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A348973 Numerator of ratio A129283(n) / A003959(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative. 4
1, 1, 1, 8, 1, 11, 1, 20, 15, 17, 1, 7, 1, 23, 23, 16, 1, 13, 1, 22, 31, 35, 1, 17, 35, 41, 27, 5, 1, 61, 1, 112, 47, 53, 47, 2, 1, 59, 55, 2, 1, 83, 1, 23, 7, 71, 1, 40, 63, 95, 71, 6, 1, 45, 71, 37, 79, 89, 1, 19, 1, 95, 57, 256, 83, 127, 1, 70, 95, 43, 1, 19, 1, 113, 65, 13, 95, 149, 1, 128, 189, 125, 1, 13, 107 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

It is known that A129283(n) <= A003959(n) for all n (see A348970 for a proof), which implies that each ratio a(n)/A348974(n) is at most 1: 1/1, 1/1, 1/1, 8/9, 1/1, 11/12, 1/1, 20/27, 15/16, 17/18, 1/1, 7/9, 1/1, 23/24, 23/24, 16/27, 1/1, 13/16, 1/1, 22/27, 31/32, 35/36, 1/1, 17/27, 35/36, 41/42, 27/32, 5/6, 1/1, 61/72, 1/1, 112/243, etc.

LINKS

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

FORMULA

a(n) = A129283(n) / A348972(n) = A129283(n) / gcd(A003959(n), A129283(n)).

MATHEMATICA

f1[p_, e_] := e/p; f2[p_, e_] := (p + 1)^e; a[n_] := Numerator[n*(1 + Plus @@ f1 @@@ (f = FactorInteger[n]))/Times @@ f2 @@@ f]; Array[a, 100] (* Amiram Eldar, Nov 06 2021 *)

PROG

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

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

A348973(n) = { my(u=n+A003415(n)); (u/gcd(A003959(n), u)); };

CROSSREFS

Cf. A003415, A003959, A129283, A348970, A348972, A348974 (denominators).

Cf. also A345059.

Sequence in context: A302152 A160925 A345059 * A099614 A032012 A092702

Adjacent sequences: A348970 A348971 A348972 * A348974 A348975 A348976

KEYWORD

nonn,frac

AUTHOR

Antti Karttunen, Nov 06 2021

STATUS

approved

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Last modified January 29 20:26 EST 2023. Contains 359926 sequences. (Running on oeis4.)