A280076 -id:A280076 - cp's OEIS frontend

A007955 Product of divisors of n.

1, 2, 3, 8, 5, 36, 7, 64, 27, 100, 11, 1728, 13, 196, 225, 1024, 17, 5832, 19, 8000, 441, 484, 23, 331776, 125, 676, 729, 21952, 29, 810000, 31, 32768, 1089, 1156, 1225, 10077696, 37, 1444, 1521, 2560000, 41, 3111696, 43, 85184, 91125, 2116, 47, 254803968, 343

Offset: 1

R. Muller

All terms of this sequence occur only once. See the second T. D. Noe link for a proof. - T. D. Noe, Jul 07 2008
Every natural number has a unique representation in terms of divisor products. See the W. Lang link. - Wolfdieter Lang, Feb 08 2011
a(n) = n only if n is prime or 1 (or, if n is in A008578). - Alonso del Arte, Apr 18 2011
Sometimes called the "divisorial" of n. - Daniel Forgues, Aug 03 2012
a(n) divides EulerPhi(x^n-y^n) (see A. Rotkiewicz link). - Michel Marcus, Dec 15 2012
The proof that all the terms of this sequence occur only once (mentioned above) was given by Niven in 1984. - Amiram Eldar, Aug 16 2020

			Divisors of 10 = [1, 2, 5, 10]. So, a(10) = 2*5*10 = 100. - _Indranil Ghosh_, Mar 22 2017

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 57.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 83.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Wolfdieter Lang, Divisor Product Representation for Natural Numbers.
M. Le, On Smarandache Divisor Products, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 144-145.
F. Luca, On the product of divisors of n and sigma(n), J. Ineq. Pure Appl. Math. 4 (2) 2003, Article 46.
T. D. Noe, The Divisor Product is Unique
A. Rotkiewicz, On the numbers Phi(a^n +/- b^n), Proc. Amer. Math. Soc. 12 (1961), 419-421.
Rodica Simon and Frank W. Schmid, Problem E 2946, The American Mathematical Monthly, Vol. 89, No. 5 (1982), p. 333, Ivan Niven, Product of all Positive Divisors of n, solution to problem E 2946, ibid., Vol. 91, No. 10 (1984), p. 650.
F. Smarandache, Only Problems, Not Solutions!.
Eric Weisstein's World of Mathematics, Divisor Product.
Zhu Weiyi, On the divisor product sequences, Smarandache Notions J., Vol. 14 (2004), pp. 144-146.
OEIS Wiki, Divisorial.

Cf. A000005, A000196, A001222, A007422, A007956, A027750, A030628, A046523, A069264, A072046, A111398, A162947, A224381, A243103, A280076, A283995.
Cf. A000203 (sums of divisors).
Cf. A000010 (comments on product formulas).

List(List([1..50],n->DivisorsInt(n)),Product); # Muniru A Asiru, Feb 17 2019

a007955 = product . a027750_row  -- Reinhard Zumkeller, Feb 06 2012

f := function(n); t1 := &*[d : d in Divisors(n) ]; return t1; end function;

A007955 := proc(n) mul(d,d=numtheory[divisors](n)) ; end proc: # R. J. Mathar, Mar 17 2011
seq(isqrt(n^numtheory[tau](n)), n=1..50); # Gary Detlefs, Feb 15 2019

Array [ Times @@ Divisors[ # ]&, 100 ]
a[n_] := n^(DivisorSigma[0, n]/2); Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Nov 21 2013 *)

a(n)=if(issquare(n,&n),n^numdiv(n^2),n^(numdiv(n)/2)) \\ Charles R Greathouse IV, Feb 11 2011

from sympy import prod, divisors
print([prod(divisors(n)) for n in range(1, 51)]) # Indranil Ghosh, Mar 22 2017

from math import isqrt
from sympy import divisor_count
def A007955(n):
    d = divisor_count(n)
    return isqrt(n)**d if d % 2 else n**(d//2) # Chai Wah Wu, Jan 05 2022

[prod(divisors(n)) for n in (1..100)] # Giuseppe Coppoletta, Dec 16 2014

[n^(sigma(n,0)/2) for n in (1..49)] # Stefano Spezia, Jul 14 2025

;; A naive stand-alone implementation:
(define (A007955 n) (let loop ((d n) (m 1)) (cond ((zero? d) m) ((zero? (modulo n d)) (loop (- d 1) (* m d))) (else (loop (- d 1) m)))))
;; Faster, if A000005 and A000196 are available:
(define (A007955 n) (A000196 (expt n (A000005 n))))
;; Antti Karttunen, Mar 22 2017

a(n) = n^(d(n)/2) = n^(A000005(n)/2). Since a(n) = Product_(d|n) d = Product_(d|n) n/d, we have a(n)*a(n) = Product_(d|n) d*(n/d) = Product_(d|n) n = n^(tau(n)), whence a(n) = n^(tau(n)/2).
a(p^k) = p^A000217(k). - Enrique Pérez Herrero, Jul 22 2011
a(n) = A078599(n) * A178649(n). - Reinhard Zumkeller, Feb 06 2012
a(n) = A240694(n, A000005(n)). - Reinhard Zumkeller, Apr 10 2014
From Antti Karttunen, Mar 22 2017: (Start)
a(n) = A000196(n^A000005(n)). [From the original formula.]
A001222(a(n)) = A069264(n). [See Geoffrey Critzer's Feb 03 2015 comment in the latter sequence.]
A046523(a(n)) = A283995(n).
(End)
a(n) = Product_{k=1..n} gcd(n,k)^(1/phi(n/gcd(n,k))) = Product_{k=1..n} (n/gcd(n,k))^(1/phi(n/gcd(n,k))) where phi = A000010. - Richard L. Ollerton, Nov 07 2021
From Bernard Schott, Jan 11 2022: (Start)
a(n) = n^2 iff n is in A007422.
a(n) = n^3 iff n is in A162947.
a(n) = n^4 iff n is in A111398.
a(n) = n^5 iff n is in A030628.
a(n) = n^(3/2) iff n is in A280076. (End)
From Amiram Eldar, Oct 29 2022: (Start)
a(n) = n * A007956(n).
Sum_{k=1..n} 1/a(k) ~ log(log(n)) + c + O(1/log(n)), where c is a constant (Weiyi, 2004; Sandor and Crstici, 2004). (End)
a(n) = Product_{k=1..n} (n * (1 - ceiling(n/k - floor(n/k))))/k + ceiling(n/k - floor(n/k)). - Adriano Steffler, Feb 08 2024

A347438 Number of unordered factorizations of n with alternating product 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 06 2021

nonn

Also the number of unordered factorizations of n with alternating sum 0.
Also the number of unordered factorizations of n with all even multiplicities.
This is the even-length case of A347437, the odd-length case being A347441.
An unordered factorization of n is a weakly increasing sequence of positive integers > 1 with product n.
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

			The a(n) factorizations for n = 16, 64, 144, 256, 576:
  4*4      8*8          12*12        16*16            24*24
  2*2*2*2  2*2*4*4      2*2*6*6      2*2*8*8          3*3*8*8
           2*2*2*2*2*2  3*3*4*4      4*4*4*4          4*4*6*6
                        2*2*2*2*3*3  2*2*2*2*4*4      2*2*12*12
                                     2*2*2*2*2*2*2*2  2*2*2*2*6*6
                                                      2*2*3*3*4*4
                                                      2*2*2*2*2*2*3*3

Antti Karttunen, Table of n, a(n) for n = 1..65537
PlanetMath, alternating sum

Positions of zeros are A000037.
Positions of nonzero terms are A000290.
The restriction to perfect squares is A001055 (ordered: A273013).
The restriction to powers of 2 is A035363.
The additive version is A119620, ranked by A028982.
Positions of non-1's are A213367 \ {1}.
Positions of 1's are A280076 = {1} \/ A001248.
Sorted first positions are 1, 2, and all terms of A330972 squared.
Allowing any alternating product <= 1 gives A339846.
Allowing any alternating product > 1 gives A339890.
Allowing any integer alternating product gives A347437.
Allowing any integer reciprocal alternating product gives A347439.
Allowing any alternating product < 1 gives A347440.
Allowing any alternating product >= 1 gives A347456.
A046099 counts factorizations with no alternating permutations.
A071321 gives the alternating sum of prime factors (reverse: A071322).
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344606 counts alternating permutations of prime factors.
A347441 counts odd-length factorizations with integer alternating product.
A347460 counts possible alternating products of factorizations.
Cf. A000041, A005117, A025047, A038548, A062312, A088218, A316523, A332269, A344607, A347442, A347446, A347463.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[facs[n],altprod[#]==1&]],{n,100}]

A347438(n, m=n, k=0, t=1) = if(1==n, (1==t), my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A347438(n/d, d, 1-k, t*(d^((-1)^k))))); (s)); \\ Antti Karttunen, Oct 30 2021

a(2^n) = A035363(n).

a(n^2) = A001055(n).

Name and comments clarified (with unordered) by Jacob Sprittulla, Oct 05 2021

A360676 Sum of the left half (exclusive) of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 04 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 810 are {1,2,2,2,2,3}, with left half (exclusive) {1,2,2}, so a(810) = 5.
The prime indices of 3675 are {2,3,3,4,4}, with left half (exclusive) {2,3}, so a(3675) = 5.

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of 0's are 1 and A000040.
Positions of first appearances are 1 and A001248.
These partitions are counted by A360675, right A360672.
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
A360616 gives half of bigomega (exclusive), inclusive A360617.
A360673 counts multisets by right sum (exclusive), inclusive A360671.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A026424, A280076, A359912, A360006, A360457, A360674.

f:= proc(n) local F,i,t;
  F:= [seq(numtheory:-pi(t[1])$t[2], t = sort(ifactors(n)[2],(a,b) -> a[1] < b[1]))];
  add(F[i],i=1..floor(nops(F)/2))
end proc:
map(f, [$1..100]); # Robert Israel, Feb 02 2025

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Take[prix[n],Floor[Length[prix[n]]/2]]],{n,100}]

A360676(n) + A360679(n) = A001222(n).

A360677(n) + A360678(n) = A001222(n).

A360678 Sum of the left half (inclusive) of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 2, 2, 1, 5, 2, 6, 1, 2, 2, 7, 3, 8, 2, 2, 1, 9, 2, 3, 1, 4, 2, 10, 3, 11, 3, 2, 1, 3, 2, 12, 1, 2, 2, 13, 3, 14, 2, 4, 1, 15, 3, 4, 4, 2, 2, 16, 3, 3, 2, 2, 1, 17, 2, 18, 1, 4, 3, 3, 3, 19, 2, 2, 4, 20, 3, 21, 1, 5, 2, 4, 3, 22, 3, 4, 1

Offset: 1

Gus Wiseman, Mar 05 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 810 are {1,2,2,2,2,3}, with left half (inclusive) {1,2,2}, so a(810) = 5.
The prime indices of 3675 are {2,3,3,4,4}, with left half (inclusive) {2,3,3}, so a(3675) = 8.

Positions of first appearances are 1 and A001248.
Positions of 1's are A001747.
These partitions are counted by A360675 with rows reversed.
The exclusive version is A360676.
The right version is A360679.
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
A360616 gives half of bigomega (exclusive), inclusive A360617.
A360673 counts multisets by right sum (exclusive), inclusive A360671.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A026424, A280076, A359912, A360006, A360457.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Take[prix[n],Ceiling[Length[prix[n]]/2]]],{n,100}]

A360676(n) + A360679(n) = A001222(n).

A360677(n) + A360678(n) = A001222(n).

A360616 Half the number of prime factors of n (counted with multiplicity, A001222), rounded down.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 08 2023

nonn

			The prime indices of 378 are {1,2,2,2,4}, so a(378) = floor(5/2) = 2.

Positions of 0's are 1 and A000040.
Positions of first appearances are A000302 = 2^(2k) for k >= 0.
Positions of 1's are A168645.
Rounding up instead of down gives A360617.
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
A360673 counts multisets by right sum (exclusive), inclusive A360671.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A001248, A026424, A280076, A359912, A360006, A360457, A360674.

Mathematica
```
Table[Floor[PrimeOmega[n]/2],{n,100}]
```

A024525 a(n) = 1^2 + prime(1)^2 + prime(2)^2 + ... + prime(n)^2.

Original entry on oeis.org

1, 5, 14, 39, 88, 209, 378, 667, 1028, 1557, 2398, 3359, 4728, 6409, 8258, 10467, 13276, 16757, 20478, 24967, 30008, 35337, 41578, 48467, 56388, 65797, 75998, 86607, 98056, 109937, 122706, 138835, 155996, 174765, 194086, 216287, 239088, 263737, 290306, 318195, 348124

Offset: 0

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A024450.

Partial sums of A280076.

[1] cat [n le 1 select 5 else Self(n-1) + NthPrime(n)^2: n in [1..80]]; // G. C. Greubel, Jan 30 2025

Join[{1},Accumulate[Prime[Range[40]]^2]+1] (* Harvey P. Dale, Oct 24 2015 *)

def A024525(n):
    if n<2: return (1,5)[n]
    else: return A024525(n-1) + nth_prime(n)**2
print([A024525(n) for n in range(81)]) # G. C. Greubel, Jan 30 2025

a(n) = 1 + A024450(n), for n >= 1.

a(n) = a(n-1) + prime(n)^2, with a(0) = 1, a(1) = 5. - G. C. Greubel, Jan 30 2025

Corrected by Harvey P. Dale, Oct 24 2015

A349126 Sum of A064989 and its Dirichlet inverse, where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

Original entry on oeis.org

2, 0, 0, 1, 0, 4, 0, 1, 4, 6, 0, 2, 0, 10, 12, 1, 0, 4, 0, 3, 20, 14, 0, 2, 9, 22, 8, 5, 0, 0, 0, 1, 28, 26, 30, 4, 0, 34, 44, 3, 0, 0, 0, 7, 12, 38, 0, 2, 25, 9, 52, 11, 0, 8, 42, 5, 68, 46, 0, 6, 0, 58, 20, 1, 66, 0, 0, 13, 76, 0, 0, 4, 0, 62, 18, 17, 70, 0, 0, 3, 16, 74, 0, 10, 78, 82, 92, 7, 0, 12, 110, 19, 116

Offset: 1

Antti Karttunen, Nov 13 2021

Question: Are all terms nonnegative?

Answer: All terms certainly are >= 0. See Sebastian Karlsson's Nov 13 2021 multiplicative formula for A349125. - Antti Karttunen, Apr 20 2022

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A001248, A064989, A030059, A030229, A280076, A349125.
Cf. also A322581, A349135.
Coincides with A349349 on odd numbers.

f1[p_, e_] := If[p == 2, 1, NextPrime[p, -1]^e]; a1[1] = 1; a1[n_] := Times @@ f1 @@@ FactorInteger[n]; f2[p_, e_] := If[e == 1, If[p == 2, -1, -NextPrime[p, -1]], 0]; a2[1] = 1; a2[n_] := Times @@ f2 @@@ FactorInteger[n]; a[n_] := a1[n] + a2[n]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)

A349126(n) = (A064989(n)+A349125(n)); \\ Needs also code from A349125.

A349126(n) = if(1==n,2,-sumdiv(n, d, if(1==d||n==d,0,A064989(d)*A349125(n/d)))); \\ (This demonstrates the "cut convolution" formula) - Antti Karttunen, Nov 13 2021

a(n) = A064989(n) + A349125(n).
a(1) = 2, and for n > 1, a(n) = -Sum_{d|n, 1A064989(d) * A349125(n/d).
For all n >= 1, a(A030059(n)) = 0, a(A030229(n)) = 2*A064989(A030229(n)).
For all n >= 1, a(A001248(n)) = A280076(n).

A299788 a(n) = denominator of Product_{d|n} (sigma(d)/d) where sigma(k) = the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 2, 3, 8, 5, 1, 7, 64, 27, 25, 11, 3, 13, 49, 25, 1024, 17, 27, 19, 1000, 441, 121, 23, 16, 125, 169, 729, 7, 29, 625, 31, 32768, 121, 289, 1225, 5832, 37, 361, 1521, 6400, 41, 2401, 43, 1331, 3375, 529, 47, 3072, 343, 31250, 289, 17576, 53, 6561, 3025, 49

Offset: 1

Jaroslav Krizek, Jan 21 2019

			For n=4; Product_{d|4} (sigma(d)/d) = (sigma(1)/1)*(sigma(2)/2)*(sigma(4)/4) = (1/1)*(3/2)*(7/4) = 21/8; a(4) = 8.

Cf. A000203, A318491, A318492, A322673 (numerator).

[Denominator(&*[&+[c: c in Divisors(d)] / d: d in Divisors(n)]): n in [1..100]];

Array[Denominator@ Product[DivisorSigma[1, d]/d, {d, Divisors@ #}] &, 51]

a(n) = my(p=1); fordiv(n, d, p *= sigma(d)/d); denominator(p); \\ Michel Marcus, Jan 21 2019

a(n) = 1 for n = 1, 6, ... (no other n <= 5*10^6).
a(n) = n for n = primes (A000040).
a(n) = n * sqrt(n) for n in A280076 (union of 1 and squares of primes (A001248)).

A306345 Absolute difference between the number of prime divisors and the number of composite divisors of n.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 0, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 0, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 5, 2, 1, 1, 5, 1, 1, 1, 3

Offset: 1

Felix Fröhlich, Feb 08 2019

nonn

Conjecture: a(n) = 0 iff n is a term of A280076 = union of A001248 and {1}.

Conjecture is true, since having an n with k distinct prime factors such that a(n) = 0 requires that 2k+1 can be factored into k parts > 1, and 1 is the only positive k for which this is possible. - Charlie Neder, Feb 12 2019

			For n = 24: The set of divisors of 24 is {1, 2, 3, 4, 6, 8, 12, 24}. The prime divisors are {2, 3} and the composite divisors are {4, 6, 8, 12, 24}. The cardinalities of the sets are 2 and 5, respectively, and abs(2-5) = 3, so a(24) = 3.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A001248, A055212, A280076.

Array[Abs[2 PrimeNu@ # - DivisorSigma[0, #] + 1] &, 105] (* Michael De Vlieger, Feb 17 2019 *)

a(n) = my(d=divisors(n), p=0, c=0); for(k=2, #d, if(ispseudoprime(d[k]), p++, c++)); abs(p-c)

a(n) = abs(2*omega(n) - numdiv(n) + 1); \\ Michel Marcus, Feb 12 2019

a(n) = abs(A001221(n) - A055212(n)).

a(n) = abs(2*A001221(n) - A000005(n) + 1). - Michel Marcus, Feb 12 2019

a(1)=0 prepended by David A. Corneth, Feb 12 2019

A332646 Numbers m with a divisor d such that d^tau(d) = m.

Original entry on oeis.org

1, 4, 9, 25, 49, 64, 121, 169, 289, 361, 529, 729, 841, 961, 1296, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4096, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10000, 10201, 10609, 11449, 11881, 12769, 15625, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569

Offset: 1

Jaroslav Krizek, Feb 18 2020

nonn

Possible values for function n^tau(n) (A062758).

Supersequence of A189991 (numbers with prime factorization p^4*q^4; d = pq), A001248 (numbers with prime factorization p^2; d = p), A030516 (numbers with prime factorization p^6; d = p^2) and A280076.

			64 is a term because 4^3 = 64; 4 divides 64; tau(4) = 3.

Cf. A000005, A001248, A030516, A062758, A189991, A280076.

[n: n in [1..100000] | #[d: d in Divisors(n) | d^NumberOfDivisors(d) eq n] ge 1];

divPowerQ[n_] := AnyTrue[Divisors[n], #^DivisorSigma[0, #] == n &]; Select[Range[27000], divPowerQ] (* Amiram Eldar, Feb 18 2020 *)

isok(m) = fordiv(m, d, if (d^numdiv(d) == m, return (1))); \\ Michel Marcus, Feb 18 2020

cp's OEIS Frontend

A007955 Product of divisors of n.

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A347438 Number of unordered factorizations of n with alternating product 1.

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A360676 Sum of the left half (exclusive) of the prime indices of n.

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A360678 Sum of the left half (inclusive) of the prime indices of n.

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A360616 Half the number of prime factors of n (counted with multiplicity, A001222), rounded down.

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A024525 a(n) = 1^2 + prime(1)^2 + prime(2)^2 + ... + prime(n)^2.

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A349126 Sum of A064989 and its Dirichlet inverse, where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p.

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