A000026 -id:A000026 - cp's OEIS frontend

A304412 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*(k_j + 1)).

1, 6, 8, 9, 12, 48, 16, 12, 12, 72, 24, 72, 28, 96, 96, 15, 36, 72, 40, 108, 128, 144, 48, 96, 18, 168, 16, 144, 60, 576, 64, 18, 192, 216, 192, 108, 76, 240, 224, 144, 84, 768, 88, 216, 144, 288, 96, 120, 24, 108, 288, 252, 108, 96, 288, 192, 320, 360, 120, 864, 124, 384, 192, 21, 336, 1152, 136, 324

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(36) = a(2^2*3^2) = (2 + 1)*(2 + 1) * (3 + 1)*(2 + 1) = 108.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A000005, A000026, A000040, A000203, A000302 (numbers n such that a(n) is odd), A001221, A001615, A003959, A005117, A007947, A008864, A045967, A048250, A064549, A064840, A304407, A304408, A304409, A304411.

Cf. A065463.

a[n_] := Times @@ ((#[[1]] + 1) (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 68}]
Table[DivisorSigma[0, n] Total[Select[Divisors[n], SquareFreeQ]], {n, 68}]

a(n)={numdiv(n)*sumdiv(n, d, moebius(d)^2*d)} \\ Andrew Howroyd, Jul 24 2018

a(n) = A000005(n)*A048250(n) = A000005(n)*A000203(A007947(n)).
a(p^k) = (p + 1)*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*Product_{p|n} (p + 1) if n is a squarefree (A005117), where omega() = A001221.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^(s-1) - 1/p^(2*s-1)). - Amiram Eldar, Sep 17 2023
From Vaclav Kotesovec, May 06 2025: (Start)
Let f(s) = Product_{p prime} (p^s-p)^2 * (p^(2*s)+2*p^(s+1)-p) / (p^(2*s) * (p^s-1)^2).
Dirichlet g.f.: zeta(s-1)^2 * f(s).
Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
f(2) = A065463 = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442200999165592736603350326637210188586431417098049414226842591...,
f'(2) = f(2) * Sum_{p prime} 2*(2*p^2-1)*log(p) / ((p^2-1)*(p^2+p-1)) = f(2) * 1.799151495460164053607059266860868724519705035904425832307664926571...
and gamma is the Euler-Mascheroni constant A001620. (End)

A182938 If n = Product (p_j^e_j) then a(n) = Product (binomial(p_j, e_j)).

Original entry on oeis.org

1, 2, 3, 1, 5, 6, 7, 0, 3, 10, 11, 3, 13, 14, 15, 0, 17, 6, 19, 5, 21, 22, 23, 0, 10, 26, 1, 7, 29, 30, 31, 0, 33, 34, 35, 3, 37, 38, 39, 0, 41, 42, 43, 11, 15, 46, 47, 0, 21, 20, 51, 13, 53, 2, 55, 0, 57, 58, 59, 15, 61, 62, 21, 0, 65, 66

Offset: 1

Peter Luschny, Jan 16 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / n^2 for n = 1..1000000

Cf. A000026, A001414, A008473, A008474, A008475, A008476, A008477, A028310, A069799.

Cf. A027748, A124010, A007318, A328745.

a182938 n = product $ zipWith a007318'
   (a027748_row n) (map toInteger $ a124010_row n)
-- Reinhard Zumkeller, Feb 18 2012

A182938 := proc(n) local e,j; e := ifactors(n)[2]:
mul (binomial(e[j][1], e[j][2]), j=1..nops(e)) end:
seq (A182938(n), n=1..100);

a[n_] := Times @@ (Map[Binomial @@ # &, FactorInteger[n], 1]);
Table[a[n], {n, 1, 100}] (* Kellen Myers, Jan 16 2011 *)

a(n)=prod(i=1,#n=factor(n)~,binomial(n[1,i],n[2,i])) \\ M. F. Hasler

for(n=1, 100, print1(direuler(p=2, n, (1 + X)^p)[n], ", ")) \\ Vaclav Kotesovec, Mar 28 2025

a(A185359(n)) = 0. - Reinhard Zumkeller, Feb 18 2012
Dirichlet g.f.: Product_{p prime} (1 + p^(-s))^p. - Ilya Gutkovskiy, Oct 26 2019
Conjecture: Sum_{k=1..n} a(k) ~ c * n^2, where c = 0.33754... - Vaclav Kotesovec, Mar 28 2025

Given terms checked with new PARI code by M. F. Hasler, Jan 16 2011

A304407 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*k_j).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 3, 4, 4, 10, 4, 12, 6, 8, 4, 16, 4, 18, 8, 12, 10, 22, 6, 8, 12, 6, 12, 28, 8, 30, 5, 20, 16, 24, 8, 36, 18, 24, 12, 40, 12, 42, 20, 16, 22, 46, 8, 12, 8, 32, 24, 52, 6, 40, 18, 36, 28, 58, 16, 60, 30, 24, 6, 48, 20, 66, 32, 44, 24, 70, 12, 72, 36, 16

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(60) = a(2^2*3*5) = (2 - 1)*2 * (3 - 1)*1 * (5 - 1)*1 = 16.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A000010, A000026, A000040, A002110, A003958, A005117, A005361, A005867, A006093, A007947, A023900, A048250, A059975, A068997, A081294 (numbers n such that a(n) is odd), A087656, A090624, A152649, A173557, A304117, A304408, A304409, A304411, A304412.

Cf. A001221, A076479.

seq(mul((p-1)*padic[ordp](n, p), p in numtheory[factorset](n)), n=1..100); # Ridouane Oudra, Jun 06 2025

a[n_] := Times @@ ((#[[1]] - 1) #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 75}]
Table[EulerPhi[Last[Select[Divisors[n], SquareFreeQ]]] DivisorSigma[0, n/Last[Select[Divisors[n], SquareFreeQ]]], {n, 75}]

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p-1)*e)} \\ Andrew Howroyd, Jul 24 2018

a(n) = A005361(n)*abs(A023900(n)) = A005361(n)*A173557(n) = A005361(n)*A000010(A007947(n)).
a(p^k) = (p - 1)*k where p is a prime and k > 0.
a(n) = phi(n) if n is a squarefree (A005117), where phi() = A000010.
a(A002110(k)) = A005867(k).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} (1 - 4/p^2 + 3/p^3 + 1/p^4 - 1/p^5) = 0.2644703894... . - Amiram Eldar, Nov 30 2022
a(n) = (-1)^A001221(n) * (Sum_{d1|n} Sum_{d2|n} mu(d1)*gcd(d1,d2)). - Ridouane Oudra, Jun 06 2025

A304117 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j)*k_j), where pi() = A000720.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, May 06 2018

			a(36) = 8 because 36 = 2^2*3^2 = prime(1)^2*prime(2)^2 and 1*2*2*2 = 8.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Extended graphical example
Index entries for sequences computed from indices in prime factorization
Index entries for sequences computed from exponents in factorization of n

Cf. A000026, A000040, A000720, A001414, A002110, A003963, A005361, A056239, A156061, A304037.

a[n_] := Times @@ (PrimePi[#[[1]]] #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 1, 80}]

a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = primepi(f[k,1])*f[k,2]; f[k, 2] = 1); factorback(f); \\ Michel Marcus, May 06 2018

a(n) = A005361(n)*A156061(n).
a(p^k) = A000720(p)*k where p is a prime.
a(A002110(m)^k) = k^m*m!.
As an example:
a(A000040(k)) = k.
a(A006450(k)) = A000040(k).
a(A001248(k)) = a(A031215(k)) = A005843(k).
a(A030078(k)) = a(A031336(k)) = A008585(k)
a(A061742(k)) = A000165(k).
a(A115964(k)) = A032031(k).
a(A002110(k)) = A000142(k).
a(A080696(k)) = A002110(k).

A381178 Irregular triangle read by rows, where row n lists the elements of the multiset of bases and exponents (including exponents = 1) in the prime factorization of n.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 1, 5, 1, 1, 2, 3, 1, 7, 2, 3, 2, 3, 1, 1, 2, 5, 1, 11, 1, 2, 2, 3, 1, 13, 1, 1, 2, 7, 1, 1, 3, 5, 2, 4, 1, 17, 1, 2, 2, 3, 1, 19, 1, 2, 2, 5, 1, 1, 3, 7, 1, 1, 2, 11, 1, 23, 1, 2, 3, 3, 2, 5, 1, 1, 2, 13, 3, 3, 1, 2, 2, 7, 1, 29, 1, 1, 1, 2, 3, 5, 1, 31

Offset: 2

Paolo Xausa, Feb 27 2025

Terms in each row are sorted; cf. A035306, where they are given in (base, exponent) groups.

			Triangle begins:
   [2]  1, 2;
   [3]  1, 3;
   [4]  2, 2;
   [5]  1, 5;
   [6]  1, 1, 2, 3;
   [7]  1, 7;
   [8]  2, 3;
   [9]  2, 3;
  [10]  1, 1, 2, 5;
  ...
The prime factorization of 10 is 2^1*5^1 and the multiset of these bases and exponents is {1, 1, 2, 5}.
The prime factorization of 132 is 2^2*3^1*11^1 and the multiset of these bases and exponents is {1, 1, 2, 2, 3, 11}.

Paolo Xausa, Table of n, a(n) for n = 2..11293 (rows 2..2500 of triangle, flattened).

Cf. A000026 (row products), A001221 (row lengths, divided by 2), A008474 (row sums).
Cf. A081812 (right border), A381212 (first column), A381576 (second column).
Cf. A035306, A381203, A381204, A381398, A381401, A381403, A381404.

A381178row[n_] := Sort[Flatten[FactorInteger[n]]];
Array[A381178row, 30, 2]

A193551 Smallest number with n as multiplicative projection.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 16, 27, 10, 11, 12, 13, 14, 15, 256, 17, 24, 19, 20, 21, 22, 23, 36, 3125, 26, 19683, 28, 29, 30, 31, 65536, 33, 34, 35, 72, 37, 38, 39, 80, 41, 42, 43, 44, 135, 46, 47, 144, 823543, 160, 51, 52, 53, 216, 55, 112, 57, 58, 59, 60, 61, 62

Offset: 1

Reinhard Zumkeller, Aug 27 2011

nonn

A000026(a(n)) = n and A000026(m) <> n for m < a(n);
a(p^k) = p^(p^(k-1)), p prime, k > 0; the sequence is not multiplicative, but for coprime odd numbers u, v: a(u*v) = a(u) * a(v);
A078779 gives fixed points: a(A078779(n)) = A078779(n).

import Data.List (elemIndex, findIndices)
import Data.Maybe (fromJust)
a193551 n = (fromJust $ elemIndex n a000026_list) + 1

A225395 Replace each prime number with its rank in the recursive prime factorization of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 2, 6, 4, 6, 1, 7, 2, 8, 3, 8, 5, 9, 2, 3, 6, 4, 4, 10, 6, 11, 1, 10, 7, 12, 2, 12, 8, 12, 3, 13, 8, 14, 5, 6, 9, 15, 2, 4, 3, 14, 6, 16, 4, 15, 4, 16, 10, 17, 6, 18, 11, 8, 1, 18, 10, 19, 7, 18, 12, 20, 2, 21, 12, 6, 8, 20, 12, 22, 3, 2, 13, 23, 8, 21, 14, 20, 5, 24, 6, 24, 9, 22, 15, 24, 2, 25, 4, 10, 3, 26, 14, 27, 6, 24, 16, 28

Offset: 1

Paul Tek, May 06 2013

a(A000040(n)) = n, hence all natural numbers appear in this sequence.
a(2n) = n.
It appears that a(35) = 12 is the only instance where a composite index yields a larger value than any smaller index. Checked to 10^7. - Charles R Greathouse IV, Jul 30 2016

			The number 9967 is the 1228th prime number.
Hence a(9967) = 1228.
The recursive prime factorization of 31250 is 2*5^(2*3).
The numbers 2, 3 and 5 are respectively the 1st, 2nd and 3rd prime numbers.
Hence a(31250) = a(2*5^(2*3)) = 1*3^(1*2) = 9.

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, Perl program for this sequence

Cf. A000040, A000026, A156061.

a225395 n = product $ zipWith (^)
    (map a049084 $ a027748_row n) (map a225395 $ a124010_row n)
-- Reinhard Zumkeller, May 10 2013

a[1] = 1; a[p_?PrimeQ] := a[p] = PrimePi[p]; a[n_] := a[n] = Times @@ (PrimePi[#[[1]]]^a[#[[2]]]& /@ FactorInteger[n]); Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 07 2013 *)

a(n)=if(n<3, return(1)); my(f=factor(n)); prod(i=1,#f~, primepi(f[i,1])^a(f[i,2])) \\ Charles R Greathouse IV, Jul 30 2016

Perl
```
# See Tek link.
```

Multiplicative, with a(prime(i)^j) = i^a(j).

a(n) = prod(A049084(A027748(k))^a(A124010(k)): k=1..A001221(n)). - Reinhard Zumkeller, May 10 2013

A303278 If n = Product_j p_j^k_j where the p_j are distinct primes then a(n) = (Product_j k_j)^(Product_j p_j).

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 9, 8, 1, 1, 64, 1, 1, 1, 16, 1, 64, 1, 1024, 1, 1, 1, 729, 32, 1, 27, 16384, 1, 1, 1, 25, 1, 1, 1, 4096, 1, 1, 1, 59049, 1, 1, 1, 4194304, 32768, 1, 1, 4096, 128, 1024, 1, 67108864, 1, 729, 1, 4782969, 1, 1, 1, 1073741824, 1, 1, 2097152, 36, 1, 1, 1, 17179869184, 1, 1, 1, 46656, 1, 1, 32768

Offset: 1

Ilya Gutkovskiy, Apr 20 2018

nonn

This is different from A008477, which is Product_j k_j^p_j. - N. J. A. Sloane, May 01 2021

			a(36) = a(2^2 * 3^2) = (2*2)^(2*3) = 4^6 = 4096.

Antti Karttunen, Table of n, a(n) for n = 1..4096

Cf. A000026, A000142, A005117 (indices of ones), A005361, A007947, A008477, A034386, A039696, A135291, A285769, A303277.

Table[Times@@Transpose[FactorInteger[n]][[2]]^Last[Select[Divisors[n], SquareFreeQ]], {n, 75}]

a(n) = my(f=factor(n)); factorback(f[, 2])^factorback(f[, 1]); \\ Michel Marcus, Apr 21 2018

a(n) = tau(n/rad(n))^rad(n) = A005361(n)^A007947(n).
a(p^k) = k^p where p is a prime.
a(A000142(k)) = A135291(k)^A034386(k).

Definition clarified by N. J. A. Sloane, May 01 2021

A304410 Numbers k such that k = Product (p_j^e_j) = Product (p_j*(e_j + 1)).

Original entry on oeis.org

1, 8, 9, 72, 13440, 21120, 24960, 29568, 32640, 34944, 36480, 44160, 45696, 49280, 51072, 54912, 55680, 58240, 59520, 61824, 71040, 71808, 76160, 77952, 78720, 80256, 82560, 83328, 84864, 85120, 90240, 91520, 94848, 97152, 99456, 101760, 103040, 110208, 113280, 114816, 115584, 117120, 119680

Offset: 1

Ilya Gutkovskiy, May 12 2018

nonn

Numbers k such that A000005(k)*A007947(k) = k.
Fixed points of A304409.
All terms are refactorable numbers (A033950).

			13440 is a term because 13440 = 2^7*3*5*7 = 2*(7 + 1) * 3*(1 + 1) * 5*(1 + 1) * 7*(1 + 1).

Cf. A000005, A000026, A007947, A008478, A033950, A036762, A048768, A078779, A190473, A304409.

a[n_] := Times @@ (#[[1]] (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Select[Range[120000], a[#] == # &]

isok(k) = {my(f = factor(k)); numdiv(f) * vecprod(f[, 1]) == k;} \\ Amiram Eldar, Jan 31 2025

A039786 phi(a(n)) is equal to the multiplicative projection of (a(n)-1).

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 23, 29, 31, 43, 47, 53, 59, 61, 67, 71, 79, 83, 103, 107, 131, 139, 149, 157, 167, 173, 179, 191, 211, 223, 227, 229, 239, 263, 269, 277, 283, 293, 311, 317, 331, 347, 349, 359, 367, 373, 383, 389, 419, 421, 431, 439, 443, 461, 463, 467, 479

Offset: 1

Olivier Gérard

nonn

Only primes (except 9) may qualify.

			phi(29)=28, 28=2^2*7^1, 2*2*7*1=28.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000010, A000026.

is(n)=my(f=factor(n)); eulerphi(f)==prod(i=1,#f~,f[i,1]*f[i,2])-1 \\ Charles R Greathouse IV, Mar 11 2014

a(1) inserted by Charles R Greathouse IV, Mar 11 2014

cp's OEIS Frontend

A304412 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*(k_j + 1)).

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A182938 If n = Product (p_j^e_j) then a(n) = Product (binomial(p_j, e_j)).

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A304407 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*k_j).

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A304117 If n = Product (p_j^k_j) then a(n) = Product (pi(p_j)*k_j), where pi() = A000720.

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A381178 Irregular triangle read by rows, where row n lists the elements of the multiset of bases and exponents (including exponents = 1) in the prime factorization of n.

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A193551 Smallest number with n as multiplicative projection.

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A225395 Replace each prime number with its rank in the recursive prime factorization of n.

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