A321362 a(n) is the least k such that in the prime power factorization of k! the exponents of primes p_1, ..., p_n are odd, while the exponent of p_(n+1) is even.

2, 3, 5, 119, 57, 220, 1131, 2986, 1505, 3211, 21300, 26795, 11820, 14575, 67385, 221051, 33782, 132512, 819236, 1478432, 1630903, 26736550, 1095752, 41815849, 24813938, 31982450, 142574286, 860986855, 602660826, 2638930495, 2664421881, 1309662955, 33767540563

Offset

1,1

Comments

Analog to A240537 where odd and even are swapped.

Links

Giovanni Resta, Table of n, a(n) for n = 1..37 (terms < 10^12)

Rémy Sigrist, C program for A321362

Example

For k=2, 2!=2 with factorization 2 where the first exponent is odd.
For k=3, 3!=6 with factorization 2*3 where the first 2 exponents are odd.
For k=5, 5!=120 with factorization 2*3*5 where the first 3 exponents are odd.
In each case, there are no lesser numbers with the same property.

Mathematica

aQ[k_, n_] := Module[{e=FactorInteger[k!][[;; , 2]]}, Length[e]>=n && AllTrue[ e[[1;; n]], OddQ ] && If[Length[e]>n, EvenQ[e[[n+1]]], True]]; a[n_] := Module[ {k=2}, While[!aQ[k, n], k++]; k ]; Array[a, 10] (* Amiram Eldar, Nov 07 2018 *)

Prog

(PARI) isok(v, n) = {if (#v < n, return (0)); for (i=1, n, if (!(v[i] % 2), return(0)); ); (#v == n) || !(v[n+1] % 2); }
newv(v, i) = {if (isprime(i), return(concat(v, 1))); f = factor(i); for (k=1, #f~, v[primepi(f[k, 1])] += f[k, 2]; ); return (v); }
a(n) = {my(v =[1], i = 2); while (!isok(v, n), i++; v = newv(v, i)); i; }
(C) See Links section.

Crossrefs

Cf. A115627, A240537.

Sequence in context: A205668 A118505 A067799 * A230372 A234900 A117702

Adjacent sequences: A321359 A321360 A321361 * A321363 A321364 A321365

Keyword

nonn

Author

Michel Marcus, Nov 07 2018

Extensions

a(16)-a(32) from Rémy Sigrist, Nov 08 2018

a(33) from Giovanni Resta, Nov 09 2018

Status

approved