A124010 -id:A124010 - cp's OEIS frontend

A056169 Number of unitary prime divisors of n.

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

Offset: 1

Labos Elemer, Jul 27 2000

The zeros of this sequences are the powerful numbers (A001694). There are no arbitrarily long subsequences with a given upper bound; for example, every sequence of 4 values includes one divisible by 2 but not 4, so there are no more than 3 consecutive zeros. Similarly, there can be no more than 23 consecutive values with none divisible by both 2 and 3 but neither 4 nor 9 (so a(n) >= 2), etc. In general, this gives an upper bound that is a (relatively) small multiple of the k-th primorial number (prime(k)#). One suspects that the actual upper bounds for such subsequences are quite a bit lower; e.g., Erdős conjectured that there are no three consecutive powerful numbers. - Franklin T. Adams-Watters, Aug 08 2006
In particular, for every A048670(k)*A002110(k) consecutive terms, at least one is greater than or equal to k. - Charlie Neder, Jan 03 2019
Following Catalan's conjecture (which became Mihăilescu's theorem in 2002), the first case of two consecutive zeros in this sequence is for a(8) and a(9), because 8 = 2^3 and 9 = 3^2, and there are no other consecutive zeros for consecutive powers. However, there are other pairs of consecutive zeros at powerful numbers (A001694, A060355). The next example is a(288) = a(289) = 0, because 288 = 2^5 * 3^2 and 289 = 17^2, then also a(675) and a(676). - Bernard Schott, Jan 06 2019
a(2k-1) is the number of primes p such that p || x + y and p^2 || x^(2k-1) + y^(2k-1) for some positive integers x and y. For any positive integers x, y and k > 1, there is no prime p such that p || x + y and p^2 || x^(2k) + y^(2k). - Jinyuan Wang, Apr 08 2020

			9 = 3^2 so a(9) = 0; 10 = 2 * 5 so a(10) = 2; 11 = 11^1 so a(11) = 1.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Catalan's Conjecture.
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A001694, A002110, A034444, A056170, A055231, A076445, A162642, A275812, A295659, A295662, A295664, A001694.
See also A060355.
Cf. A077761, A085548.

a056169 = length . filter (== 1) . a124010_row
-- Reinhard Zumkeller, Sep 10 2013

a:= n-> nops(select(i-> i[2]=1, ifactors(n)[2])):
seq(a(n), n=1..120);  # Alois P. Heinz, Mar 27 2017

Join[{0},Table[Count[Transpose[FactorInteger[n]][[2]],1],{n,2,110}]] (* Harvey P. Dale, Mar 15 2012 *)
Table[DivisorSum[n, 1 &, And[PrimeQ@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Nov 28 2017 *)

a(n)=my(f=factor(n)[,2]); sum(i=1,#f,f[i]==1) \\ Charles R Greathouse IV, Apr 29 2015

from sympy import factorint
def a(n):
    f=factorint(n)
    return 0 if n==1 else sum(1 for i in f if f[i]==1)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 19 2017

;; With memoization-macro definec.
(definec (A056169 n) (if (= 1 n) 0 (+ (if (= 1 (A067029 n)) 1 0) (A056169 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

A prime factor of n is unitary iff its exponent is 1 in prime factorization of n. In general, gcd(p, n/p) = 1 or = p.
Additive with a(p^e) = 1 if e = 1, 0 otherwise.
a(n) = #{k: A124010(n,k) = 1, k = 1..A001221}. - Reinhard Zumkeller, Sep 10 2013
From Antti Karttunen, Nov 28 2017: (Start)
a(1) = 0; for n > 1, a(n) = A063524(A067029(n)) + a(A028234(n)).
a(n) = A001221(A055231(n)) = A001222(A055231(n)).
a(n) = A001221(n) - A056170(n) = A001221(n) - A001221(A000188(n)).
a(n) = A001222(n) - A275812(n).
a(n) = A162642(n) - A295662(n).
a(n) <= A162642(n) <= a(n) + A295659(n).
a(n) <= A295664(n).
(End)
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B - C), where B is Mertens's constant (A077761) and C = Sum_{p prime} (1/p^2) = 0.452247... (A085548). - Amiram Eldar, Sep 28 2023

A071178 Exponent of the largest prime factor of n.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jun 10 2002

a(n) = the multiplicity of the largest part in the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436. Example: a(18) = 2; indeed, the partition having Heinz number 18 = 2*3*3 is [1,2,2]. - Emeric Deutsch, Jun 04 2015

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A006530, A053585, A067029, A067255, A070003, A124010, A215366.

a071178 = last . a124010_row -- Reinhard Zumkeller, Aug 27 2011

with(numtheory): with(padic):
a:= n-> `if`(n=1, 0, ordp(n, max(factorset(n)[]))):
seq(a(n), n=1..120);  # Alois P. Heinz, Jun 04 2015

a[n_] := FactorInteger[n] // Last // Last; Table[a[n], {n, 1, 120}] (* Jean-François Alcover, Jun 12 2015 *)
Join[{0},Table[FactorInteger[n][[-1,2]],{n,2,120}]] (* Harvey P. Dale, Aug 02 2025 *)

a(n) = if(n == 1, 0, my(e = factor(n)[, 2]); e[#e]); \\ Amiram Eldar, Oct 02 2024

from sympy import factorint
def A071178(n): return max(factorint(n).items())[1] if n>1 else 0 # Chai Wah Wu, Oct 10 2023

a(n) = A124010(n, A001221(n)); A053585(n) = A006530(n)^a(n). - Reinhard Zumkeller, Aug 27 2011
a(n) = A067255(n, A001222(n)). - Reinhard Zumkeller, Jun 11 2013
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 (since the asymptotic density of A070003 is 0). - Amiram Eldar, Oct 02 2024

A368100 Numbers of which it is possible to choose a different prime factor of each prime index.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 39, 41, 43, 47, 51, 53, 55, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 107, 109, 111, 113, 119, 123, 127, 129, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163

Offset: 1

Gus Wiseman, Dec 12 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 2849 are {4,5,12}, with prime factors {{2,2},{5},{2,2,3}}, and of the two choices (2,5,2) and (2,5,3) the latter has all different terms, so 2849 is in the sequence.
The terms together with their prime indices of prime indices begin:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  23: {{2,2}}
  29: {{1,3}}
  31: {{5}}
  33: {{1},{3}}
  35: {{2},{1,1}}
  37: {{1,1,2}}
  39: {{1},{1,2}}

The complement is A355529, odd A355535, binary A367907.
Positions of positive terms in A367771.
The version for binary indices is A367906, positive positions in A367905.
For a unique choice we have A368101, binary A367908.
The version for divisors instead of factors is A368110, complement A355740.
A058891 counts set-systems, covering A003465, connected A323818.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
Cf. A092918,  A355737, A355739, A355741, A355744, A355745, A367902.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], Select[Tuples[prix/@prix[#]], UnsameQ@@#&]!={}&]

A035316 Sum of the square divisors of n.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 21, 1, 10, 1, 5, 1, 1, 1, 5, 26, 1, 10, 5, 1, 1, 1, 21, 1, 1, 1, 50, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 21, 50, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 85, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 26, 5, 1, 1, 1, 21, 91, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1

Offset: 1

N. J. A. Sloane

The Dirichlet generating function is zeta(s)*zeta(2s-2). The sequence is the Dirichlet convolution of A000012 with the sequence defined by n*A010052(n). - R. J. Mathar, Feb 18 2011

Inverse Möbius transform of n * c(n), where c(n) is the characteristic function of squares (A010052). - Wesley Ivan Hurt, Jun 20 2024

Nick Hobson, Table of n, a(n) for n = 1..1000
A. Dixit, B. Maji, and A. Vatwani, Voronoi summation formula for the generalized divisor function sigma_z^k(n), arXiv:2303.09937 [math.NT], 2023, sigma_(z=2,k=2,n).
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT] (2011), Remark 15.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

Cf. A001157, A010052, A027748, A124010, A113061 (sum cube divs).
Sum of the k-th powers of the square divisors of n for k=0..10: A046951 (k=0), this sequence (k=1), A351307 (k=2), A351308 (k=3), A351309 (k=4), A351310 (k=5), A351311 (k=6), A351313 (k=7), A351314 (k=8), A351315 (k=9), A351315 (k=10).
Cf. A008836, A000203.

a035316 n = product $
   zipWith (\p e -> (p ^ (e + 2 - mod e 2) - 1) `div` (p ^ 2 - 1))
           (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Jul 28 2014

A035316 := proc(n)
    local a,pe,p,e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ;
        e := pe[2] ;
        if type(e,'even') then
            e := e+2 ;
        else
            e := e+1 ;
        end if;
        a := a*(p^e-1)/(p^2-1) ;
    end do:
    a ;
end proc:
seq(A035316(n),n=1..100) ; # R. J. Mathar, Oct 10 2017

Table[ Plus @@ Select[ Divisors@ n, IntegerQ@ Sqrt@ # &], {n, 93}] (* Robert G. Wilson v, Feb 19 2011 *)
f[p_, e_] := (p^(2*(1 + Floor[e/2])) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 01 2020 *)

vector(93, n, sumdiv(n, d, issquare(d)*d))

a(n)=my(f=factor(n));prod(i=1,#f[,1],(f[i,1]^(f[i,2]+2-f[i,2]%2)-1)/(f[i,1]^2-1)) \\ Charles R Greathouse IV, May 20 2013

Multiplicative with a(p^e)=(p^(e+2)-1)/(p^2-1) for even e and a(p^e)=(p^(e+1)-1)/(p^2-1) for odd e. - Vladeta Jovovic, Dec 05 2001
G.f.: Sum_{k>0} k^2*x^(k^2)/(1-x^(k^2)). - Vladeta Jovovic, Dec 13 2002
a(n^2) = A001157(n). - Michel Marcus, Jan 14 2014
L.g.f.: -log(Product_{k>=1} (1 - x^(k^2))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
Sum_{k=1..n} a(k) ~ Zeta(3/2)*n^(3/2)/3 - n/2. - Vaclav Kotesovec, Feb 04 2019
a(n) = Sum_{k=1..n} k * (floor(sqrt(k)) - floor(sqrt(k-1))) * (1 - ceiling(n/k) + floor(n/k)). - Wesley Ivan Hurt, Jun 13 2021
a(n) = Sum_{d|n} d * c(d), where c = A010052. - Wesley Ivan Hurt, Jun 20 2024
a(n) = Sum_{d|n} lambda(d)*d*sigma(n/d), where lambda = A008836. - Ridouane Oudra, Jul 18 2025

A008475 If n = Product (p_j^k_j) then a(n) = Sum (p_j^k_j) (a(1) = 0 by convention).

Original entry on oeis.org

0, 2, 3, 4, 5, 5, 7, 8, 9, 7, 11, 7, 13, 9, 8, 16, 17, 11, 19, 9, 10, 13, 23, 11, 25, 15, 27, 11, 29, 10, 31, 32, 14, 19, 12, 13, 37, 21, 16, 13, 41, 12, 43, 15, 14, 25, 47, 19, 49, 27, 20, 17, 53, 29, 16, 15, 22, 31, 59, 12, 61, 33, 16, 64, 18, 16, 67, 21, 26, 14, 71, 17, 73

Offset: 1

Olivier Gérard

For n>1, a(n) is the minimal number m such that the symmetric group S_m has an element of order n. - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 26 2001

If gcd(u,w) = 1, then a(u*w) = a(u) + a(w); behaves like logarithm; compare A001414 or A056239. - Labos Elemer, Mar 31 2003

			a(180) = a(2^2 * 3^2 * 5) = 2^2 + 3^2 + 5 = 18.

F. J. Budden, The Fascination of Groups, Cambridge, 1972; pp. 322, 573.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter IV, p. 147.
T. Z. Xuan, On some sums of large additive number theoretic functions (in Chinese), Journal of Beijing normal university, No. 2 (1984), pp. 11-18.

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
John Bamberg, Grant Cairns and Devin Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, Vol. 110, No. 3 (March 2003), pp. 202-209.
Roger B. Eggleton and William P. Galvin, Upper Bounds on the Sum of Principal Divisors of an Integer, Mathematics Magazine, Vol. 77, No. 3 (Jun., 2004), pp. 190-200.

Cf. A001414, A000961, A005117, A051613, A072691, A081402, A081403, A081404, A027748, A124010, A001221, A028233, A034684, A053585, A159077, A023888, A078771, A092509, A286875.

See A222416 for the variant with a(1)=1.

a008475 1 = 0
a008475 n = sum $ a141809_row n
-- Reinhard Zumkeller, Jan 29 2013, Oct 10 2011

A008475 := proc(n) local e,j; e := ifactors(n)[2]:
add(e[j][1]^e[j][2], j=1..nops(e)) end:
seq(A008475(n), n=1..60); # Peter Luschny, Jan 17 2010

f[n_] := Plus @@ Power @@@ FactorInteger@ n; f[1] = 0; Array[f, 73]

for(n=1,100,print1(sum(i=1,omega(n), component(component(factor(n),1),i)^component(component(factor(n),2),i)),","))

a(n)=local(t);if(n<1,0,t=factor(n);sum(k=1,matsize(t)[1],t[k,1]^t[k,2])) /* Michael Somos, Oct 20 2004 */

A008475(n) = { my(f=factor(n)); vecsum(vector(#f~,i,f[i,1]^f[i,2])); }; \\ Antti Karttunen, Nov 17 2017

from sympy import factorint
def a(n):
    f=factorint(n)
    return 0 if n==1 else sum([i**f[i] for i in f]) # Indranil Ghosh, May 20 2017

Additive with a(p^e) = p^e.
a(A000961(n)) = A000961(n); a(A005117(n)) = A001414(A005117(n)).
a(n) = Sum_{k=1..A001221(n)} A027748(n,k) ^ A124010(n,k) for n>1. - Reinhard Zumkeller, Oct 10 2011
a(n) = Sum_{k=1..A001221(n)} A141809(n,k) for n > 1. - Reinhard Zumkeller, Jan 29 2013
Sum_{k=1..n} a(k) ~ (Pi^2/12)* n^2/log(n) + O(n^2/log(n)^2) (Xuan, 1984). - Amiram Eldar, Mar 04 2021

A050361 Number of factorizations into distinct prime powers greater than 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1

Offset: 1

Christian G. Bower, Oct 15 1999

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

The number of unordered factorizations of n into 1 and exponentially odd prime powers, i.e., p^e where p is a prime and e is odd (A246551). - Amiram Eldar, Jun 12 2025

			From _Gus Wiseman_, Jul 30 2022: (Start)
The A000688(216) = 9 factorizations of 216 into prime powers are:
  (2*2*2*3*3*3)
  (2*2*2*3*9)
  (2*2*2*27)
  (2*3*3*3*4)
  (2*3*4*9)
  (2*4*27)
  (3*3*3*8)
  (3*8*9)
  (8*27)
Of these, the a(216) = 4 strict cases are:
  (2*3*4*9)
  (2*4*27)
  (3*8*9)
  (8*27)
(End)

Antti Karttunen, Table of n, a(n) for n = 1..100000 (first 10000 terms from Reinhard Zumkeller)
Index entries for sequences computed from exponents in factorization of n.

Cf. A000009, A050360, A050362, A050363, A050364, A101296, A246551.
Cf. A124010.
This is the strict case of A000688.
Positions of 1's are A004709, complement A046099.
The case of primes (instead of prime-powers) is A008966, non-strict A000012.
The non-strict additive version allowing 1's A023893, ranked by A302492.
The non-strict additive version is A023894, ranked by A355743.
The additive version (partitions) is A054685, ranked by A356065.
The additive version allowing 1's is A106244, ranked by A302496.
A001222 counts prime-power divisors.
A005117 lists all squarefree numbers.
A034699 gives maximal prime-power divisor.
A246655 lists all prime-powers (A000961 includes 1), towers A164336.
A296131 counts twice-factorizations of type PQR, non-strict A295935.
Cf. A001970, A002110, A025487, A055887, A063834, A076610, A085970, A279786, A302590, A302601, A354911, A355742.

a050361 = product . map a000009 . a124010_row
-- Reinhard Zumkeller, Aug 28 2014

A050361 := proc(n)
    local a,f;
    if n = 1 then
        1;
    else
        a := 1 ;
        for f in ifactors(n)[2] do
            a := a*A000009(op(2,f)) ;
        end do:
    end if;
end proc: # R. J. Mathar, May 25 2017

Table[Times @@ PartitionsQ[Last /@ FactorInteger[n]], {n, 99}] (* Arkadiusz Wesolowski, Feb 27 2017 *)

A000009(n,k=(n-!(n%2))) = if(!n,1,my(s=0); while(k >= 1, if(k<=n, s += A000009(n-k,k)); k -= 2); (s));
A050361(n) = factorback(apply(A000009,factor(n)[,2])); \\ Antti Karttunen, Nov 17 2019

Dirichlet g.f.: Product_{n is a prime power >1}(1 + 1/n^s).
Multiplicative with a(p^e) = A000009(e).
a(A002110(k))=1.
a(n) = A050362(A101296(n)). - R. J. Mathar, May 26 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.26020571070524171076..., where f(x) = (1-x) * Product_{k>=1} (1 + x^k). - Amiram Eldar, Oct 03 2023

A353833 Numbers whose multiset of prime indices has all equal run-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 40, 41, 43, 47, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173, 179

Offset: 1

Gus Wiseman, May 23 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

			The prime indices of 12 are {1,1,2}, with run-sums (2,2), so 12 is in the sequence.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

For parts instead of run-sums we have A000961, counted by A000005.
For run-lengths instead of run-sums we have A072774, counted by A047966.
These partitions are counted by A304442.
These are the positions of powers of primes in A353832.
The restriction to nonprimes is A353834.
For distinct instead of equal run-sums we have A353838, counted by A353837.
The version for compositions is A353848, counted by A353851.
A001222 counts prime factors, distinct A001221.
A005811 counts runs in binary expansion, distinct run-lengths A165413.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A300273 ranks collapsible partitions, counted by A275870.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353840-A353846 deal with iterated run-sums for partitions.
A353862 gives greatest run-sum of prime indices, least A353931.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A007947, A071625, A073093, A181819, A238279, A304660, A323014, A333755, A353839.

Select[Range[100],SameQ@@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&]

A049419 a(1) = 1; for n > 1, a(n) = number of exponential divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 4, 1, 1

Offset: 1

Yasutoshi Kohmoto

The exponential divisors of a number x = Product p(i)^r(i) are all numbers of the form Product p(i)^s(i) where s(i) divides r(i) for all i.
Wu gives a complicated Dirichlet g.f.
a(1) = 1 by convention. This is also required for a function to be multiplicative. - N. J. A. Sloane, Mar 03 2009
The inverse Moebius transform seems to be in A124315. The Dirichlet inverse appears to be related to A166234. - R. J. Mathar, Jul 14 2014

			a(8)=2 because 2 and 2^3 are e-divisors of 8.
The sets of e-divisors start as:
  1:{1}
  2:{2}
  3:{3}
  4:{2, 4}
  5:{5}
  6:{6}
  7:{7}
  8:{2, 8}
  9:{3, 9}
  10:{10}
  11:{11}
  12:{6, 12}
  13:{13}
  14:{14}
  15:{15}
  16:{2, 4, 16}
  17:{17}
  18:{6, 18}
  19:{19}
  20:{10, 20}
  21:{21}
  22:{22}
  23:{23}
  24:{6, 24}

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Andrew V. Lelechenko, Exponential and infinitary divisors, arXiv:1405.7597 [math.NT], 2014, sequence tau^(e).
David Moews, A database of aliquot cycles.
J. O. M. Pedersen, Tables of Aliquot Cycles.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
László Tóth and Nicuşor Minculete, Exponential unitary divisors, arXiv:0910.2798 [math.NT], 2009.
Tim Trudgian, The sum of the unitary divisor function, arXiv:1312.4615 [math.NT], 2013-2014, Section 3.
Eric Weisstein's World of Mathematics, e-Divisor.
Jie Wu, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, J. Theor. Nombr. Bordeaux 7 (1) (1995) 133-141.

Row lengths of A322791.
Cf. A049599, A061389, A051377 (sum of e-divisors).
Partial sums are in A099593.
Cf. A124010, A000005, A049599, A072911, A124315, A166234, A327837, A361012.

A049419:=n->Product(List(Collected(Factors(n)), p -> Tau(p[2]))); List([1..10^4], n -> A049419(n)); # Muniru A Asiru, Oct 29 2017

a049419 = product . map (a000005 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

A049419 := proc(n)
    local a;
    a := 1 ;
    for pf in ifactors(n)[2] do
        a := a*numtheory[tau](op(2,pf)) ;
    end do:
    a ;
end proc:
seq(A049419(n),n=1..20) ; # R. J. Mathar, Jul 14 2014

a[1] = 1; a[p_?PrimeQ] = 1; a[p_?PrimeQ, e_] := DivisorSigma[0, e]; a[n_] := Times @@ (a[#[[1]], #[[2]]] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Jan 30 2012, after Vladeta Jovovic *)

a(n) = vecprod(apply(numdiv, factor(n)[,2])); \\ Amiram Eldar, Mar 27 2023

Multiplicative with a(p^e) = tau(e). - Vladeta Jovovic, Jul 23 2001

Sum_{k=1..n} a(k) ~ A327837 * n. - Vaclav Kotesovec, Feb 27 2023

More terms from Jud McCranie, May 29 2000

A067255 Irregular triangle read by rows: row n gives exponents in prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Jeppe Stig Nielsen, Feb 20 2002

Row lengths are given by A061395(n), n >= 2: [1, 2, 1, 3, 2, 4, 1, 2, ... ].

This sequence contains every finite sequence of nonnegative integers. - Franklin T. Adams-Watters, Jun 22 2005

			1 = 2^0
2 = 2^1
3 = 2^0 3^1
4 = 2^2
5 = 2^0 3^0 5^1
6 = 2^1 3^1
... and reading the exponents gives the sequence.
Since for example 99=2^0*3^2*5^0*7^0*11^1, we use this symbol for ninety-nine: 99: {0,2,0,0,1}. Concatenating all the symbols for 1,2,3,4,5,6,..., we get the sequence.

Reinhard Zumkeller, Rows n = 1..250 of triangle, flattened
Jeppe Stig Nielsen, See this explanation.

Cf. A133457.
Cf. A001222 (row sums), A061395 (lengths of rows n >= 2).
Cf. A007814 (left edge), A071178 (right edge).
Other versions: A054841 (rows reversed and concatenated into a decimal number), A060175 (square array), A082786 (regular triangle), A124010 (with 0's removed, excepting row 1), A143078 (another irregular triangle).

a067255 n k = a067255_tabf !! (n-1) !! (k-1)
a067255_row 1 = [0]
a067255_row n = f n a000040_list where
   f 1 _      = []
   f u (p:ps) = g u 0 where
     g v e = if m == 0 then g v' (e + 1) else e : f v ps
             where (v',m) = divMod v p
a067255_tabf = map a067255_row [1..]
-- Reinhard Zumkeller, Jun 11 2013

f[n_] := If[n == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ n]; Array[f, 29] // Flatten (* Michael De Vlieger, Mar 08 2019 *)

A257993 Least gap in the partition having Heinz number n; index of the least prime not dividing n.

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3

Offset: 1

Emeric Deutsch, May 18 2015

nonn

The "least gap" of a partition is the least positive integer that is not a part of the partition. For example, the least gap of the partition [7,4,2,2,1] is 3.
We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.
In the Maple program the subprogram B yields the partition with Heinz number n.
Sum of least gaps of all partitions of m = A022567(m).
From Antti Karttunen, Aug 22 2016: (Start)
Index of the least prime not dividing n. (After a formula given by Heinz.)
Least k such that A002110(k) does not divide n.
One more than the number of trailing zeros in primorial base representation of n, A049345.
(End)
The least gap is also called the mex (minimal excludant) of the partition. - Gus Wiseman, Apr 20 2021

			a(18) = 3 because the partition having Heinz number 18 = 2*3*3 is [1,2,2], having least gap equal to 3.

G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004, Cambridge.
Miklós Bóna, A Walk Through Combinatorics, World Scientific Publishing Co., 2002.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
P. J. Grabner and A. Knopfmacher, Analysis of some new partition statistics, Ramanujan J., 12, 2006, 439-454.
Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.
Wikipedia, Mex (mathematics).
Index entries for sequences related to primorial base.

Cf. A215366, A002110, A022567, A049345, A053669, A055396, A064648, A276086, A276094, A276152.
Positions of 1's are A005408.
Positions of 2's are A047235.
The number of gaps is A079067.
The version for crank is A257989.
The triangle counting partitions by this statistic is A264401.
One more than A276084.
The version for greatest difference is A286469 or A286470.
A maximal instead of minimal version is A339662.
Positions of even terms are A342050.
Positions of odd terms are A342051.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A238709 counts partitions by sum and least difference.
A333214 lists positions of adjacent unequal prime gaps.
A339737 counts partitions by sum and greatest gap.
Cf. A001223, A001511, A005117, A026794, A029707, A072233, A079068, A098743, A124010, A279945, A325351, A325352.

with(numtheory): a := proc (n) local B, q: B := proc (n) local nn, j, m: nn := op(2, ifactors(n)): for j to nops(nn) do m[j] := op(j, nn) end do: [seq(seq(pi(op(1, m[i])), q = 1 .. op(2, m[i])), i = 1 .. nops(nn))] end proc: for q while member(q, B(n)) = true do  end do: q end proc: seq(a(n), n = 1 .. 150);
# second Maple program:
a:= n-> `if`(n=1, 1, (s-> min({$1..(max(s)+1)} minus s))(
        {map(x-> numtheory[pi](x[1]), ifactors(n)[2])[]})):
seq(a(n), n=1..100);  # Alois P. Heinz, May 09 2016
# faster:
A257993 := proc(n) local p, c; c := 1; p := 2;
while n mod p = 0 do p := nextprime(p); c := c + 1 od: c end:
seq(A257993(n), n=1..100); # Peter Luschny, Jun 04 2017

A053669[n_] := For[p = 2, True, p = NextPrime[p], If[CoprimeQ[p, n], Return[p]]]; a[n_] := PrimePi[A053669[n]]; Array[a, 100] (* Jean-François Alcover, Nov 28 2016 *)
Table[k = 1; While[! CoprimeQ[Prime@ k, n], k++]; k, {n, 100}] (* Michael De Vlieger, Jun 22 2017 *)

a(n) = forprime(p=2,, if (n % p, return(primepi(p)))); \\ Michel Marcus, Jun 22 2017

from sympy import nextprime, primepi
def a053669(n):
    p = 2
    while True:
        if n%p!=0: return p
        else: p=nextprime(p)
def a(n): return primepi(a053669(n)) # Indranil Ghosh, May 12 2017

(define (A257993 n) (let loop ((n n) (i 1)) (let* ((p (A000040 i)) (d (modulo n p))) (if (not (zero? d)) i (loop (/ (- n d) p) (+ 1 i))))))
;; Antti Karttunen, Aug 22 2016

a(n) = A000720(A053669(n)). - Alois P. Heinz, May 18 2015
From Antti Karttunen, Aug 22-30 2016: (Start)
a(n) = 1 + A276084(n).
a(n) = A055396(A276086(n)).
A276152(n) = A002110(a(n)).
(End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{k>=1} 1/A002110(k) = 1.705230... (1 + A064648). - Amiram Eldar, Jul 23 2022
a(n) << log n/log log n. - Charles R Greathouse IV, Dec 03 2022

A simpler description added to the name by Antti Karttunen, Aug 22 2016

cp's OEIS Frontend

A056169 Number of unitary prime divisors of n.

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A071178 Exponent of the largest prime factor of n.

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A368100 Numbers of which it is possible to choose a different prime factor of each prime index.

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A035316 Sum of the square divisors of n.

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A008475 If n = Product (p_j^k_j) then a(n) = Sum (p_j^k_j) (a(1) = 0 by convention).

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A050361 Number of factorizations into distinct prime powers greater than 1.

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