A077466 -id:A077466

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A077465

Values of n such that A006046(n)/n^theta, where theta=log(3)/log(2), is a local minimum, computed according to Harborth's recurrence.

+10
10

1, 3, 5, 11, 21, 43, 87, 173, 347, 693, 1387, 2775, 5549, 11099, 22197, 44395, 88789, 177579, 355159, 710317, 1420635, 2841269, 5682539, 11365079, 22730157, 45460315, 90920629, 181841259, 363682519, 727365037, 1454730075

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Harborth's recurrence can miss local minima that are 2 less than values in this sequence. A complete listing of cumulative minima is given by A084230.

LINKS

Table of n, a(n) for n=1..31.

H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.

Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant

CROSSREFS

Cf. A006046, A077464, A077466, A077467, A084230.

KEYWORD

nonn,changed

AUTHOR

Eric W. Weisstein, Nov 05 2002

STATUS

approved

A077464

Stolarsky-Harborth constant; lim inf_{n->oo} F(n)/n^theta, where F(n) is the number of odd binomial coefficients in the first n rows and theta=log(3)/log(2).

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

The limit supremum of F(n)/n^theta is 1. - Charles R Greathouse IV, Oct 30 2016

Named by Finch (2003) after Kenneth B. Stolarsky and Heiko Harborth. Stolarsky (1977) evaluated that its value is in the interval [0.72, 0.815], and Harborth (1977) calculated the value 0.812556. - Amiram Eldar, Dec 03 2020

REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, pp. 145-151.

LINKS

Table of n, a(n) for n=0..100.

Heiko Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc., Vol. 62, No. 1 (1977), pp. 19-22.

Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Periodic minimum in the count of binomial coefficients not divisible by a prime, arXiv:2408.06817 [math.NT], 2024. See pp. 2, 4.

Kenneth B. Stolarsky, Digital sums and binomial coefficients, Notices of the American Mathematical Society, Vol. 22, No. 6 (1975), A-669, entire volume.

Kenneth B. Stolarsky, Power and Exponential Sums of Digital Sums Related to Binomial Coefficient Parity, SIAM J. Appl. Math., Vol. 32, No. 4 (1977), pp. 717-730.

Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant.

Eric Weisstein's World of Mathematics, Pascal's Triangle.

FORMULA

Equals lim inf_{n->oo} A006046(n)/n^A020857. - Amiram Eldar, Dec 03 2020

EXAMPLE

0.812556559016006387694882...

CROSSREFS

Cf. A006046, A020857, A077465, A077466, A077467.

KEYWORD

nonn,cons,changed

AUTHOR

Eric W. Weisstein, Nov 06 2002

STATUS

approved

A077467

Sum of binary digits of A077465(n).

+10
4

1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 18, 18, 19, 19, 20, 20, 21, 22, 22, 23, 23, 24, 25, 25, 26, 26, 27, 27, 28, 29, 29, 30, 30, 31, 32, 32, 33, 33, 34, 34, 35, 36, 36, 37, 37, 38, 39, 39, 40, 40, 41, 41, 42, 43, 43, 44

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..74.

H. Harborth, Number of Odd Binomial Coefficients, Proc. Amer. Math. Soc. 62, 19-22, 1977.

Eric Weisstein's World of Mathematics, Stolarsky-Harborth Constant

CROSSREFS

Cf. A006046, A077464, A077465, A077466.

KEYWORD

nonn,base,changed

AUTHOR

Eric W. Weisstein, Nov 05 2002

STATUS

approved

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