A141564 Subtract 1 from all bases and exponents which are greater than 1 in the prime number decomposition of n.

0, 1, 2, 1, 4, 2, 6, 1, 2, 4, 10, 2, 12, 6, 8, 1, 16, 2, 18, 4, 12, 10, 22, 2, 4, 12, 4, 6, 28, 8, 30, 1, 20, 16, 24, 2, 36, 18, 24, 4, 40, 12, 42, 10, 8, 22, 46, 2, 6, 4, 32, 12, 52, 4, 40, 6, 36, 28, 58, 8, 60, 30, 12, 1, 48, 20, 66, 16, 44, 24, 70, 2, 72, 36, 8, 18, 60, 24, 78, 4, 8

Offset: 1

Author

Juri-Stepan Gerasimov, Aug 14 2008

Keywords

• nonn

Comments

Start from the prime number decomposition of n, that is the list 1, 2, 3, 2^2, 5, 2*3, 7, 2^3, 3^2, 2*5, 11, 2^2*3... Subtract 1 from all visible bases and exponents (visible in the sense that exponents are not written down if they equal 1), to give 1-1, 2-1, 3-1, (2-1)^(2-1), 5-1, (2-1)*(3-1), 7-1, (2-1)^(3-1), (3-1)^(2-1), (2-1)*(5-1), 11-1, (2-1)^(2-1)*(3-1)... Evaluate this modified product to yield a(n).

Crossrefs

Cf. A000040, A002808.

Programs

Maple

A := proc(n) local a,p,e,q,ifs ; if n = 1 then RETURN(0) ; fi; ifs := ifactors(n)[2] ; a := 1; for p in ifs do q := op(1,p)-1 ; if op(2,p) > 1 then e := op(2,p)-1 ; else e := 1 ; fi; a := a*q^e ; od: RETURN(a) ; end: for n from 1 to 120 do printf("%d,",A(n)) ; od: # R. J. Mathar, Aug 21 2008

Mathematica

f[{p_,e_}]:=If[e==1,p-1,(p-1)^(e-1)];a[n_]:=Times@@f/@FactorInteger[n];Array[a,81] (* James C. McMahon, Jul 26 2025 *)

Extensions

Corrected and extended by R. J. Mathar, Aug 21 2008