A063938 -id:A063938 - cp's OEIS frontend

A079334 Numbers k such that k divides tau(k) and k+1 divides tau(k+1), where tau(k)=A000594(k) is Ramanujan's tau function; i.e., k and k+1 are in A063938.

1, 2, 3, 4, 5, 6, 7, 8, 9, 14, 15, 20, 24, 27, 35, 48, 49, 63, 80, 90, 91, 125, 160, 161, 224, 440, 728, 735, 2024, 2400, 2744, 4095, 4374, 12879, 13824, 20735, 30624

Offset: 1

Dean Hickerson, Jan 03 2003

No other terms < 212000. - Robert G. Wilson v, Jan 06 2002

No other terms < 30000000. - Dana Jacobsen, Sep 06 2015

Eric Weisstein's World of Mathematics, Tau Function.

Cf. A000594, A063938.

Mathematica
```
(* First do <
				
```

tauvec(N) = Vec(q*eta(q + O(q^N))^24)
v=tauvec(10000); for(n=1,#v-1,if(Mod(v[n],n) == 0 && Mod(v[n+1],n+1) == 0,print1(n", "))) \\ Dana Jacobsen, Sep 06 2015

use ntheory ":all"; my @p = grep { !(ramanujan_tau($) % $) } 1..10000; for (0 .. $#p-1) { say $p[$] if $p[$]+1 == $p[$+1] } # _Dana Jacobsen, Sep 06 2015

A079333 Duplicate of A063938.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40

Offset: 1

dead

A002473 7-smooth numbers: positive numbers whose prime divisors are all <= 7.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 125, 126, 128, 135, 140, 144, 147, 150, 160, 162, 168, 175, 180, 189, 192

Offset: 1

N. J. A. Sloane

Also called humble numbers; sometimes also called highly composite numbers, but this usually refers to A002182.
Successive numbers k such that phi(210k) = 48k. - Artur Jasinski, Nov 05 2008
The divisors of 10! (A161466) are a finite subsequence. - Reinhard Zumkeller, Jun 10 2009
Numbers n such that A198487(n) > 0 and A107698(n) > 0. - Jaroslav Krizek, Nov 04 2011
A262401(a(n)) = a(n). - Reinhard Zumkeller, Sep 25 2015
Numbers which are products of single-digit numbers. - N. J. A. Sloane, Jul 02 2017
Phi(a(n)) is 7-smooth. In fact, the Euler Phi function applied to p-smooth numbers, for any prime p, is p-smooth. - Richard Locke Peterson, May 09 2020
Also those integers k, such that, for every prime p > 5, p^(12k) - 1 == 0 (mod 5040k). - Federico Provvedi, Jun 06 2022
The nonprimes with this property are all terms except for 2, 3, 5 and 7, i.e.: (1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, ...); the composite terms are all but the first one of this subsequence. ["Trivial" data provided mainly for search purpose.] - M. F. Hasler, Jun 06 2023

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 52.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 5841 terms from N. J. A. Sloane)
Raphael Schumacher, The Formulas for the Distribution of the 3-Smooth, 5-Smooth, 7-Smooth and all other Smooth Numbers, arXiv preprint arXiv:1608.06928 [math.NT], 2016.
University of Ulm, The first 5842 terms.
Eric Weisstein's World of Mathematics, Smooth Number.
Wikipedia, Smooth number

Subsequence of A080672, complement of A068191. Subsequences: A003591, A003594, A003595, A195238, A059405.
Not the same as A063938. For p-smooth numbers with other values of p, see A003586, A051037, A051038, A080197, A080681, A080682, A080683.
Cf. A002182, A067374, A210679, A238985 (zeroless terms), A006530.
Cf. A262401.

import Data.Set (singleton, deleteFindMin, fromList, union)
a002473 n = a002473_list !! (n-1)
a002473_list = f $ singleton 1 where
   f s = x : f (s' `union` fromList (map (* x) [2,3,5,7]))
         where (x, s') = deleteFindMin s
-- Reinhard Zumkeller, Mar 08 2014, Apr 02 2012, Apr 01 2012

[n: n in [1..200] | PrimeDivisors(n) subset PrimesUpTo(7)]; // Bruno Berselli, Sep 24 2012

Select[Range[250], Max[Transpose[FactorInteger[ # ]][[1]]]<=7&]
aa = {}; Do[If[EulerPhi[210 n] == 48 n, AppendTo[aa, n]], {n, 1, 1200}]; aa (* Artur Jasinski, Nov 05 2008 *)
mxExp = 8; Select[Union[Times @@@ Flatten[Table[Tuples[{2, 3, 5, 7}, n], {n, mxExp}], 1]], # <= 2^mxExp &] (* Harvey P. Dale, Aug 13 2012 *)
mx = 200; Sort@ Flatten@ Table[ 2^i*3^j*5^k*7^l, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}, {k, 0, Log[5, mx/(2^i*3^j)]}, {l, 0, Log[7, mx/(2^i*3^j*5^k)]}] (* Robert G. Wilson v, Aug 17 2012 *)

test(n)=m=n; forprime(p=2,7, while(m%p==0,m=m/p)); return(m==1)
for(n=1,200,if(test(n),print1(n",")))

is_A002473(n)=n<11||vecmax(factor(n,8)[,1])<8 \\ M. F. Hasler, Jan 16 2015

list(lim)=my(v=List(),t); for(a=0,logint(lim\1,7), for(b=0,logint(lim\7^a,5), for(c=0,logint(lim\7^a\5^b,3), t=3^c*5^b*7^a; while(t<=lim, listput(v,t); t<<=1)))); Set(v) \\ Charles R Greathouse IV, Feb 22 2017

import heapq
from itertools import islice
from sympy import primerange
def A002473gen(p=7): # generate all p-smooth terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(A002473gen(), 65))) # Michael S. Branicky, Nov 19 2022

from sympy import integer_log
def A002473(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n+x
        for i in range(integer_log(x,7)[0]+1):
            i7 = 7**i
            m = x//i7
            for j in range(integer_log(m,5)[0]+1):
                j5 = 5**j
                r = m//j5
                for k in range(integer_log(r,3)[0]+1):
                    c -= (r//3**k).bit_length()
        return c
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

A006530(a(n)) <= 7. - Reinhard Zumkeller, Apr 01 2012

Sum_{n>=1} 1/a(n) = Product_{primes p <= 7} p/(p-1) = (2*3*5*7)/(1*2*4*6) = 35/8. - Amiram Eldar, Sep 22 2020

More terms from James Sellers, Dec 23 1999
Additional comments from Michel Lecomte, Jun 09 2007
Edited by M. F. Hasler, Jan 16 2015

A007659 Primes p such that Ramanujan number tau(p) is divisible by p.

Original entry on oeis.org

2, 3, 5, 7, 2411, 7758337633

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Primes at which cusp form Delta_12 (see A007332) is not ordinary.

a(5) was found by Newman (1972). - Amiram Eldar, Jan 06 2025

Morris Newman, A table of tau(p) modulo p, p prime, 3 <= p <= 16067, National Bureau of Standards, 1972.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 275.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Fernando Q. Gouvêa, Non-ordinary primes: a story, Experimental Mathematics 6(3) (1997), 195-205; alternative link.
Nik Lygeros and Olivier Rozier, A new solution for the equation tau(p)=0 (mod p). Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.4.
Nik Lygeros and Olivier Rozier, A new solution for the equation tau(p)=0 mod p. Number Theory mailing list (NMBRTHRY), 2010.
Morris Newman, A table of tau(p) modulo p, p prime, 3 <= p <= 16067, Review, Mathematics of Computation, Vol. 27, No. 121 (1973), pp. 215-216.

Cf. A000594, A007332. A proper subset of A063938.

Select[ Prime[ Range[ 5133]], Mod[ RamanujanTau[ # ], # ] == 0 &] (* Dean Hickerson, Jan 03 2003 *)
Select[Prime[Range[400]],Divisible[RamanujanTau[#],#]&] (* The program generates the first 5 terms of the sequence. *) (* Harvey P. Dale, Jun 06 2022 *)

a(6) = 7758337633 from N. Lygeros and O. Rozier, Mar 16 2010. - N. J. A. Sloane, Mar 16 2010

Edited by Max Alekseyev, Jul 11 2010

A273650 a(n) = A000594(n) mod n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 8, 0, 0, 0, 10, 0, 7, 0, 0, 20, 1, 0, 0, 16, 0, 0, 24, 0, 21, 0, 21, 32, 0, 0, 31, 22, 27, 0, 30, 0, 31, 24, 0, 22, 27, 0, 0, 0, 21, 28, 29, 0, 45, 0, 54, 4, 14, 0, 49, 54, 0, 0, 30, 24, 64, 36, 45, 0, 19, 0, 67, 70, 0, 32, 42, 54, 37, 0, 0, 18

Offset: 1

Seiichi Manyama, May 27 2016

nonn

			tau(10) mod 10 = (-115920) mod 10 = 0,
tau(11) mod 11 = 534612 mod 11 = 1.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000594, A063938, A295654.

a[n_] := Mod[RamanujanTau[n], n]; Array[a, 100] (* Amiram Eldar, Jan 08 2025 *)

a(n)=ramanujantau(n)%n \\ assumes the GRH; Charles R Greathouse IV, May 27 2016

from sympy import divisor_sigma
def A273650(n): return -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 # Chai Wah Wu, Nov 08 2022

a(n) = A000594(n) mod n.
From Amiram Eldar, Jan 08 2025: (Start)
a(A063938(n)) = 0.
abs(a(A295654(n))) = 1. (End)

A296991 Numbers k such that k^2 divides tau(k), where tau(k) = A000594(k) is Ramanujan's tau function.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 21, 24, 27, 32, 36, 40, 42, 48, 54, 64, 72, 81, 84, 96, 108, 120, 128, 135, 144, 162, 168, 189, 192, 216, 243, 256, 270, 280, 288, 324, 336, 360, 378, 384, 432, 448, 486, 512, 540, 576, 640, 648, 672, 729, 756, 768, 828, 840, 864

Offset: 1

Seiichi Manyama, Dec 22 2017

nonn

2^k is a term for k >= 0.

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

Cf. A000594, A063938, A296992, A296993.

fQ[n_] := Mod[RamanujanTau@n, n^2] == 0; Select[Range@875, fQ] (* Robert G. Wilson v, Dec 23 2017 *)

is(n) = Mod(ramanujantau(n), n^2)==0 \\ Felix Fröhlich, Dec 24 2017

from itertools import count, islice
from sympy import divisor_sigma
def A296991_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: not -24*((m:=n+1>>1)**2*(0 if n&1 else m*(35*m - 52*n)*divisor_sigma(m)**2)+sum(i**3*(70*i - 140*n)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m))) % n**2, count(max(startvalue,1)))
A296991_list = list(islice(A296991_gen(),20)) # Chai Wah Wu, Nov 08 2022

A296992 Largest number m such that n^m divides tau(n), where tau(n) = A000594(n) is Ramanujan's tau function.

Original entry on oeis.org

3, 2, 3, 1, 3, 1, 3, 2, 1, 0, 2, 0, 1, 1, 3, 0, 2, 0, 1, 2, 0, 0, 3, 1, 0, 2, 1, 0, 1, 0, 3, 0, 0, 1, 2, 0, 0, 0, 2, 0, 2, 0, 0, 1, 0, 0, 2, 1, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 0, 0, 1, 2, 0, 0, 2, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0

Offset: 2

Seiichi Manyama, Dec 22 2017

nonn

			tau(2) =   -24 and 2^3 divides   24, so a(2) = 3.
tau(3) =   252 and 3^2 divides  252, so a(3) = 2.
tau(4) = -1472 and 4^3 divides 1472, so a(4) = 3.

Amiram Eldar, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Tau Function.

Cf. A063938 (a(n)>=1), A296991 (a(n)>=2), A296993 (a(n)>=3).

Cf. A191599 (a(n)=0), A297000 (a(n)=1), A297001 (a(n)=2).

f[n_] := Block[{m = 0}, While[Mod[RamanujanTau@n, n^m] == 0, m++]; m - 1]; Array[f, 93, 2] (* Robert G. Wilson v, Dec 23 2017 *)
a[n_] := IntegerExponent[RamanujanTau[n], n]; Array[a, 100, 2] (* Amiram Eldar, Jan 09 2025 *)

a(n) = valuation(ramanujantau(n), n); \\ Amiram Eldar, Jan 09 2025

A299157 Numbers k such that 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, 11, 13, 17, 19, 20, 22, 23, 27, 29, 31, 41, 45, 47, 53, 55, 59, 68, 71, 76, 77, 79, 83, 87, 89, 91, 97, 104, 107, 114, 127, 137, 139, 149, 160, 167, 171, 177, 179, 183, 191, 195, 199, 209, 223, 229, 239, 240, 243, 251, 269, 275, 293, 297, 321, 343

Offset: 1

Seiichi Manyama, Feb 04 2018

nonn

Numbers k such that A299163(k) = 0.

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

Cf. A000594, A063938, A079334, A296991, A299158, A299163.

q[k_] := Divisible[RamanujanTau[k], k+1]; Select[Range[350], q] (* Amiram Eldar, Jan 08 2025 *)

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

A296993 Numbers k such that k^3 divides tau(k), where tau(k) = A000594(k) is Ramanujan's tau function.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 24, 32, 64, 96, 128, 256, 288, 384, 512, 1024, 1536, 2048, 4096, 6144, 8192, 16384, 18432, 24576, 32768, 65536, 98304, 131072, 172032, 262144, 276480, 393216, 524288, 1048576, 1179648, 1572864, 1935360, 2097152, 2621440, 3538944, 4194304

Offset: 1

Seiichi Manyama, Dec 22 2017

nonn

2^k is a term for k >= 0.

Eric Weisstein's World of Mathematics, Tau Function.

Cf. A000594, A063938, A296991, A296992.

from itertools import count, islice
from sympy import divisor_sigma
def A296993_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: not -24*((m:=n+1>>1)**2*(0 if n&1 else (m*(35*m - 52*n) + 18*n**2)*divisor_sigma(m)**2)+sum((i*(i*(i*(70*i - 140*n) + 90*n**2)))*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m))) % n**3, count(max(startvalue,1)))
A296993_list = list(islice(A296993_gen(),10)) # Chai Wah Wu, Nov 08 2022

A297000 Numbers k such that A296992(k) = 1.

Original entry on oeis.org

5, 7, 10, 14, 15, 20, 25, 28, 30, 35, 45, 49, 50, 56, 60, 63, 70, 75, 80, 88, 90, 91, 92, 98, 100, 105, 112, 115, 125, 126, 140, 147, 150, 160, 161, 175, 180, 182, 196, 200, 207, 210, 224, 225, 230, 240, 245, 250, 252, 264, 273, 276, 294, 300, 315, 320, 322, 343

Offset: 1

Seiichi Manyama and Robert G. Wilson v, Dec 23 2017

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2000
Eric Weisstein's World of Mathematics, Tau Function.

Cf. A000594, A063938, A296992, A297001.

1 + Position[Table[-1 + SelectFirst[Range[0, 5], ! Divisible[RamanujanTau@ n, n^#] &], {n, 2, 350}], 1][[All, 1]] (* Michael De Vlieger, Dec 23 2017 *)

isok(k) = if(k == 1, 0, valuation(ramanujantau(k), k) == 1); \\ Amiram Eldar, Jan 09 2025

cp's OEIS Frontend

A079334 Numbers k such that k divides tau(k) and k+1 divides tau(k+1), where tau(k)=A000594(k) is Ramanujan's tau function; i.e., k and k+1 are in A063938.

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A079333 Duplicate of A063938.

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A002473 7-smooth numbers: positive numbers whose prime divisors are all <= 7.

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A007659 Primes p such that Ramanujan number tau(p) is divisible by p.

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A273650 a(n) = A000594(n) mod n.

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A296991 Numbers k such that k^2 divides tau(k), where tau(k) = A000594(k) is Ramanujan's tau function.

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A296992 Largest number m such that n^m divides tau(n), where tau(n) = A000594(n) is Ramanujan's tau function.

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A299157 Numbers k such that k+1 divides tau(k), where tau(k) = A000594(k) is Ramanujan's tau function.