A242138 -id:A242138 - cp's OEIS frontend

A242139 Decimal equivalents of A242138.

5, 9, 10, 17, 18, 21, 27, 33, 34, 36, 42, 45, 51, 54, 65, 66, 68, 73, 85, 99, 102, 119, 129, 130, 132, 136, 146, 153, 165, 170, 187, 195, 198, 204, 219, 221, 231, 238, 257, 258, 260, 264, 273, 292, 297, 325, 330, 341, 363, 365, 387, 390, 396, 429, 438, 455

Offset: 1

Felix Fröhlich, May 05 2014

isvper(v) = {nv = #v; if (isprime(nv), return (0)); fordiv(nv, d, if ((d > 1) && (d < nv), dv = vector(d, i, v[i]); pdv = []; for (k=1, nv/d, pdv = concat(pdv, dv)); if (pdv == v, return (1)););); return (0);}
isok(n) = {vn = binary(n); if (vecmin(vn) == vecmax(vn), return (0)); if (isvper(vn), return (1)); nbmax = #vn; for (k=1, nbmax, vn = concat(0, vn); if (isvper(vn), return (1));); return (0);} \\ Michel Marcus, Aug 25 2014

Missing terms 99, 102, 119 added and more terms from Michel Marcus, Aug 25 2014

A353339 Number of integers b with n > b > 1 such that the base-b representation of n is periodic.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 2, 2, 2, 4, 1, 2, 2, 2, 3, 2, 3, 4, 1, 3, 4, 2, 1, 4, 1, 4, 3, 4, 1, 4, 3, 3, 3, 4, 2, 5, 2, 2, 2, 5, 1, 5, 2, 3, 4, 2, 1, 5, 1, 5, 4, 4, 1, 5, 2, 4, 3, 2, 1, 7, 1, 3, 5, 3, 5, 5, 1, 5, 2, 5, 1, 5, 3, 3, 3, 3, 2, 5, 1, 6, 2, 4, 1, 7, 4, 3, 2

Offset: 1

Felix Fröhlich, Apr 14 2022

			For n = 10: The base-2, base-3, base-4 and base-9 representations of 10 are 1010, 0101, 22 and 11, respectively, and these are the only representations that are periodic, so a(10) = 4.

Cf. A121016, A242138, A242139, A321513.

is(n, b) = for (w=1, oo, my (d=digits(n, b^w)); if (#d<=1, return (0), #Set(d)==1, return (1))) \\ after Rémy Sigrist in A321513
a(n) = my(i=0); for(b=2, n-1, if(is(n, b), i++)); i

A352886 Number of B-periodic numbers of bit pseudo-length n.

Original entry on oeis.org

1, 0, 4, 0, 7, 3, 16, 0, 37, 0, 64, 18, 127, 0, 283, 0, 517, 66, 1024, 0, 2167, 15, 4096, 255, 8197, 0, 16906, 0, 32767, 1026, 65536, 78, 133087, 0, 262144, 4098, 524407, 0, 1056730, 0, 2097157, 16635, 4194304, 0, 8421247, 63, 16777711, 65538, 33554437, 0

Offset: 4

Felix Fröhlich, Apr 07 2022

For the definition of "B-periodic numbers" and "bit pseudo-length", see Dobeš, Kureš, 2010, p. 294. The first few terms are given in the table on p. 295.

The sequence counts periodic binary numbers of length n where the least-significant bit is 0 (see Dobeš, Kureš, 2010, p. 294).

			For n = 6: The B-periodic numbers of bit pseudo-length 6 are 101010, 100100, 010010 and 110110, so a(6) = 4.

J. Dobeš and M. Kureš, Search for Wieferich Primes through the use of Periodic Binary Strings, Serdica Journal of Computing, Vol. 4, No. 3 (2010), 293-300.

Cf. A008683, A027375, A242138, A242139.

c(n) = sumdiv(n, d, moebius(d)*2^(n/d))
a(n) = (2^n - c(n) - 2)/2

a(n) = (2^n - c(n) - 2)/2, where c(n) = Sum_{d|n} A008683(d)*2^(n/d).

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A242139 Decimal equivalents of A242138.

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A353339 Number of integers b with n > b > 1 such that the base-b representation of n is periodic.

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A352886 Number of B-periodic numbers of bit pseudo-length n.

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