A138591 Sums of two or more consecutive nonnegative integers.

1, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset

1,2

Comments

Closely related to but different from A057716. - N. J. A. Sloane, May 16 2008

These are called polite numbers [From Howard Berman (howard_berman(AT)hotmail.com), Oct 29 2008] by those who require nonnegative integers in the definition as opposed to positive integers. With the latter requirement, 1 = 0 + 1 does not count as a polite number. [This difference of definition pointed out by Ant King (Nov 19 2010)] There is no disagreement that 1 belongs in this sequence, but there is disagreement as to whether it counts as a polite number. - Ant King, Nov 19 2010

Of course sums of two or more consecutive nonpositive integers have the same absolute values (noted while inserting "nonnegative" in title). All integers are sums of two or more consecutive integers without such restriction. - Rick L. Shepherd, Jun 03 2014

In K-12 education, these are known as "staircase numbers." The "1" is often omitted. - Gordon Hamilton, Mar 17 2015

Complement of A155559. - Ray Chandler, Mar 23 2016

References

J. M. Rodriguez Caballero, A characterization of the hypotenuses of primitive Pythagorean triangles ..., Amer. Math. Monthly 126 (2019), 74-77.

A. Wah and H. Picciotto, Algebra: Themes, Tools, Concepts, 1994, page 190.

Links

Ray Chandler, Table of n, a(n) for n = 1..10000

Tom M. Apostol, Sums of Consecutive Positive Integers, The Mathematical Gazette, Vol. 87, No. 508, (March 2003).

Ron Knott, An Introduction to Runsums

Mathedpage, Staircases

NRICH, Polite numbers

Melfried Olson, Sequentially so, Mathematics Magazine 52:5, pp. 297-298.

Erzsébet Orosz, On odd-summing numbers, Acia Academiae Paedagogicae Agriensis, Seciio Maihemaiicae 31 (2004), pp. 125-129.

PlanetMath, Polite number

Wai Yan Pong, Sums of consecutive integers, The College Mathematics Journal, 38 (2007), 119-123. - Parthasarathy Nambi, May 20 2009

Wikipedia, Polite number

Formula

a(n) = n + A000523(n + A000523(n)). - Charles R Greathouse IV, Aug 12 2010

Example

0+1=1, 1+2=3, 2+3=5, 1+2+3=6, 3+4=7, 4+5=9, 1+2+3+4=10, ...

Mathematica

1 + # + Floor[Log[2, # + 1 + Log[2, # + 1]]] &/@Range[0, 70] (* Ant King, Nov 18 2010 *)

Prog

(PARI) a(n)=n+logint(n+logint(n, 2), 2) \\ Charles R Greathouse IV, Sep 01 2015
(PARI) is(n)=n>>valuation(n, 2)>1 || n==1 \\ Charles R Greathouse IV, Aug 01 2016
(Python)
def A138591(n): return len(bin(n+len(bin(n))-3)) + n - 3 # Chai Wah Wu, Feb 18 2022
(C#) BigInteger a(BigInteger n) => (n + n.GetBitLength() - 1).GetBitLength() + n - 1; // Delbert L. Johnson, Mar 12 2023

Crossrefs

Cf. A057716, A155559.

Sequence in context: A229858 A269020 A057716 * A136492 A062506 A308468

Adjacent sequences: A138588 A138589 A138590 * A138592 A138593 A138594

Keyword

nonn,easy

Author

Vladimir Joseph Stephan Orlovsky, May 13 2008

Extensions

More terms from Carl R. White, Jul 22 2009

Status

approved