A000005 -id:A000005 - cp's OEIS frontend

A035116 a(n) = tau(n)^2, where tau(n) = A000005(n).

1, 4, 4, 9, 4, 16, 4, 16, 9, 16, 4, 36, 4, 16, 16, 25, 4, 36, 4, 36, 16, 16, 4, 64, 9, 16, 16, 36, 4, 64, 4, 36, 16, 16, 16, 81, 4, 16, 16, 64, 4, 64, 4, 36, 36, 16, 4, 100, 9, 36, 16, 36, 4, 64, 16, 64, 16, 16, 4, 144, 4, 16

Offset: 1

N. J. A. Sloane

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 59.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 304.

T. D. Noe, Table of n, a(n) for n = 1..1000
Mircea Merca, The Lambert series factorization theorem, The Ramanujan Journal, January 2017; DOI: 10.1007/s11139-016-9856-3.

Cf. A000005, A048691, A061391.

Cf. A066446, A184389, A061502.

a035116 = (^ 2) . a000005'  -- Reinhard Zumkeller, Sep 08 2015

[ NumberOfDivisors(n)^2 : n in [1..100] ];

A035116 := proc(n) numtheory[tau](n)^2 ; end proc:
seq(A035116(n),n=1..40) ; # R. J. Mathar, Apr 02 2011

DivisorSigma[0, Range[100]]^2 (* Vladimir Joseph Stephan Orlovsky, Jul 20 2011 *)

PARI
```
A035116(n)=numdiv(n)^2;
```

Dirichlet g.f.: zeta(s)^4/zeta(2s).
tau(n)^2 = Sum_{d|n} tau(d^2), Dirichlet convolution of A048691 and A000012 (i.e.: inverse Mobius transform of A048691).
Multiplicative with a(p^e) = (e+1)^2. - Vladeta Jovovic, Dec 03 2001
G.f.: Sum_{n>=1} A000005(n^2)*x^n/(1-x^n). - Mircea Merca, Feb 25 2014
a(n) = A066446(n) + A184389(n). - Reinhard Zumkeller, Sep 08 2015
Let b(n), n > 0, be the Dirichlet inverse of a(n). Then b(n) is multiplicative with b(p^e) = (-1)^e*(Sum_{i=0..e} binomial(3,i)) for prime p and e >= 0, where binomial(n,k)=0 if n < k; abs(b(n)) is multiplicative and has the Dirichlet g.f.: (zeta(s))^4/(zeta(2*s))^3. - Werner Schulte, Feb 07 2021

Additional comments from Vladeta Jovovic, Apr 29 2001

A034884 Numbers k such that k < d(k)^2, where d(k) = A000005(k).

Original entry on oeis.org

2, 3, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 72, 80, 84, 90, 96, 108, 120, 126, 132, 140, 144, 168, 180, 192, 210, 216, 240, 252, 288, 300, 336, 360, 420, 480, 504, 540, 720, 840, 1260

Offset: 1

Erich Friedman

See comment in A175495. - Vladimir Shevelev, May 07 2013
The deficient terms are 2, 3, 4, 8, 10, 14, 15, 16, 32; the first perfect or abundant number not listed is 66 = 2 * 3 * 11; the only term not 7-smooth is 132 = 2^2 * 3 * 11; the largest not divisible by 6 is 140 = 2^2 * 5 * 7. - Peter Munn, Sep 19 2021
The union of this sequence and A276734 has 74 total terms which are all k with floor(sqrt(k)) <= d(k). - Bill McEachen, Apr 07 2023

Cf. A000005, A035033, A035034, A035035, A033950, A036763, A175495, A276734.

Select[Range[1300],#Harvey P. Dale, Apr 11 2014 *)

isok(n) = (n < numdiv(n)^2) \\ Michel Marcus, Jun 07 2013

Labos Elemer added the last three terms and observes that this sequence is now complete.

A050226 Numbers m such that m divides Sum_{k = 1..m} A000005(k).

Original entry on oeis.org

1, 4, 5, 15, 42, 44, 47, 121, 336, 340, 347, 930, 2548, 6937, 6947, 51322, 379097, 379131, 379133, 2801205, 20698345, 56264090, 56264197, 152941920, 152942012, 8350344420, 61701166395, 455913379395, 455913379831, 1239301050694, 3368769533660, 3368769533812

Offset: 1

Labos Elemer, Dec 20 1999

			For k = 15 the sum is 1 + 2 + 2 + 3 + 2 + 4 + 2 + 4 + 3 + 4 + 2 + 6 + 2 + 4 + 4 = 45 which is divisible by 15.

Julian Havil, "Gamma: Exploring Euler's Constant", Princeton University Press, Princeton and Oxford, pp. 112-113, 2003.

Donovan Johnson, Table of n, a(n) for n = 1..39 (quotients <= 40)

Cf. A000005, A006218, A057494, A085567, A085829.

s = 0; Do[ s = s + DivisorSigma[ 0, n ]; If[ Mod[ s, n ] == 0, Print[ n ] ], {n, 1, 2*10^9} ]
k=10^6; a[1]=1;a[n_]:=a[n]=DivisorSigma[0,n]+a[n-1]; nd=a/@Range@k; Select[Range@k,Divisible[nd[[#]],#]&] (* Ivan N. Ianakiev, Apr 30 2016 *)
Module[{nn=400000},Select[Thread[{Range[nn],Accumulate[DivisorSigma[0,Range[nn]]]}],Divisible[#[[2]],#[[1]]]&]][[All,1]] (* The program generates the first 19 terms of the sequence. To generate more, increase the nn constant. *) (* Harvey P. Dale, Jul 03 2022 *)

lista(nn) = {my(s = 0); for (n=1, nn, s += numdiv(n); if (!(s % n), print1(n, ", ")););} \\ Michel Marcus, Dec 14 2015

def A050226_list(len):
    a, L = 0, []
    for n in (1..len):
        a += sigma(n,0)
        if n.divides(a): L.append(n)
    return L
A050226_list(10000) # Peter Luschny, Dec 18 2015

m is in the sequence if Sum_{i = 1..m} d(i) = m*k, k an integer, where d(i) = number of divisors of i.

More terms from Robert G. Wilson v, Sep 21 2000
Further terms from Naohiro Nomoto, Aug 03 2001
a(26)-a(30) from Donovan Johnson, Dec 21 2008

A119899 Integers i such that bigomega(i) (A001222) and tau(i) (A000005) are both even.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 24, 26, 33, 34, 35, 38, 39, 40, 46, 51, 54, 55, 56, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 104, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 135, 136, 140, 141, 142, 143, 145, 146, 150

Offset: 1

Antti Karttunen, Jun 04 2006

nonn

Also numbers whose alternating sum of prime indices is < 0. Equivalently, numbers with even bigomega whose conjugate prime indices are not all even. This is the intersection of A028260 and A000037. - Gus Wiseman, Jun 20 2021

			From _Gus Wiseman_, Jun 20 2021: (Start)
The sequence of terms together with their prime indices begins:
       6: {1,2}          51: {2,7}          86: {1,14}
      10: {1,3}          54: {1,2,2,2}      87: {2,10}
      14: {1,4}          55: {3,5}          88: {1,1,1,5}
      15: {2,3}          56: {1,1,1,4}      90: {1,2,2,3}
      21: {2,4}          57: {2,8}          91: {4,6}
      22: {1,5}          58: {1,10}         93: {2,11}
      24: {1,1,1,2}      60: {1,1,2,3}      94: {1,15}
      26: {1,6}          62: {1,11}         95: {3,8}
      33: {2,5}          65: {3,6}          96: {1,1,1,1,1,2}
      34: {1,7}          69: {2,9}         104: {1,1,1,6}
      35: {3,4}          74: {1,12}        106: {1,16}
      38: {1,8}          77: {4,5}         111: {2,12}
      39: {2,6}          82: {1,13}        115: {3,9}
      40: {1,1,1,3}      84: {1,1,2,4}     118: {1,17}
      46: {1,9}          85: {3,7}         119: {4,7}
(End)

Superset: A119847. Subset: A006881. The intersection of A028260 and A000037.
Positions of negative terms in A316524.
The partitions with these Heinz numbers are counted by A344608.
Complement of A344609.
Cf. A000041, A000070, A000097, A027187, A103919, A116406, A239829, A239830, A343941, A344607, A344651.

Select[Range[200],And@@EvenQ[{PrimeOmega[#],DivisorSigma[0,#]}]&] (* Harvey P. Dale, Jan 24 2013 *)

A009191 a(n) = gcd(n, d(n)), where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 4, 3, 2, 1, 6, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 9, 1, 2, 1, 8, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 8, 1, 2, 1, 12, 1, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 12, 1, 2, 3, 2, 1, 2, 1, 10, 1, 2, 1, 12, 1, 2, 1, 8, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 3, 1, 1, 2, 1, 8, 1

Offset: 1

David W. Wilson

nonn

a(A046642(n)) = 1.
First occurrence of k: 1, 2, 9, 8, 400, 12, 3136, 24, 36, 80, 123904, 60, 692224, 448, 2025, 384, .... Conjecture: each k is present. - Robert G. Wilson v, Mar 27 2013
Conjecture is true. See David A. Corneth's comment in A324553. - Antti Karttunen, Mar 06 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from T. D. Noe)

Cf. A000005, A009194, A009195, A009205, A009213, A009230, A049820, A125168, A138010, A286540, A303781, A318459, A319337, A322979, A322980, A323073.

Cf. A046642 (positions of ones), A324553 (position of the first occurrence of each n).

a009191 n = gcd n $ a000005 n
-- Reinhard Zumkeller, May 09 2013, Aug 14 2011

f[n_] := GCD[n, DivisorSigma[0, n]]; Array[f, 105] (* Robert G. Wilson v, Mar 27 2013 *)

a(n)=gcd(numdiv(n),n) \\ Charles R Greathouse IV, Mar 26 2013

a(n) = gcd(n, A000005(n)) = gcd(n, A049820(n)). - Antti Karttunen, Sep 25 2018

A045765 k - d(k) never takes these values, where d(k) = A000005(k).

Original entry on oeis.org

7, 8, 13, 19, 20, 24, 25, 28, 33, 36, 37, 40, 43, 49, 50, 52, 55, 56, 63, 64, 66, 67, 68, 74, 75, 79, 85, 88, 98, 100, 103, 108, 109, 112, 113, 116, 117, 123, 124, 126, 131, 132, 133, 134, 136, 140, 143, 145, 150, 153, 156, 159, 160, 163, 164, 167, 168

Offset: 1

David W. Wilson

nonn

Complement of A236562. - Jaroslav Krizek, Feb 09 2014
Positions of zeros in A060990, leaf-nodes in the tree generated by edge-relation A049820(child) = parent. - Antti Karttunen, Oct 06 2015
Since A000005(x) <= 1 + x/2, k is in the sequence if there are no x <= 2*(k+1) with k = x - d(x). - Robert Israel, Oct 12 2015
This can be improved as: k is in the sequence if there are no x <= k + A002183(2+A261100(k)) with k = x - d(x). Cf. also A070319, A262686. - Antti Karttunen, Oct 12 2015
Luca (2005) proved that this seqeunce is infinite. - Amiram Eldar, Jul 26 2025

Antti Karttunen, Table of n, a(n) for n = 1..10000
Florian Luca, On numbers not of the form n-omega(n), Acta Mathematica Hungarica, Vol. 106, No. 1 (2005), pp. 117-135.

Top row of A262898.
Cf. A000005, A002183, A049820, A060990, A070319, A236562, A236565, A261100, A262511, A262686, A262901, A262902, A262903, A262909, A263081.
Cf. A263091 (primes in this sequence), A263095 (squares).
Cf. A259934 (gives the infinite trunk of the same tree, conjectured to be unique).

N:= 1000: # to get all terms <= N
sort(convert({$1..N} minus {seq(x - numtheory:-tau(x), x=1..2*(1+N))},list)); # Robert Israel, Oct 12 2015

lim = 10000; Take[Complement[Range@ lim, Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, lim}]], 57] (* Michael De Vlieger, Oct 13 2015 *)

allocatemem((2^31)+(2^30));
uplim = 36756720 + 640; \\ = A002182(53) + A002183(53).
v060990 = vector(uplim);
for(n=3, uplim, v060990[n-numdiv(n)]++);
A060990 = n -> if(!n,2,v060990[n]);
uplim2 = 36756720;
n=0; k=1; while(n <= uplim2, if(0==A060990(n), write("b045765_big.txt", k, " ", n); k++); n++;);
\\ Antti Karttunen, Oct 09 2015

(define A045765 (ZERO-POS 1 1 A060990))
;; Using also IntSeq-library of Antti Karttunen, Oct 06 2015

A051950 Differences between values of tau(n) (A000005): a(n) = tau(n)-tau(n-1).

Original entry on oeis.org

1, 0, 1, -1, 2, -2, 2, -1, 1, -2, 4, -4, 2, 0, 1, -3, 4, -4, 4, -2, 0, -2, 6, -5, 1, 0, 2, -4, 6, -6, 4, -2, 0, 0, 5, -7, 2, 0, 4, -6, 6, -6, 4, 0, -2, -2, 8, -7, 3, -2, 2, -4, 6, -4, 4, -4, 0, -2, 10, -10, 2, 2, 1, -3, 4, -6, 4, -2, 4, -6, 10, -10, 2, 2, 0, -2, 4, -6, 8, -5, -1, -2, 10, -8, 0, 0, 4, -6, 10

Offset: 2

Asher Auel, Dec 16 1999

N. J. A. Sloane, Table of n, a(n) for n = 2..20000 [First 10000 terms from T. D. Noe]

Cf. A000005, A058197, A058198, A058199.

a051950 n = a051950_list !! n
a051950_list = 1 : zipWith (-) (tail a000005_list) a000005_list
-- Reinhard Zumkeller, Feb 04 2013

Differences[DivisorSigma[0, Range[100]]] (* Alonso del Arte, Oct 14 2013, based on T. D. Noe's program for A182394 *)

G.f.: (1-x)*sum(k >= 1, x^k/(1-x^k)). - Benoit Cloitre, Apr 24 2003

Edited (and offset corrected) by N. J. A. Sloane, Oct 05 2017

A062069 a(n) = sigma(d(n)), where d(k) is the number of divisors function (A000005) and sigma(k) is the sum of divisors function (A000203).

Original entry on oeis.org

1, 3, 3, 4, 3, 7, 3, 7, 4, 7, 3, 12, 3, 7, 7, 6, 3, 12, 3, 12, 7, 7, 3, 15, 4, 7, 7, 12, 3, 15, 3, 12, 7, 7, 7, 13, 3, 7, 7, 15, 3, 15, 3, 12, 12, 7, 3, 18, 4, 12, 7, 12, 3, 15, 7, 15, 7, 7, 3, 28, 3, 7, 12, 8, 7, 15, 3, 12, 7, 15, 3, 28, 3, 7, 12, 12, 7, 15, 3, 18, 6, 7, 3, 28, 7, 7, 7, 15, 3

Offset: 1

Amarnath Murthy, Jun 13 2001

nonn

a(1) = 1, a(p) = 3 for p = primes (A000040), a(pq) = 7 for pq = product of two distinct primes (A006881), a(pq...z) = 2^(k+1)-1 = A000225(k+1) for pq...z = product of k (k > 2) distinct primes p,q,...,z (A120944), a(p^k) = sigma(k+1) = A000203(k+1) for p^k = prime powers (A000961(n) for n > 1). Sequence {1,3,4,12} is finite sequence of numbers n such that sigma(tau(n)) = n. [Jaroslav Krizek, Jul 16 2009]
For semiprime n, a(n) is either 4 or 7. Also a(n) = d(n) + omega(n) + mu(n), the sum of three core sequences A000005, A001221 and A008683. When n is semiprime, a(n) is completely defined by the Mobius function as: a(n) = 4 + 3*mu(n). a(n) also has the fractal-like identities a(d(n)) = d(n) and a(n) = sigma(a(d(n))). - Wesley Ivan Hurt, Sep 02 2013
If n is a triprime (A014612), d(n) is 4, 6, or 8 and a(n) = sigma(d(n)) is 7, 12, or 15 respectively. Then a(n) = -d(n)^2/4 + 5*d(n) - 9. - Wesley Ivan Hurt, Sep 08 2013

			sigma(d(12)) = sigma(6) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000005, A000203, A001221, A008683, A062068.

A062069:= (n-> numtheory[sigma](numtheory[tau](n))):
seq (A062069(n), n=1..40); # Jani Melik, Jan 25 2011

Table[DivisorSigma[1, DivisorSigma[0, n]], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

v=[]; for(n=1,150,v=concat(v, sigma(numdiv(n)))); v

{ for (n=1, 1000, write("b062069.txt", n, " ", sigma(numdiv(n))) ) } \\ Harry J. Smith, Jul 31 2009

a(n) = A000203(A000005(n)). - Wesley Ivan Hurt, Sep 09 2013

More terms from Jason Earls, Jun 19 2001

A036459 Number of iterations required to reach stationary value when repeatedly applying d, the number of divisors function (A000005).

Original entry on oeis.org

0, 0, 1, 2, 1, 3, 1, 3, 2, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 4, 2, 3, 3, 4, 1, 4, 1, 4, 3, 3, 3, 3, 1, 3, 3, 4, 1, 4, 1, 4, 4, 3, 1, 4, 2, 4, 3, 4, 1, 4, 3, 4, 3, 3, 1, 5, 1, 3, 4, 2, 3, 4, 1, 4, 3, 4, 1, 5, 1, 3, 4, 4, 3, 4, 1, 4, 2, 3, 1, 5, 3, 3, 3, 4, 1, 5, 3, 4, 3, 3, 3, 5, 1, 4, 4

Offset: 1

Labos Elemer

nonn

Iterating d for n, the prestationary prime and finally the fixed value of 2 is reached in different number of steps; a(n) is the number of required iterations.

Each value n > 0 occurs an infinite number of times. For positions of first occurrences of n, see A251483. - Ivan Neretin, Mar 29 2015

			If n=8, then d(8)=4, d(d(8))=3, d(d(d(8)))=2, which means that a(n)=3. In terms of the number of steps required for convergence, the distance of n from the d-equilibrium is expressed by a(n). A similar method is used in A018194.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
B. L. Mayer and L. H. A. Monteiro, On the divisors of natural and happy numbers: a study based on entropy and graphs, AIMS Mathematics (2023) Vol. 8, Issue 6, 13411-13424.

Equals A060937 - 1. Cf. A007624, A036450, A036452, A036453, A036455, A030630.

Table[ Length[ FixedPointList[ DivisorSigma[0, # ] &, n]] - 2, {n, 105}] (* Robert G. Wilson v, Mar 11 2005 *)

for(x = 1,150, for(a=0,15, if(a==0,d=x, if(d<3,print(a-1),d=numdiv(d) )) ))

a(n)=my(t);while(n>2,n=numdiv(n);t++);t \\ Charles R Greathouse IV, Apr 07 2012

a(n) = a(d(n)) + 1 if n > 2.

a(n) = 1 iff n is an odd prime.

A048785 a(0) = 0; a(n) = tau(n^3), where tau = number of divisors (A000005).

Original entry on oeis.org

0, 1, 4, 4, 7, 4, 16, 4, 10, 7, 16, 4, 28, 4, 16, 16, 13, 4, 28, 4, 28, 16, 16, 4, 40, 7, 16, 10, 28, 4, 64, 4, 16, 16, 16, 16, 49, 4, 16, 16, 40, 4, 64, 4, 28, 28, 16, 4, 52, 7, 28, 16, 28, 4, 40, 16, 40, 16, 16, 4, 112, 4, 16, 28, 19, 16, 64, 4, 28, 16

Offset: 0

N. J. A. Sloane

The inverse Mobius transform of A074816. - R. J. Mathar, Feb 09 2011

a(n) is also the number of ordered triples (i,j,k) of positive integers such that i|n, j|n, k|n and i,j,k are pairwise relatively prime. - Geoffrey Critzer, Jan 11 2015

			a(6) = 16 because there are 16 divisors of 6^3 = 216: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216.
Also there are 16 ordered triples of divisors of 6 that are pairwise relatively prime: (1,1,1), (1,1,2), (1,1,3), (1,1,6), (1,2,1), (1,2,3), (1,3,1), (1,3,2), (1,6,1), (2,1,1), (2,1,3), (2,3,1), (3,1,1), (3,1,2), (3,2,1), (6,1,1).

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A000005, A001221, A048691, A074816, A144943, A353551.

seq(numtheory:-tau(n^3), n=0..100); # Robert Israel, Jan 11 2015

Join[{0,1},Table[Product[3 k + 1, {k, FactorInteger[n][[All, 2]]}], {n, 2, 69}]] (* Geoffrey Critzer, Jan 11 2015 *)
Join[{0},DivisorSigma[0,Range[70]^3]] (* Harvey P. Dale, Jan 23 2016 *)

A048785(n) = if(!n,n,numdiv(n^3)); \\ Antti Karttunen, May 19 2017

print1("0, "); for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, May 15 2021
print1("0, "); for(n=1, 100, print1(direuler(p=2, n, (1 - 3*X^2 + 2*X^3)/(1 - X)^4)[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

from math import prod
from sympy import factorint
def A048785(n): return 0 if n == 0 else prod(3*e+1 for e in factorint(n).values()) # Chai Wah Wu, May 10 2022

from sympy import divisor_count
def A048785(n): return divisor_count(n**3) # Karl-Heinz Hofmann, May 10 2022

a(n) = Sum_{d|n} 3^omega(d), where omega(x) is the number of distinct prime factors in the factorization of x. - Benoit Cloitre, Apr 14 2002
Multiplicative with a(p^e) = 3e+1. - Mitch Harris, Jun 09 2005
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(3^omega(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 26 2018
For n>0, a(n) = Sum_{d|n} mu(d)^2*tau(d)*tau(n/d). - Ridouane Oudra, Nov 18 2019
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 2/p^s). - Vaclav Kotesovec, May 15 2021
Dirichlet g.f.: zeta(s)^4 * Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)). - Vaclav Kotesovec, Aug 20 2021

cp's OEIS Frontend

A035116 a(n) = tau(n)^2, where tau(n) = A000005(n).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Formula

Extensions

A034884 Numbers k such that k < d(k)^2, where d(k) = A000005(k).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A050226 Numbers m such that m divides Sum_{k = 1..m} A000005(k).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

Extensions

A119899 Integers i such that bigomega(i) (A001222) and tau(i) (A000005) are both even.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A009191 a(n) = gcd(n, d(n)), where d(n) is the number of divisors of n (A000005).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Formula

A045765 k - d(k) never takes these values, where d(k) = A000005(k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Scheme

A051950 Differences between values of tau(n) (A000005): a(n) = tau(n)-tau(n-1).

Views

Author

Keywords

Links

Crossrefs

Programs