A122841 -id:A122841 - cp's OEIS frontend

A072078 Number of 3-smooth divisors of n.

1, 2, 2, 3, 1, 4, 1, 4, 3, 2, 1, 6, 1, 2, 2, 5, 1, 6, 1, 3, 2, 2, 1, 8, 1, 2, 4, 3, 1, 4, 1, 6, 2, 2, 1, 9, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 10, 1, 2, 2, 3, 1, 8, 1, 4, 2, 2, 1, 6, 1, 2, 3, 7, 1, 4, 1, 3, 2, 2, 1, 12, 1, 2, 2, 3, 1, 4, 1, 5, 5, 2, 1, 6, 1, 2, 2, 4, 1, 6, 1, 3, 2, 2, 1

Offset: 1

Reinhard Zumkeller, Jun 13 2002

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A000005, A003586, A007814, A007949, A072079, A122841, A322026.

[(Valuation(n,2)+1)*(Valuation(n,3)+1): n in [1..120]]; // Vincenzo Librandi, Mar 24 2015

a[n_] := DivisorSum[n, MoebiusMu[6*#]*DivisorSigma[0, n/#] &]; Array[a, 100] (* or *) a[n_] := ((1+IntegerExponent[n, 2])*(1+IntegerExponent[n, 3])); Array[a, 100] (* Amiram Eldar, Dec 03 2018 from the pari codes *)

a(n)=sumdiv(n,d,moebius(6*d)*numdiv(n/d)) \\ Benoit Cloitre, Jun 21 2007

A072078(n) = ((1+valuation(n,2))*(1+valuation(n,3))); \\ Antti Karttunen, Dec 03 2018

a(n) = A000005(A065331(n)).
a(n) = (A007814(n) + 1)*(A007949(n) + 1).
1/Product_{k>0} (1 - x^k + x^(2*k))^a(k) is g.f. for A000041(). - Vladeta Jovovic, Jun 07 2004
From Christian G. Bower, May 20 2005: (Start)
Multiplicative with a(2^e) = a(3^e) = e+1, a(p^e) = 1, p>3.
Dirichlet g.f.: 1/((1-1/2^s)*(1-1/3^s))^2 * Product{p prime > 3}(1/(1-1/p^s)). [corrected by Vaclav Kotesovec, Nov 20 2021] (End)
a(n) = Sum_{d divides n} mu(6d)*tau(n/d). - Benoit Cloitre, Jun 21 2007
Dirichlet g.f.: zeta(s)/((1-1/2^s)*(1-1/3^s)). - Ralf Stephan, Mar 24 2015; corrected by Vaclav Kotesovec, Nov 20 2021
Sum_{k=1..n} a(k) ~ 3*n. - Vaclav Kotesovec, Nov 20 2021

More terms from Benoit Cloitre, Jun 21 2007

A322026 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j) and A007949(i) = A007949(j), for all i, j, where A007814 and A007949 give the 2- and 3-adic valuations of n.

Original entry on oeis.org

1, 2, 3, 4, 1, 5, 1, 6, 7, 2, 1, 8, 1, 2, 3, 9, 1, 10, 1, 4, 3, 2, 1, 11, 1, 2, 12, 4, 1, 5, 1, 13, 3, 2, 1, 14, 1, 2, 3, 6, 1, 5, 1, 4, 7, 2, 1, 15, 1, 2, 3, 4, 1, 16, 1, 6, 3, 2, 1, 8, 1, 2, 7, 17, 1, 5, 1, 4, 3, 2, 1, 18, 1, 2, 3, 4, 1, 5, 1, 9, 19, 2, 1, 8, 1, 2, 3, 6, 1, 10, 1, 4, 3, 2, 1, 20, 1, 2, 7, 4, 1, 5, 1, 6, 3

Offset: 1

Antti Karttunen, Dec 03 2018

nonn

Restricted growth sequence transform of the ordered pair [A007814(n), A007949(n)].
For all i, j:
  A305900(i) = A305900(j) => a(i) = a(j),
  a(i) = a(j) => A122841(i) = A122841(j),
  a(i) = a(j) => A244417(i) = A244417(j),
  a(i) = a(j) => A322316(i) = A322316(j) => A072078(i) = A072078(j).
If and only if a(k) > a(i) for all k > i then k is in A003586, - David A. Corneth, Dec 03 2018
That is, A003586 gives the positions of records (1, 2, 3, 4, 5, ...) in this sequence.
Sequence A126760 (without its initial zero) and this sequence are ordinal transforms of each other.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Jon Maiga, Computer-generated formulas for A322026, Sequence Machine.

Cf. A003586 (positions of records, the first occurrence of n), A007814, A007949, A065331, A071521, A072078, A087465, A122841, A126760 (ordinal transform), A322316, A323883, A323884.

Cf. also A247714 and A255975.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A007814(n) = valuation(n,2);
A007949(n) = valuation(n,3);
v322026 = rgs_transform(vector(up_to, n, [A007814(n), A007949(n)]));
A322026(n) = v322026[n];

A065331(n) = (3^valuation(n, 3)<A065331
A071521(n) = { my(t=1/3); sum(k=0, logint(n, 3), t*=3; logint(n\t, 2)+1); }; \\ From A071521.
A322026(n) = A071521(A065331(n)); \\ Antti Karttunen, Sep 08 2024

For s = A003586(n), a(s) = n = a((6k+1)*s) = a((6k-1)*s), where s is the n-th 3-smooth number and k > 0. - David A. Corneth, Dec 03 2018
A065331(n) = A003586(a(n)). - David A. Corneth, Dec 04 2018
From Antti Karttunen, Sep 08 2024: (Start)
a(n) = Sum{k=1..n} [A126760(k)==A126760(n)], where [ ] is the Iverson bracket.
a(n) = A071521(A065331(n)). [Found by Sequence Machine and also by LODA miner]
a(n) = A323884(25*n). [Conjectured by Sequence Machine]
(End)

A244413 Exponent of highest power of 8 dividing n.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Jun 27 2014

This is the member g = 8 in the g-family of sequences, g integer >= 2, call it phi(g,n), n >= 1. In the Mahler reference, Lemma 2, pp. 6-7, this exponent is called f = -phi if g divides r = n (s = 1 there), and f = 0 if g does not divide r = n (s = 1 there).

Kurt Mahler, p-adic numbers and their functions, 2nd ed., Cambridge University press, 1981.

Antti Karttunen, Table of n, a(n) for n = 1..65536

Cf. A007814, A007949, A235127, A112765, A122841, A214411.

Table[IntegerExponent[n, 8], {n, 1, 100}] (* Amiram Eldar, Sep 14 2020 *)

A244413(n) = valuation(n,8); \\ Antti Karttunen, Oct 07 2017

def A244413(n): return (~n&n-1).bit_length()//3 # Chai Wah Wu, Jul 09 2022

n = 8^a(n)*m with a(n) nonnegative integer such that 8 does not divide m, for n >= 1.
O.g.f.: Sum_{k>=1} x^(8^k)/(1-x^(8^k)).
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/7. - Amiram Eldar, Jan 17 2022
a(n) = floor(A007814(n)/3). - Alan Michael Gómez Calderón, Jul 25 2024

A054895 a(n) = Sum_{k>0} floor(n/6^k).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16

Offset: 0

Henry Bottomley, May 23 2000

nonn

Different from the highest power of 6 dividing n! (cf. A054861). - Hieronymus Fischer, Aug 14 2007

Partial sums of A122841. - Hieronymus Fischer, Jun 06 2012

			  a(10^0) = 0.
  a(10^1) = 1.
  a(10^2) = 18.
  a(10^3) = 197.
  a(10^4) = 1997.
  a(10^5) = 19996.
  a(10^6) = 199995.
  a(10^7) = 1999995.
  a(10^8) = 19999994.
  a(10^9) = 199999993.

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Cf. A011371 and A054861 for analogs involving powers of 2 and 3.

Cf. A053827, A054861, A054899, A067080, A098844, A122841, A132030.

a054895 n = a054895_list !! n
a054895_list = scanl (+) 0 a122841_list
-- Reinhard Zumkeller, Nov 10 2013

function A054895(n)
  if n eq 0 then return n;
  else return A054895(Floor(n/6)) + Floor(n/6);
  end if; return A054895;
end function;
[A054895(n): n in [0..100]]; // G. C. Greubel, Feb 09 2023

Table[t=0; p=6; While[s=Floor[n/p]; t=t+s; s>0, p *= 6]; t, {n,0,100}]

def A054895(n):
    if (n==0): return 0
    else: return A054895(n//6) + (n//6)
[A054895(n) for n in range(104)] # G. C. Greubel, Feb 09 2023

a(n) = floor(n/6) + floor(n/36) + floor(n/216) + floor(n/1296) + ...
a(n) = (n - A053827(n))/5.
From Hieronymus Fischer, Aug 14 2007: (Start)
a(n) = a(floor(n/6)) + floor(n/6).
a(6*n) = n + a(n).
a(n*6^m) = n*(6^m-1)/5 + a(n).
a(k*6^m) = k*(6^m-1)/5, for 0 <= k < 6, m >= 0.
Asymptotic behavior:
a(n) = (n/5) + O(log(n)).
a(n+1) - a(n) = O(log(n)); this follows from the inequalities below.
a(n) <= (n-1)/5; equality holds for powers of 6.
a(n) >= ((n-5)/5) - floor(log_6(n)); equality holds for n=6^m-1, m>0.
lim inf (n/5 - a(n)) = 1/5, for n-->oo.
lim sup (n/5 - log_6(n) - a(n)) = 0, for n-->oo.
lim sup (a(n+1) - a(n) - log_6(n)) = 0, for n-->oo.
G.f.: (1/(1-x))*Sum_{k > 0} x^(6^k)/(1-x^(6^k)). (End)

An incorrect formula was deleted by N. J. A. Sloane, Nov 18 2008

Examples added by Hieronymus Fischer, Jun 06 2012

A234959 Highest power of 6 dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 36, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 6

Offset: 1

Tom Edgar, Jan 01 2014

The generalized binomial coefficients produced by this sequence provide an analog to Kummer's Theorem using arithmetic in base 6.

			Since 12 = 6 * 2, a(12) = 6. Likewise, since 6 does not divide 13, a(13) = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.

Cf. A006519, A038500, A122841, A234957, A243758.

a234959 = f 1 where
   f y x = if m == 0 then f (y * 6) x' else y  where (x', m) = divMod x 6
-- Reinhard Zumkeller, Feb 09 2015

6^Table[IntegerExponent[n, 6], {n, 84}] (* Alonso del Arte, Jan 01 2014 *)

a(n)=6^valuation(n,6) \\ Charles R Greathouse IV, Aug 05 2015

n=200 #change n for more terms
[6^(valuation(i,6)) for i in [1..n]]

a(n) = 6^(valuation(n,6)).
a(n) = 6^A122841(n). - Joerg Arndt, Jan 02 2014
G.f.: x/(1 - x) + 5 * Sum_{k>=1} 6^(k-1)*x^(6^k)/(1 - x^(6^k)). - Ilya Gutkovskiy, Jul 10 2019

A244417 Exponents of 6 in appearing in the 6-adic value of 1/n, n>=1 (A244416).

Original entry on oeis.org

0, 1, 1, 2, 0, 1, 0, 3, 2, 1, 0, 2, 0, 1, 1, 4, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 3, 2, 0, 1, 0, 5, 1, 1, 0, 2, 0, 1, 1, 3, 0, 1, 0, 2, 2, 1, 0, 4, 0, 1, 1, 2, 0, 3, 0, 3, 1, 1, 0, 2, 0, 1, 2, 6, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 2, 0, 1, 0, 4, 4, 1, 0, 2, 0, 1, 1, 3, 0, 2, 0, 2, 1, 1, 0, 5, 0, 1, 2, 2

Offset: 1

Wolfdieter Lang, Jul 02 2014

For the definition of 'g-dic value of 1/n' see a comment on A244416. In the Mahler reference, p. 7, the present exponent of 6 is there called f = f(1/n) for g = 6.

			See A244416.

Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.

Antti Karttunen, Table of n, a(n) for n = 1..19683

Cf. A122841, A244416, A007814 (g=2), A007949 (g=3), A244415 (g=4), A112765 (g=5), A051903, A065331.

Cf. also A322026, A322316.

a[n_] := Max[IntegerExponent[n, {2, 3}]]; Array[a, 100] (* Amiram Eldar, Aug 19 2024 *)

A244417(n) = max(valuation(n,2), valuation(n,3)); \\ Antti Karttunen, Dec 04 2018

a(n) = 0 if n is congruent 1 or 5 (mod 6). a(n) = max(A007814(n), A007949(n)) if n == 0 (mod 6). a(n) = A007814(n) if n == 2 or 4 (mod 6) and a(n) = A007949(n) if n == 3 (mod 6).
a(n) = max(A007814(n), A007949(n)), in all cases. - Antti Karttunen, Dec 04 2018
From Amiram Eldar, Aug 19 2024: (Start)
a(n) = A051903(A065331(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 13/10. (End)

A373216 Expansion of Sum_{k>=0} x^(6^k) / (1 - x^(6^k)).

Original entry on oeis.org

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

Offset: 1

Seiichi Manyama, May 28 2024

nonn

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

Cf. A001511, A051064, A055457, A115362, A373217.

Cf. A122841, A373220.

PARI
```
a(n) = valuation(n, 6)+1;
```

G.f. A(x) satisfies A(x) = x/(1 - x) + A(x^6).
a(6*n+1) = a(6*n+2) = ... = (6*n+5) = 1 and a(6*n+6) = 1 + a(n+1) for n >= 0.
a(n) = A122841(n) + 1.
G.f.: Sum_{i>=1, j>=0} x^(i*6^j). - Seiichi Manyama, Mar 23 2025
a(n) = A122841(6*n). - R. J. Mathar, Jun 28 2025

A249344 A(n,k) = exponent of the largest power of n-th prime which divides k, square array read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 28 2014

Square array A(n,k), where n = row, k = column, read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ... (transpose of array A060175).
A(n,k) is the (p_n)-adic valuation of k, where p_n is the n-th prime, A000040(n).
Each row is effectively a ruler function, s, with s(1) = 0. - Peter Munn, Apr 30 2022

			The top-left corner of the array:
  0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, ...
  0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, ...
  0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, ...
  0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, ...
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, ...
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, ...
  ...
A(1,8) = 3, because 2^3 is the largest power of 2 (= p_1 = A000040(1)) that divides 8.
a(2,9) = 2, because 3^2 is the largest power of 3 (= p_2) that divides 9.
a(3,15) = 1, because 5^1 is the largest power of 5 (= p_3) that divides 15.

Transpose: A060175.
Row 1: A007814.
Row 2: A007949.
Row 3: A112765.
Row 4: A214411.
Cf. also A000040, A002260, A004736, A085604, A090622, A115627, A249421.
Completely additive sequences where more than one prime is mapped to 1, all other primes to 0: A065339, A083025, A087436, A169611.
Ruler functions, s, with s(1) = 0 that are not rows here: A122840, A122841, A235127, A244413.

A[n_, k_] := IntegerExponent[k, Prime[n]]; Table[A[k, n - k + 1], {n, 1, 15}, {k, 1, n}] // Flatten (* Amiram Eldar, Oct 01 2023 *)

a(n, k) = valuation(k, prime(n)); \\ Michel Marcus, Jun 24 2017

from sympy import prime
def a(n, k):
    p=prime(n)
    i=z=0
    while p**i<=k:
        if k%(p**i)==0: z=i
        i+=1
    return z
for n in range(1, 10): print([a(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Jun 24 2017

(define (A249344 n) (A249344bi (A002260 n) (A004736 n)))
(define (A249344bi row col) (let ((p (A000040 row))) (let loop ((n col) (i 0)) (cond ((not (zero? (modulo n p))) i) (else (loop (/ n p) (+ i 1)))))))

Row n, as a sequence, is completely additive with A(n, prime(n)) = 1, A(n, prime(m)) = 0 for m <> n. - Peter Munn, Apr 30 2022

Sum_{k=1..m} A(n,k) ~ (1/(prime(n)-1)) * m. - Amiram Eldar, Oct 01 2023

A102679 Number of digits >= 7 in decimal representation of n.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Feb 03 2005

a(n) = 0 iff n is in A007093 (numbers in base 7). - Bernard Schott, Feb 12 2023

Hieronymus Fischer, Table of n, a(n) for n = 0..10000

Cf. A007093, A027868, A054899, A055640, A055641, A102669-A102685, A117804, A122840, A122841, A160093, A160094, A196563, A196564. Partial sums see A102680.

Cf. A000120, A000788, A023416, A059015 (for base 2).

p:=proc(n) local b,ct,j: b:=convert(n,base,10): ct:=0: for j from 1 to nops(b) do if b[j]>=7 then ct:=ct+1 else ct:=ct fi od: ct: end: seq(p(n),n=0..125); # Emeric Deutsch, Feb 23 2005

From Hieronymus Fischer, Jun 10 2012: (Start)
a(n) = Sum_{j=1..m+1} (floor(n/10^j + 3/10) - floor(n/10^j)), where m = floor(log_10(n)).
G.f.: g(x) = (1/(1-x))*Sum_{j>=0} (x^(7*10^j) - x^(10*10^j))/(1 - x^10^(j+1)). (End)

More terms from Emeric Deutsch, Feb 23 2005

A309891 a(n) is the total number of trailing zeros in the representations of n over all bases b >= 2.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 5, 3, 3, 1, 6, 1, 3, 3, 8, 1, 6, 1, 6, 3, 3, 1, 9, 3, 3, 5, 6, 1, 7, 1, 10, 3, 3, 3, 11, 1, 3, 3, 9, 1, 7, 1, 6, 6, 3, 1, 13, 3, 6, 3, 6, 1, 9, 3, 9, 3, 3, 1, 12, 1, 3, 6, 14, 3, 7, 1, 6, 3, 7, 1, 15, 1, 3, 6, 6, 3, 7, 1, 13, 8, 3, 1, 12

Offset: 1

Rémy Sigrist, Aug 21 2019

a(n) depends only on the prime signature of n.

a(n) is the sum of the k-adic valuations of n for k >= 2. - Friedjof Tellkamp, Jan 25 2025

			For n = 12: 12 has 2 trailing zeros in base 2 (1100), 1 trailing zero in bases 3, 4, 6 and 12 (110, 30, 20, 10) and no trailing zero in other bases, hence a(12) = 1*2 + 4*1 = 6.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001221, A006218, A007814, A007949, A033093, A056239, A112765, A122841, A169594, A214411, A235127, A244413, A286561, A293514, A369180.
Cf. A111003.
First differences of A078632.

Table[DivisorSum[n, IntegerExponent[n, #] &, # > 1 &], {n, 84}] (* Jon Maiga, Aug 25 2019 *)

a(n) = sumdiv(n, d, if (d>1, valuation(n,d), 0))

a(n) = {if(n == 1, return(0)); my(f = factor(n)[, 2], res = 0, t = 2, of = f, nf = f >> 1, nd(v) = prod(i = 1, #v, v[i] + 1)); while(Set(of) != [0], res += (nd(of) - nd(nf)) * (t-1); of = nf; t++; nf = f \ t); res} \\ David A. Corneth, Aug 22 2019

a(n) = Sum_{d|n, d>1} A286561(n,d), where A286561 gives the d-valuation of n.
a(p) = 1 for any prime number p.
a(p^k) = A006218(k) for any k >= 0 and any prime number p.
a(n) = 2^A001221(n) - 1 for any squarefree number n.
a(n) = 3 for any semiprime number n.
a(m*n) >= a(m) + a(n).
a(n) >= A007814(n) + A007949(n) + A235127(n) + A112765(n) + A122841(n) + A214411(n) + A244413(n).
a(n) = A056239(A293514(n)). - Antti Karttunen, Aug 22 2019
a(n) <= A033093(n). - Michel Marcus, Aug 22 2019
a(n) = A169594(n) - 1. - Jon Maiga, Aug 25 2019
From Friedjof Tellkamp, Feb 27 2024: (Start)
G.f.: Sum_{k>=2, j>=1} x^(k^j)/(1-x^(k^j)).
Dirichlet g.f.: zeta(s) * Sum_{k>=1} (zeta(k*s) - 1).
Sum_{n>=1} a(n)/n^2 = Pi^2/8 (A111003). (End)

cp's OEIS Frontend

A072078 Number of 3-smooth divisors of n.

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A322026 Lexicographically earliest infinite sequence such that a(i) = a(j) => A007814(i) = A007814(j) and A007949(i) = A007949(j), for all i, j, where A007814 and A007949 give the 2- and 3-adic valuations of n.

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A244413 Exponent of highest power of 8 dividing n.

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A054895 a(n) = Sum_{k>0} floor(n/6^k).

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A234959 Highest power of 6 dividing n.

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A244417 Exponents of 6 in appearing in the 6-adic value of 1/n, n>=1 (A244416).

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