 14, 15, 56, 57, 59, 70, 85, 105, 107, 116, 136, 209, 267, 295, 323, 352, 393, 415, 442, 530, 551, 645, 646, 760, 855, 1197, 1288, 1342, 1415, 1472, 1496, 1625, 1765, 1953, 2002, 2255, 2485, 2847, 2945, 3039, 3382, 3591, 3745, 3905, 4233, 4264, 4313

Offset: 1

Seiichi Manyama, Feb 05 2018

nonn

Numbers k such that A299204(k) = 0.

Seiichi Manyama, Table of n, a(n) for n = 1..500
Eric Weisstein's World of Mathematics, Tau Function.

For the sequence when n is prime see A299172.

Cf. A000594, A063938, A299157, A299204.

Select[Range[2, 5000], Divisible[RamanujanTau[#], #-1] &] (* Amiram Eldar, Jan 10 2025 *)

isok(n) = (ramanujantau(n) % (n-1)) == 0; \\ Michel Marcus, Feb 05 2018

A063940 Composite numbers k such that Ramanujan's function tau(k) (A000594) is not divisible by k.

Original entry on oeis.org

22, 26, 33, 34, 38, 39, 44, 46, 51, 52, 55, 57, 58, 62, 65, 66, 68, 69, 74, 76, 77, 78, 82, 85, 86, 87, 93, 94, 95, 99, 102, 104, 106, 110, 111, 114, 116, 117, 118, 119, 121, 122, 123, 124, 129, 130, 132, 133, 134, 136, 138, 141, 142, 143, 145, 146, 148, 152, 153

Offset: 1

Robert G. Wilson v, Aug 31 2001

nonn

			22 is a term because Ramanujan's tau(22) = 18643272 and 18643272 mod 22 = 10.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000594, A063938, A007659.

Select[ Range[ 70 ], Mod[ CoefficientList[ Take[ Expand[ Product[ (1 - x^k)^24, {k, 1, 70} ] ], 70 ], x ][ [ # ] ], # ] != 0 && ! PrimeQ[ # ] & ]
(* First do *) <Dean Hickerson, Jan 03 2003 *)

A191599 Numbers k that do not divide Ramanujan's tau(k).

Original entry on oeis.org

11, 13, 17, 19, 22, 23, 26, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 99, 101, 102, 103, 104, 106, 107, 109, 110, 111, 113, 114

Offset: 1

Luis H. Gallardo, Jun 08 2011

nonn

This sequence has its first 45 terms in common with A068191.
Subsequence of A068191.
Complement of A063938. - Omar E. Pol, Aug 28 2011

			For n=1, a(1)=11 since 11 does not divide tau(11) = 534612.

T. D. Noe, Table of n, a(n) for n = 1..1000
B. Ramakrishnan and Brundaban Sahu, Rankin-Cohen brackets and Van der Pol-type identities for the Ramanujan's tau function, arXiv:0711.3512 [math.NT], 2007, pp. 14. See Corollary 2.11

Cf. A000594, A068191.

with(numtheory): tn := proc(n) modp(-840*sum(k^4*sigma(k)*sigma(n-k),k=1..n-1),n); end; ser := proc(a,b) local n,lis; lis := []; for n from a to b do if tn(n) <> 0 then lis := [op(lis),n]; fi; od; lis; end;

Select[Range[120], !Divisible[RamanujanTau[#], #]&] (* Jean-François Alcover, Nov 29 2017 *)

isok(k) = ramanujantau(k) % k; \\ Michel Marcus, Aug 14 2021

A295654 Numbers k such that tau(k) +- 1 is congruent to 0 (mod k), where tau is the Ramanujan tau function (A000594).

Original entry on oeis.org

1, 11, 23, 691, 5807, 85583, 189751, 37264081

Offset: 1

Seiichi Manyama, Nov 25 2017

Compare with A063938.

a(9) > 8*10^7. - Seiichi Manyama, Jan 01 2018

			tau(11) = 534612 and 11 | (534612 - 1).
tau(23) = 18643272 and 23 | (18643272 - 1).
tau(691) = -2747313442193908 and 691 | (-2747313442193908 - 1).
tau(5807) = 237456233554906855056 and 5807 | (237456233554906855056 + 1).
tau(85583) = 90954516543892718450139576 and 85583 | (90954516543892718450139576 - 1).
tau(189751) = 4685230754227867924094547904 and 189751 | (4685230754227867924094547904 + 1).
tau(37264081) = 831105005803795341334403814220760726696052 and 37264081 | (831105005803795341334403814220760726696052 - 1).

Eric Weisstein's World of Mathematics, Tau Function.
Wikipedia, Ramanujan tau function.

Cf. A000594, A063938, A273650, A295645.

fQ[n_] := Block[{t = RamanujanTau@n}, Mod[t, n] == 1 || Mod[t, n] + 1 == n]; (* Robert G. Wilson v, Nov 25 2017 *)

from itertools import count, islice
from sympy import divisor_sigma
def A295654_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: n==1 or abs(-840*(pow(m:=n+1>>1,2,n)*(0 if n&1 else pow(m*divisor_sigma(m),2,n))+(sum(pow(i,4,n)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m))<<1)) % n)==1, count(max(startvalue,1)))
A295654_list = list(islice(A295654_gen(),4)) # Chai Wah Wu, Nov 08 2022

A273650(a(n)) is 1 or n - 1.

a(8) from Seiichi Manyama, Jan 01 2018

A299158 Numbers k such that k*(k+1) divides tau(k), where tau(k) = A000594(k) is Ramanujan's tau function.

Original entry on oeis.org

2, 3, 5, 6, 7, 20, 27, 45, 91, 160, 240, 243, 343, 384, 792, 896, 2639, 1163799

Offset: 1

Seiichi Manyama, Feb 04 2018

a(19) > 5*10^6.
Numbers k such that A299165(k) = 0.
Intersection of A063938 and A299157.

Eric Weisstein's World of Mathematics, Tau Function.

Cf. A000594, A063938, A299157, A299165.

Select[Range[1000], Divisible[RamanujanTau[#], #*(#+1)] &] (* Amiram Eldar, Jan 10 2025 *)

isok(k) = !(ramanujantau(k) % (k*(k+1))); \\ Amiram Eldar, Jan 10 2025

A147317 Numbers k which divide tau(k) and are not 7-smooth (where tau(k) = A000594(k) is Ramanujan's tau function).

Original entry on oeis.org

88, 91, 92, 115, 161, 182, 207, 230, 264, 273, 276, 322, 345, 364, 414, 440, 455, 460, 483, 546, 549, 616, 644, 690, 728, 736, 792, 805, 819, 828, 910, 920, 928, 966, 1035, 1092, 1098, 1144, 1288, 1320, 1365, 1380, 1408, 1449, 1456, 1610, 1638, 1656, 1820

Offset: 1

Artur Jasinski, Nov 05 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..7427 (calculated from the b-file at A063938).

Cf. A063938, A002473.

aa = {}; Do[If[Mod[RamanujanTau[n], n] == 0 && EulerPhi[210 n] != 48 n, AppendTo[aa, n]], {n, 1, 2000}]; aa

cp's OEIS Frontend

A299205 Numbers k such that k-1 divides tau(k), where tau(k) = A000594(k) is Ramanujan's tau function.

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A063940 Composite numbers k such that Ramanujan's function tau(k) (A000594) is not divisible by k.

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A191599 Numbers k that do not divide Ramanujan's tau(k).

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A295654 Numbers k such that tau(k) +- 1 is congruent to 0 (mod k), where tau is the Ramanujan tau function (A000594).

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A299158 Numbers k such that k*(k+1) divides tau(k), where tau(k) = A000594(k) is Ramanujan's tau function.

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A147317 Numbers k which divide tau(k) and are not 7-smooth (where tau(k) = A000594(k) is Ramanujan's tau function).

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