A008474 -id:A008474 - cp's OEIS frontend

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

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

A303277 If n = Product (p_j^k_j) then a(n) = (Sum (k_j))^(Sum (p_j)).

Original entry on oeis.org

1, 1, 1, 4, 1, 32, 1, 9, 8, 128, 1, 243, 1, 512, 256, 16, 1, 243, 1, 2187, 1024, 8192, 1, 1024, 32, 32768, 27, 19683, 1, 59049, 1, 25, 16384, 524288, 4096, 1024, 1, 2097152, 65536, 16384, 1, 531441, 1, 1594323, 6561, 33554432, 1, 3125, 128, 2187, 1048576, 14348907, 1, 1024, 65536

Offset: 1

Ilya Gutkovskiy, Apr 20 2018

nonn

			a(48) = a(2^4 * 3^1) = (4 + 1)^(2 + 3) = 5^5 = 3125.

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

Cf. A000142, A000312, A001222, A002110, A007504, A008472, A008474, A008477, A022559, A034387, A039697, A088865, A263653, A285769.

Join[{1}, Table[PrimeOmega[n]^DivisorSum[n, # &, PrimeQ[#] &], {n, 2, 55}]]

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

a(n) = bigomega(n)^sopf(n) = A001222(n)^A008472(n).
a(p^k) = k^p where p is a prime.
a(A000312(k)) = a(k)*k^A008472(k).
a(A000142(k)) = A022559(k)^A034387(k).
a(A002110(k)) = k^A007504(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]

A107737 Numbers n such that, in prime decomposition of n, sum of all prime factors and their orders is prime.

Original entry on oeis.org

2, 6, 8, 9, 14, 25, 26, 30, 32, 38, 40, 45, 56, 63, 66, 70, 74, 75, 81, 86, 88, 96, 99, 100, 104, 117, 121, 130, 134, 136, 138, 144, 147, 153, 154, 158, 160, 168, 174, 184, 190, 194, 196, 206, 207, 216, 218, 238, 248, 250, 252, 254, 266, 275, 279, 280, 286, 289

Offset: 1

Zak Seidov, May 23 2005

nonn

Corresponding primes in A107738. Cf. A008474 If n = Product (p_j^k_j) then a(n) = Sum (p_j + k_j).

			n = 104 OK because 104 = 2^3 * 13^1 => (2+3)+(13+1) = 19 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Prime Factorization

Cf. A008474, A107738.

ta=Table[Plus @@ Flatten[FactorInteger[n]], {n, 300}];bb={};Do[If[PrimeQ[t=ta[[i]]], bb=Append[bb, {i, t}]], {i, 300}];tr=Transpose[bb];A107738=tr[[2]];A107737=tr[[1]]
Select[Range[2,300],PrimeQ[Total[Flatten[FactorInteger[#]]]]&] (* Harvey P. Dale, Feb 05 2017 *)

A107738 Primes as a sum of prime factors and their orders in prime decomposition of some n.

Original entry on oeis.org

3, 7, 5, 5, 11, 7, 17, 13, 7, 23, 11, 11, 13, 13, 19, 17, 41, 11, 7, 47, 17, 11, 17, 11, 19, 19, 13, 23, 71, 23, 31, 11, 13, 23, 23, 83, 13, 17, 37, 29, 29, 101, 13, 107, 29, 11, 113, 29, 37, 11, 17, 131, 31, 19, 37, 19, 29, 19, 43

Offset: 1

Zak Seidov, May 23 2005

nonn

Robert Price, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization

Primes occurring in A008474.

Cf. A107737.

ta=Table[Plus @@ Flatten[FactorInteger[n]], {n, 300}];bb={};Do[If[PrimeQ[t=ta[[i]]], bb=Append[bb, {i, t}]], {i, 300}];tr=Transpose[bb];A107737=tr[[1]];A107738=tr[[2]]

A112376 Sum of base and exponent of prime powers.

Original entry on oeis.org

3, 4, 4, 6, 8, 5, 5, 12, 14, 6, 18, 20, 24, 7, 6, 30, 32, 7, 38, 42, 44, 48, 9, 54, 60, 62, 8, 68, 72, 74, 80, 7, 84, 90, 98, 102, 104, 108, 110, 114, 13, 8, 128, 9, 132, 138, 140, 150, 152, 158, 164, 168, 15, 174, 180, 182, 192, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242

Offset: 1

Zak Seidov, Dec 04 2005

nonn

If n = p^q, where p is prime and q > 0, then p+q is in the sequence.
If n is not of that form, omit the term.
Might be a good "puzzle" sequence - guess the rule given the first ten or so terms.

			n = 3 = 3^1, so 3+1 = 4 is a term; n = 4 = 2^2, so 2+2 = 4 is again a term; n = 5 = 5^1, so we get 5+1 =6.
But 6 is not a prime power, so we skip it.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A112375, A064438.

A008474 is another version, defined for all n.

fsum[a_] := Total[Flatten[FactorInteger[a]]]; fsum/@Select[Range[242], PrimePowerQ](* James C. McMahon, Jun 08 2024 *)

for(n=1,300,fac=factor(n);if(matsize(fac)[1]==1,print1(fac[1,1]+fac[1,2],",")))

Edited and extended by Klaus Brockhaus, Jan 21 2006

Further edited by N. J. A. Sloane, Nov 19 2018

A247095 Smallest number m such that A250030(m) = n.

Original entry on oeis.org

8, 7, 6, 5, 13, 30, 29, 157, 317, 626, 3095, 6637, 26833, 145687, 461938, 1068037, 16007153, 54690697

Offset: 1

Reinhard Zumkeller, Nov 18 2014

Cf. A008474, A250030.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a247095 = (+ 5) . fromJust . (`elemIndex` a250030_list)

A008474(n)=my(f=factor(n)); sum(i=1, #f~, f[i, 1]+f[i, 2]);
f(n)=my(k); while(n!=5, n=A008474(n); k++); k; \\ A250030
a(n) = my(k=5); while(f(k) != n, k++); k; \\ Michel Marcus, Feb 05 2022

A250030(a(n)) = n and A250030(m) != n for m < a(n).

a(17)-a(18) from Michel Marcus, Feb 05 2022

A135369 a(0)=1, a(1)=2; thereafter, let n! = p(1)^b(1)...p(r)^b(r) be the prime factorization of n!. Then a(n) = Sum_{i=1..r} (p(i) + b(i)).

Original entry on oeis.org

1, 2, 3, 7, 9, 15, 17, 25, 28, 30, 32, 44, 47, 61, 63, 65, 69, 87, 90, 110, 113, 115, 117, 141, 145, 147, 149, 152, 155, 185, 188, 220, 225, 227, 229, 231, 235, 273, 275, 277, 281, 323, 326, 370, 373, 376, 378, 426, 431, 433, 436, 438, 441, 495, 499, 501, 505

Offset: 0

Ctibor O. Zizka, Feb 17 2008

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A000142, A008474.

A001222 := proc(n) numtheory[bigomega](n) ; end:
A008472 := proc(n) add(op(1,i),i=ifactors(n)[2]) ; end:
A008474 := proc(n) A001222(n)+A008472(n) ; end:
A135369 := proc(n) if n < 2 then n+1 ; else A008474(n!) ; fi ; end: seq(A135369(n),n=0..80) ; # R. J. Mathar, Feb 28 2008

Join[{1},Table[Total[Flatten[FactorInteger[n!]]],{n,60}]] (* Harvey P. Dale, Feb 26 2012 *)

a(n) = A008474(n!) if n>1. - R. J. Mathar, Feb 28 2008

More terms from R. J. Mathar, Feb 28 2008

A382330 a(n) is the number of positive integers k for which Sum_{i=1..j} (p_i+e_i) = n, where p_1^e_1...p_j^e_j is the prime factorization of k.

Original entry on oeis.org

0, 0, 1, 2, 2, 3, 4, 6, 8, 11, 15, 21, 27, 36, 47, 61, 79, 104, 133, 170, 215, 272, 343, 433, 542, 678, 845, 1050, 1300, 1608, 1981, 2437, 2988, 3655, 4460, 5433, 6603, 8014, 9705, 11731, 14155, 17055, 20509, 24624, 29512, 35313, 42184, 50315, 59916, 71248, 84598

Offset: 1

Felix Huber, Mar 23 2025

nonn

a(n) is the number of positive integers k for A008474(k) = n.

			The a(7) = 4 positive integers k are 32 = 2^5, 81 = 3^4, 25 = 5^2, 6 = 2^1*3^1 because 2 + 5 = 3 + 4 = 5 + 2 = 2 + 1 + 3 + 1 = 7 and there is no further positive integer with that property.
The a(11) = 15 positive integers k are 512 = 2^9, 6561 = 3^8, 15625 = 5^6, 2401 = 7^4, 96 = 2^5*3^1, 144 = 2^4*3^2, 216 = 2^3*3^3, 324 = 2^2*3^4, 486 = 2^1*3^5, 40 = 2^3*5^1, 100 = 2^2*5^2, 250 = 2^1*5^3, 14 = 2^1*7^1, 45 = 3^2*5^1, 75 = 3^1*5^2 because 2 + 9 = 3 + 8 = 5 + 6 = 7 + 4 = 2 + 5 + 3 + 1 = 2 + 4 + 3 + 2 = 2 + 3 + 3 + 3 = 2 + 2 + 3 + 4 = 2 + 1 + 3 + 5 = 2 + 3 + 5 + 1 = 2 + 2 + 5 + 2 = 2 + 1 + 5 + 3 = 2 + 1 + 7 + 1 = 3 + 2 + 5 + 1 = 3 + 1 + 5 + 2 = 11 and there is no further positive integer with that property.

Felix Huber, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Fundamental Theorem of Arithmetic

Cf. A008474, A219180, A377505, A377537.

# processes b and T from Alois P. Heinz (A219180).
b:= proc(n,i) option remember;
      `if`(n=0,[1],`if`(i<1,[],zip((x,y)->x+y,b(n,i-1),
       [0,`if`(ithprime(i)>n,[],b(n-ithprime(i),i-1))[]],0)))
    end:
T:= proc(n) local l;l:=b(n,NumberTheory:-pi(n));
       while nops(l)>0 and l[-1]=0 do l:=subsop(-1=NULL,l) od; l[]
    end:
A382330:=proc(n)
    local a,k,s,i,j,L;
    a:=0;k:=1;s:=0;
    while s+k<=n do
        s:=s+ithprime(k);k:=k+1
    od;
    for i to k-1 do
        for j to n-i do
            L:=[T(j)];
            if nops(L)>=i+1 then
                a:=a+L[i+1]*binomial(n-j-1,n-j-i);
            fi
        od
    od;
    return a
end proc;
seq(A382330(n),n=1..51);

A385644 Swap multiplication and exponentiation in the canonical prime factorization of n.

Original entry on oeis.org

2, 3, 4, 5, 8, 7, 6, 6, 32, 11, 64, 13, 128, 243, 8, 17, 64, 19, 1024, 2187, 2048, 23, 216, 10, 8192, 9, 16384, 29, 14134776518227074636666380005943348126619871175004951664972849610340958208, 31, 10, 177147, 131072, 78125, 4096, 37, 524288, 1594323, 7776, 41

Offset: 2

Jens Ahlström, Jul 06 2025

nonn

In the canonical prime factorization of n larger than one, swap multiplication and exponentiation and calculate the result.

			a(6) = a(2 * 3) = 2^3 = 8,
a(24) = a(2^3 * 3) = (2 * 3)^3 = 216,
a(30) = a(2 * 3 * 5) = 2^3^5 = 2^243.

Jens Ahlström, Table of n, a(n) for n = 2..65

Cf. A001414, A000026, A005361, A008474.

f[{p_,e_}]:=p*e;a[n_]:=Module[{pp=f/@FactorInteger[n]},r=pp[[-1]];Do[r=pp[[Length[pp]-i]]^r,{i,1,Length[pp]-1}];r];Array[a,40,2] (* James C. McMahon, Jul 11 2025 *)
A385644[n_] := Power @@ Times @@@ FactorInteger[n];
Array[A385644, 40, 2] (* Paolo Xausa, Jul 14 2025 *)

from sympy import factorint
from functools import reduce
def rpow(a, b):
    return b**a
def a(n):
    return reduce(rpow, [p*e for p, e in reversed(factorint(n).items())])
print([a(n) for n in range(2, 42)])

cp's OEIS Frontend

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

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A303277 If n = Product (p_j^k_j) then a(n) = (Sum (k_j))^(Sum (p_j)).

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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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Mathematica

A107737 Numbers n such that, in prime decomposition of n, sum of all prime factors and their orders is prime.

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Mathematica

A107738 Primes as a sum of prime factors and their orders in prime decomposition of some n.

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Programs

Mathematica

A112376 Sum of base and exponent of prime powers.

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A247095 Smallest number m such that A250030(m) = n.

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Extensions

A135369 a(0)=1, a(1)=2; thereafter, let n! = p(1)^b(1)...p(r)^b(r) be the prime factorization of n!. Then a(n) = Sum_{i=1..r} (p(i) + b(i)).

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A382330 a(n) is the number of positive integers k for which Sum_{i=1..j} (p_i+e_i) = n, where p_1^e_1...p_j^e_j is the prime factorization of k.