A240905 Smallest k such that the minimal factor in factorization of k! over distinct terms of A050376 is A050376(n), or a(n) = 0 if there is no such k.

2, 12, 20, 6, 10, 130, 180, 240, 480, 597, 901, 40537, 15841, 23401, 36720, 112321, 20377, 177842, 101755, 855369, 2333282, 654549, 15078026, 9206403, 5829606, 75690506, 64946036, 650025768, 1100265039, 1966444000, 6660851430, 1505032794, 7305026242, 42063043872, 54868098259

Offset: 1

Author

Vladimir Shevelev, Apr 14 2014

Keywords

• nonn
• more

Comments

a(n) is the smallest k such that the minimal infinitary divisor of k! is A050376(n).

Conjecture: a(n) > 0 for all n.

a(28) > 2.5*10^8, if it exists. - Amiram Eldar, Jun 18 2025

a(34) > 2.8*10^10 if it exists. - David A. Corneth, Jun 19 2025

Examples

			Let n = 4. A050376(4)=5. For k = 2, 3, 4, 5, 6, we have the following factorizations over distinct terms of A050376: 2! = 2, 3! = 2*3, 4! = 2*3*4, 5! = 2*3*4*5, 6! = 5*9*16. Only the last factorization begins with 5. So a(4) = 6.
From _David A. Corneth_, Jun 19 2025: (Start)
a(6) = 130. Once we checked that a(6) is > 125 we try 126. The minimal factor of k! into distinct products must be A050376(6) = 8. For 126! we have the 5-adic valuation of 31 so the minimal factor is at most 5.
To get rid of the 5 we try the next candidate > 126 that is a multiple of 5. This is 130. We can just skip 127, 128 and 129 altogether. It turns out this smallest factor for 130! is 8 giving the value for a(6). (End)
		

Links

• David A. Corneth, PARI program
• Amiram Eldar, PARI program

Crossrefs

Cf. A050376, A240537, A240606, A240619, A240620, A240668, A240669, A240670, A240672, A240695, A240751, A240755, A240764.

Programs

PARI

\\ See Corneth link

PARI

\\ See Eldar link

Extensions

More terms from Peter J. C. Moses, Apr 19 2014

a(19)-a(27) from Amiram Eldar, Jun 18 2025

a(28)-a(35) from David A. Corneth, Jun 18, Jun 21 2025