A217983 If n = floor(p/2) * p^e, for some (by necessity unique) prime p and exponent e > 0, then a(n) = p, otherwise a(n) = 1.

1, 2, 3, 2, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Johannes W. Meijer, Oct 25 2012

Keywords

• nonn

Comments

a(A130290(n) * A000040(n)^n1) = A000040(n), n >= 1 and n1 >= 1, and a(n) = 1 elsewhere. - The original name of the sequence.

The a(n) are related to the prime numbers A000040 and the number of nonzero quadratic residues modulo the n-th prime A130290, see the first formula and the Maple program.

This sequence resembles the exponential of the von Mangoldt function A014963; for the latter sequence a(A000040(n)^n1) = A000040(n), n >= 1 and n1 >= 1, and a(n) = 1 elsewhere.

Positions of the first occurrence of each successive noncomposite number (and also the records) is given by the union of {2} and A008837. - Antti Karttunen, Jan 17 2025

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100949

Crossrefs

Cf. A000040, A014963, A128060, A130290, A160479.

Cf. A000079, A000244 (after their initial 1's, the positions of 2's and 3's respectively), A020699 (positions of 5's from its third term 10 onward), A169634 (positions of 7's from the second term onward), A379956 (positions of terms > 1).

Cf. A008837, A121057.

Programs

Maple

nmax := 78: A000040 := proc(n): ithprime(n) end: A130290 := proc(n): if n =1 then 1 else (A000040(n)-1)/2 fi: end: for n from 1 to nmax do A217983(n) := 1 od: for n from 1 to nmax do for n1 from 1 to floor(log[A000040(n)](nmax)) do A217983(A130290(n) * A000040(n)^n1) := A000040(n) od: od: seq(A217983(n), n=1..nmax);

PARI

A217983(n) = { my(f=factor(n)); for(i=1,#f~,if((n/(f[i,1]^f[i,2])) == (f[i,1]\2), return(f[i,1]))); (1); }; \\ Antti Karttunen, Jan 16 2025

Formula

a(A130290(n) * A000040(n)^n1) = A000040(n), n >= 1 and n1 >= 1, and a(n)= 1 elsewhere.

a(n) = (A160479(n+1) * A128060(n+1))/(2*n+1) for n >= 2.

Extensions

Definition simplified, original definition moved to comments; more terms added by Antti Karttunen, Jan 16 2025