A155457 a(n) = exp(Lambda(n)), where Lambda(n) is the von Mangoldt function for odd (!) primes.

1, 1, 1, 3, 1, 5, 1, 7, 1, 3, 1, 11, 1, 13, 1, 1, 1, 17, 1, 19, 1, 1, 1, 23, 1, 5, 1, 3, 1, 29, 1, 31, 1, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 7, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 1, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 3, 1, 83, 1, 1, 1, 1, 1, 89

Offset

0,4

Comments

a(n) = p if n = p^k and p odd prime, k >= 1, otherwise 1.

References

Tom M. Apostol, Introduction to analytic number theory, Springer-Verlag, 1976.

Links

Table of n, a(n) for n=0..89.

Manjul Bhargava, The factorial function and generalizations, Amer. Math. Monthly, 107 (Nov. 2000), 783-799.

Angelo B. Mingarelli, Abstract factorials, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, No. 4, 43-76, (page 44).

Wikipedia, Von Mangoldt function

Formula

a(n) = 1 + Sum_{k=3..n} (k-1)*A010051(k)*(floor(k^n/n)-floor((k^n -1)/n)). - Anthony Browne, Jun 16 2016

Example

a(8) = 1 because 8 = 2^3 is not the power of an odd prime, a(49) = 7 because 49 = 7^2.

Maple

a := proc(n) local lcm; lcm := n -> ilcm(seq(i, i = 1..n)); if type(n, even) then 1 else lcm(n)/lcm(n-1) fi end;

Mathematica

a[n_] := If[IntegerQ[Log[2, n]], 1, Exp[MangoldtLambda[n]]]; Table[a[n], {n, 0, 89}] (* Jean-François Alcover, Jan 27 2014 *)

Prog

(SageMath)
def A155457(n: int) -> int:
    b, e = is_prime_power(n, get_data=True)
    return b if e != 0 and b % 2 == 1 else 1
print([A155457(n) for n in range(0, 90)])  # Peter Luschny, Jan 03 2026

Crossrefs

Cf. A014963, A010051.

Sequence in context: A090585 A309391 A147661 * A274658 A274660 A327531

Adjacent sequences: A155454 A155455 A155456 * A155458 A155459 A155460

Keyword

nonn

Author

Peter Luschny, Jan 22 2009, Jan 25 2009

Extensions

a(0)=1 prepended by Peter Luschny, Jan 03 2026

Status

approved