A048278 -id:A048278 - cp's OEIS frontend

A249452 Numbers k such that A249441(k) = 3.

15, 31, 47, 63, 95, 127, 191, 255, 383, 511, 767, 1023, 1535, 2047, 3071, 4095, 6143, 8191, 12287, 16383, 24575, 32767, 49151, 65535, 98303, 131071, 196607, 262143, 393215, 524287, 786431, 1048575, 1572863, 2097151, 3145727, 4194303, 6291455, 8388607, 12582911

Offset: 1

Vladimir Shevelev, Oct 29 2014

Or k for which none of entries in the k-th row of Pascal's triangle (A007318) is divisible by 4 (cf. comment in A249441).
Using the Kummer carries theorem, one can prove that, for n>=2, a(n) has the form of either 1...1 or 101...1 in base 2.
The sequence is a subset of so-called binomial coefficient predictors (BCP) in base 2 (see Shevelev link, Th. 6 and Cor. 8), which were found also using Kummer theorem and have a very close binary structure.

E. E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. Reine Angew. Math. 44 (1852), 93-146.
V. Shevelev, Binomial Coefficient Predictors, Journal of Integer Sequences, Vol. 14 (2011), Article 11.2.8
Index entries for linear recurrences with constant coefficients, signature (1,2,-2).

Cf. A000225, A029744, A048278, A052955, A055010, A089633, A249441.

CoefficientList[Series[(15 + 16 x - 14 x^2 - 16 x^3)/(1 - x -2 x^2 + 2 x^3), {x, 0, 70}], x] (* Vincenzo Librandi, Oct 30 2014 *)
LinearRecurrence[{1,2,-2},{15,31,47,63},40] (* Harvey P. Dale, Apr 01 2019 *)

a(n)=if(n==1, 15, (n%2+2)<<(n\2+3)-1) \\ Charles R Greathouse IV, Nov 06 2014

is(n)=(n+1)>>valuation(n+1, 2)<5 && !setsearch([1, 2, 3, 5, 7, 11, 23], n) \\ Charles R Greathouse IV, Nov 06 2014

a(n) has either form 2^k - 1 or 3*2^m-1, k, m >= 4 (cf. A000225, A055010). Since, for k>=5, 2^k-1<3*2^(k-1)-1<2^(k+1)-1, we have that, for n>=1, a(2*n) = 2^(n+4)-1; a(2*n+1) = 3*2^(n+3)-1. - Vladimir Shevelev, Oct 29 2014, Nov 06 2014
a(1) = 15, and for n>1, a(n) = A052955(n+6). [Follows from above] - Antti Karttunen, Nov 03 2014
G.f.: (15+16*x-14*x^2-16*x^3)/(1-x-2*x^2+2*x^3); a(n) = 16*A029744(n)-1. - Peter J. C. Moses, Oct 30 2014

More terms from Peter J. C. Moses, Oct 29 2014

A146527 a(n) = number of distinct composites, when each is represented in binary, that occur as substrings within the binary representation of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 2, 1, 0, 3, 1, 2, 1, 3, 2, 3, 2, 3, 2, 2, 0, 5, 5, 3, 2, 5, 2, 4, 1, 4, 4, 3, 3, 4, 3, 4, 3, 5, 4, 3, 2, 5, 3, 3, 1, 7, 6, 7, 6, 6, 4, 4, 3, 8, 8, 5, 3, 8, 4, 5, 2, 5, 5, 5, 4, 4, 4, 5, 3, 6, 4, 5, 4, 6, 5, 6, 4, 7, 6, 6, 4, 6, 4, 5, 3, 8, 7, 6, 5, 7, 4, 5, 2, 9, 8, 8, 8, 9, 7, 8, 7, 9, 8

Offset: 1

Leroy Quet, Oct 30 2008

nonn

a(n) = 0 for n equal to only the positive integers 1,2,3,5,7,11,23 (sequence A048278).

			20 in binary is 10100. The composites, when represented in binary, that can be found within 10100 are 4 = 100 in binary, 10 (decimal) = 1010 in binary and 20 itself = 10100 in binary. There are 3 of these composites, so a(20) = 3.

A078826

Extended by Ray Chandler, Nov 03 2008

A272333 Values of Fibonacci(n+1) that divides n!.

Original entry on oeis.org

1, 2, 3, 8, 21, 144, 46368

Offset: 1

Altug Alkan, Apr 26 2016

This sequence is finite.
See comment section of A048278 for motivation. Also see the formula section of this sequence for relation between this sequence and divisors of 24.
Sequence focuses on the positive values of n, so 1 only appears once as a term.

			8 is a term because 8 = Fibonacci(6) and (6-1)! is divisible by 8.

Cf. A000045, A048278, A018253.

lista(nn) = for(n=1, nn, if(n! % fibonacci(n+1) == 0, print1(fibonacci(n+1), ", ")));

a(n) = A000045(A048278(n)+1) = A000045(A018253(n+1)), for n > 0.

cp's OEIS Frontend

A249452 Numbers k such that A249441(k) = 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A146527 a(n) = number of distinct composites, when each is represented in binary, that occur as substrings within the binary representation of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A272333 Values of Fibonacci(n+1) that divides n!.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

Formula