A065371 a(1) = 1, a(prime(i)) = prime(i) - i for i > 0 and a(u * v) = a(u) * a(v) for u, v > 0.

1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 6, 1, 7, 3, 2, 1, 10, 1, 11, 2, 3, 6, 14, 1, 4, 7, 1, 3, 19, 2, 20, 1, 6, 10, 6, 1, 25, 11, 7, 2, 28, 3, 29, 6, 2, 14, 32, 1, 9, 4, 10, 7, 37, 1, 12, 3, 11, 19, 42, 2, 43, 20, 3, 1, 14, 6, 48, 10, 14, 6, 51, 1, 52, 25, 4, 11, 18, 7, 57, 2, 1, 28, 60, 3, 20, 29

Offset

1,5

Comments

a(n) > 0 and a(n) < n for all n > 1.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

a(1) = 1; for n > 1, a(n) = (A020639(n)-A055396(n)) * a(A032742(n)). - Antti Karttunen, Jan 12 2017

Example

a(210) = a(2*3*5*7) = a(2)*a(3)*a(5)*a(7) = (prime(1)-1)*(prime(2)-2)*(prime(3)-3)*(prime(4)-4) = (2-1)*(3-2)*(5-3)*(7-4) = 1*1*2*3 = 6.

Mathematica

a[n_] := a[n] =
     If[n == 1, 1,
     If[PrimeQ[n], n - PrimePi[n],
     Product[{p, e} = pe; a[p]^e, {pe, FactorInteger[n]}]]];
Array[a, 100] (* Jean-François Alcover, Nov 20 2021 *)

Prog

(Haskell)
a065371 1 = 1
a065371 n = product $ map (a014689 . a049084) $ a027746_row n
-- Reinhard Zumkeller, Apr 09 2012
(Scheme, with memoization-macro definec)
(definec (A065371 n) (cond ((= 1 n) 1) (else (* (- (A020639 n) (A055396 n)) (A065371 (A032742 n))))))
;; Antti Karttunen, Jan 12 2017
(PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k, 1]-=primepi(f[k, 1])); factorback(f); \\ Michel Marcus, Nov 20 2021

Crossrefs

Cf. A000040, A014689, A020639, A027746, A032742, A049084, A055396, A065372, A065373, A065374.

Sequence in context: A349431 A297381 A051793 * A300982 A354203 A186007

Adjacent sequences: A065368 A065369 A065370 * A065372 A065373 A065374

Keyword

mult,nonn

Author

Reinhard Zumkeller, Nov 01 2001

Status

approved