A348945 - OEIS

A348945

a(n) = A348944(n) - sigma(n), where A348944 is the arithmetic mean of A003959 and A034448, and sigma is the sum of divisors function.

0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 6, 0, 0, 0, 0, 75, 0, 0, 0, 6, 0, 0, 0, 18, 0, 0, 0, 0, 0, 0, 0, 72, 0, 0, 0, 0, 0, 18, 0, 24, 0, 0, 0, 0, 0, 0, 0, 270, 0, 0, 0, 0, 0, 0, 0, 66, 0, 0, 0, 0, 0, 0, 0, 108, 48, 0, 0, 0, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 300, 0, 0, 0, 10

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OFFSET

1,8

LINKS

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

Index entries for sequences related to sigma(n)

FORMULA

a(n) = A348944(n) - A000203(n) = ((1/2) * (A003959(n)+A034448(n))) - A000203(n).

a(n) = (1/2) * (A348029(n)-A048146(n)).

MATHEMATICA

f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (p + 1)^e; f3[p_, e_] := p^e + 1; a[1] = 0; a[n_] := (Times @@ f2 @@@ (f = FactorInteger[n]) + Times @@ f3 @@@ f) / 2 - Times @@ f1 @@@ f; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)

PROG

(PARI)

A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };

A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };

A348944(n) = ((1/2)*(A003959(n)+A034448(n)));

A348945(n) = (A348944(n)-sigma(n));

CROSSREFS

Cf. A000203, A003959, A034448, A048146, A348029, A348944, A348946.

Sequence in context: A324326 A377131 A341743 * A151755 A325739 A181004

Adjacent sequences: A348942 A348943 A348944 * A348946 A348947 A348948

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 05 2021

STATUS

approved