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

Author

Vladimir Joseph Stephan Orlovsky, May 13 2008

Keywords

• nonn
• easy

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

Exactly the positive integers without nontrivial powers of two (i.e., 2^k, k > 0). That is, the same as A057716 except for the first term of both sequences. - M. F. Hasler, Jul 04 2025

Examples

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

References

• 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
• NRICH, Polite numbers
• Melfried Olson, Sequentially so, Mathematics Magazine 52:5, pp. 297-298.
• Erzsébet Orosz, On odd-summing numbers, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004), pp. 125-129.
• Henri Picciotto's Math Education Page, Staircases
• PlanetMath, Polite number
• Wai Yan Pong, Sums of consecutive integers, The College Mathematics Journal, 38 (2007), 119-123.
• J. M. Rodriguez Caballero, A Characterization of the Hypotenuses of Primitive Pythagorean Triangles Using Partitions into Consecutive Parts, Amer. Math. Monthly 126 (2019), 74-77.
• Wikipedia, Polite number

Crossrefs

Cf. A155559 (complement), A057716 (nonpowers of 2: essentially the same), A000079 (powers of 2).

Programs

Mathematica

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

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

PARI

is_A138591(n)=hammingweight(n)>1||n==1 \\ M. F. Hasler, Jul 04 2025

Python

def A138591(n): return len(bin(n+len(bin(n))-3)) + n - 3 # Chai Wah Wu, Feb 18 2022

Python

A138591 = lambda n: n+(n+n.bit_length()-1).bit_length()-1
is_A138591 = lambda n: n.bit_count()>1 or n==1 # M. F. Hasler, Jul 04 2025
(C#) BigInteger a(BigInteger n) => (n + n.GetBitLength() - 1).GetBitLength() + n - 1; // Delbert L. Johnson, Mar 12 2023

Formula

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

Extensions

More terms from Carl R. White, Jul 22 2009