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.
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
Keywords
Examples
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.
Links
- Felix Huber, Table of n, a(n) for n = 1..5000
- Eric Weisstein's World of Mathematics, Fundamental Theorem of Arithmetic
Programs
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Maple
# 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);
Comments