cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A240504 Read (exponents of primes in the factorization of n!) modulo 2 and convert to decimal.

Original entry on oeis.org

1, 3, 3, 7, 1, 3, 11, 11, 1, 3, 11, 23, 51, 43, 43, 87, 23, 47, 15, 95, 215, 431, 47, 47, 295, 423, 391, 783, 143, 287, 1311, 1887, 847, 719, 719, 1439, 3471, 2511, 975, 1951, 7583, 15167, 14655, 12607, 4383, 8767, 575, 575, 16959, 25407, 24895, 49791, 639, 10879
Offset: 2

Views

Author

Vladimir Shevelev, Apr 06 2014

Keywords

Examples

			Since 9! = 2^7*3^4*5*7, then we have a binary number the digits of which are the exponents modulo 2: 1011. In decimal this is 11. So a(9)=11.

Links

Crossrefs

Cf. A055204, A240502, A336510.

Programs

  • PARI
    a(n) = subst(Pol(factor(n!)[,2] % 2), x, 2); \\ Michel Marcus, Feb 15 2016
  • PARI
    a(n) = { my(res = 0); forprime(p = 2, n, res = 2*res + (val(n, p)%2) ); res }
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Feb 24 2023

Extensions

More terms from Michel Marcus, Feb 15 2016

A240505 Products of primes the squares of which are Fermi-Dirac divisors of n!

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 6, 2, 5, 5, 10, 10, 70, 210, 210, 210, 35, 35, 14, 14, 154, 154, 462, 2310, 30030, 10010, 715, 715, 4290, 4290, 4290, 4290, 36465, 7293, 4862, 4862, 92378, 277134, 277134, 277134, 1939938, 1939938, 88179, 146965, 6760390, 6760390, 20281170
Offset: 1

Views

Author

Vladimir Shevelev, Apr 06 2014

Keywords

Comments

Or equivalently, products of primes the squares of which are infinitary divisors of n!.

Links

Crossrefs

Cf. A050376, A055204, A240502, A240504.

Programs

  • PARI
    isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k,2]); bde = binary(valuation(d, f[k,1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)););); return (1);}
a(n) = {f = factor(n!); for (k=1, #f~, if ((f[k,2] > 1) && isidiv(f[k,1]^2, f), f[k,2]=1, f[k,2]=0);); factorback(f);} \\ Michel Marcus, Feb 15 2016

Extensions

a(23)-a(32) from Michel Marcus, Feb 15 2016
a(1) = 1 and more terms from Rémy Sigrist, Feb 13 2019
Showing 1-2 of 2 results.

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