A348841 -id:A348841 - cp's OEIS frontend

A055460 Number of primes with odd exponents in the prime power factorization of n!.

0, 1, 2, 2, 3, 1, 2, 3, 3, 1, 2, 3, 4, 4, 4, 4, 5, 4, 5, 4, 6, 6, 7, 5, 5, 5, 6, 5, 6, 5, 6, 7, 9, 7, 7, 7, 8, 8, 8, 8, 9, 10, 11, 10, 9, 7, 8, 7, 7, 8, 10, 9, 10, 8, 10, 12, 14, 12, 13, 11, 12, 12, 11, 11, 13, 12, 13, 12, 12, 13, 14, 13, 14, 14, 15, 14, 14, 11, 12, 13, 13, 13, 14, 16, 16, 14

Offset: 1

Labos Elemer, Jun 26 2000

nonn

The products of the corresponding primes form A055204.
Also, the number of primes dividing the squarefree part of n! (=A055204(n)).
Also, the number of prime factors in the factorization of n! into distinct terms of A050376. See the references in A241289. - Vladimir Shevelev, Apr 16 2014

			For n = 100, the exponents of primes in the factorization of n! are {97,48,24,16,9,7,5,5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1,1,1}, and there are 17 odd values: {97,9,7,5,5,3,3,1,1,1,1,1,1,1,1,1,1}, so a(100) = 17.
The factorization of 6! into distinct terms of A050376 is 5*9*16 with only one prime, so a(6)=1. - _Vladimir Shevelev_, Apr 16 2014

V. S. Shevelev, Multiplicative functions in the Fermi-Dirac arithmetic, Izvestia Vuzov of the North-Caucasus region, Nature sciences 4 (1996), 28-43 (in Russian; MR 2000f: 11097, pp. 3912-3913).

Max Alekseyev, Table of n, a(n) for n = 1..100000
S. Litsyn and V. S. Shevelev, On factorization of integers with restrictions on the exponent, INTEGERS: Electronic Journal of Combinatorial Number Theory, 7 (2007), #A33, 1-36.

Cf. A000142, A000720, A007913, A008833, A162642, A348841.

Cf. A249016 (indices of records), A249017 (values of records)

Table[Count[FactorInteger[n!][[All, -1]], m_ /; OddQ@ m] - Boole[n == 1], {n, 100}] (* Michael De Vlieger, Feb 05 2017 *)

PARI
```
a(n) = omega(core(n!))
```

a(n) = A001221(A055204(n)). - Max Alekseyev, Oct 19 2014
From Wolfdieter Lang, Nov 06 2021: (Start)
a(n) = A162642(A000142(n)).
a(n) = A000720(n) - A348841(n), (End)

Edited by Max Alekseyev, Oct 19 2014

A348842 Number of Juniper Green games with n cards.

Original entry on oeis.org

0, 1, 1, 6, 10, 35, 47, 147, 216, 452, 512, 3055, 3365, 5602, 12160, 35951, 37959, 147889, 154998, 703094, 1178850, 1467813

Offset: 1

Wolfdieter Lang, Dec 23 2021

For the rules of this two person game with cards labeled from 1 to n, for n >= 1, called JG(n), see the Ian Stewart links.
It is reported (see the FEEDBACK and the German version), that E. P. Wigner used this game in some lecture in the thirties. There the prime factorization of n! into prime powers, with the number of odd or even (>= 2) exponents, seems to have played a role (see A055460(n) and A348841(n) for the number of primes with these exponents in the factorization of n!, respectively).
The repertoire of card numbers for JG(n) that can be chosen if the latest removed card had label k is shown in A348390. Of course, only those card numbers not yet removed in earlier moves qualify. E.g., n = 4, k = 2: repertoire 1, 4.
The total number of games JG(n), for n >= 2, if the first removed card has label K = 2*k, for k = 1, 2, ... ,floor(n/2), is given in A348843.
For the irregular table which gives in row n the odd and even number of moves in the a(n) JG(n) games see A348844. This gives the number of times Alice (the first mover), respectively Bob wins.

Ian Stewart, Juniper Green, Scientific American No. 3 March 1997, pp. 118-120.
Ian Stewart, FEEDBACK, Scientific American No. 5 November 1997 p. 112.
The author's email is given, juniper green Online game with 100 cards.

Cf. A055460, A348390, A348843, A348844.

a(n) = Sum_{k=1..floor(n/2)} A348843(n, k) = Sum_{k=1..2*floor(n/2)} A348844(n, k), for n >= 2.

cp's OEIS Frontend

A055460 Number of primes with odd exponents in the prime power factorization of n!.

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