A343224 -id:A343224 - cp's OEIS frontend

A348937 a(n) = A003961(n) - A003415(n), where A003961 shifts the prime factorization of n one step towards larger primes, and A003415 gives the arithmetic derivative of n.

1, 2, 4, 5, 6, 10, 10, 15, 19, 14, 12, 29, 16, 24, 27, 49, 18, 54, 22, 39, 45, 26, 28, 91, 39, 36, 98, 67, 30, 74, 36, 163, 51, 38, 65, 165, 40, 48, 69, 121, 42, 124, 46, 69, 136, 62, 52, 293, 107, 102, 75, 97, 58, 294, 75, 205, 93, 62, 60, 223, 66, 78, 224, 537, 101, 134, 70, 99, 119, 172, 72, 519, 78, 84, 190, 127

Offset: 1

Antti Karttunen, Nov 06 2021

nonn

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

Cf. A003415, A003961, A168036, A286385, A336853, A343224, A347964, A347965.

f1[p_, e_] := e/p; f2[p_, e_] := NextPrime[p]^e; a[n_] := Times @@ f2 @@@ (f = FactorInteger[n]) - n * Plus @@ f1 @@@ f; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A348937(n) = (A003961(n) - A003415(n));

a(n) = A003961(n) - A003415(n).
a(n) = A336853(n) - A168036(n).
a(n) = A286385(n) + A343224(n).

A376414 Dirichlet inverse of sigma(n)-A003415(n), where sigma is the sum of divisors function and A003415 is the arithmetic derivative.

Original entry on oeis.org

1, -2, -3, 1, -5, 5, -7, 1, 2, 9, -11, -2, -13, 13, 14, 2, -17, -2, -19, -4, 20, 21, -23, -4, 4, 25, 2, -6, -29, -21, -31, 5, 32, 33, 34, 3, -37, 37, 38, -6, -41, -31, -43, -10, -8, 45, -47, -9, 6, -4, 50, -12, -53, -6, 54, -8, 56, 57, -59, 8, -61, 61, -12, 13, 64, -51, -67, -16, 68, -57, -71, 6, -73, 73, -8, -18

Offset: 1

Antti Karttunen, Nov 15 2024

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Dirichlet inverse of A343224.

Cf. A000203, A003415.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A343224(n) = (sigma(n) - A003415(n));
memoA376414 = Map();
A376414(n) = if(1==n,1,my(v); if(mapisdefined(memoA376414,n,&v), v, v = -sumdiv(n,d,if(dA343224(n/d)*A376414(d),0)); mapput(memoA376414,n,v); (v)));

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA343224(n/d) * a(d).

cp's OEIS Frontend

A348937 a(n) = A003961(n) - A003415(n), where A003961 shifts the prime factorization of n one step towards larger primes, and A003415 gives the arithmetic derivative of n.

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