A333182 -id:A333182 - cp's OEIS frontend

A332905 Decimal expansion of Sum_{p prime, k>=1} 1 / 2^(p^k).

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

Offset: 0

Ilya Gutkovskiy, Mar 10 2020

Prime power constant: binary expansion is the characteristic function of prime powers (A069513).

			0.48305718112590956493... = (0.0111101110...)_2.
                                |||| |||
                                2345 789

Cf. A051006, A069513, A246655, A333182.

A371134 Decimal expansion of Sum_{squarefree k>=1} k / 2^k.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 12 2024

Erdős (1981) conjectured and Chen and Ruzsa (1999) proved that this constant is irrational.

			1.69790781977962506446424089965347807016709423133847...

G. C. Greubel, Table of n, a(n) for n = 1..25000
Thomas Bloom, Is the sum Sum_{n} mu(n)^2 * n/2^n irrational?, Erdős Problems.
Yong-Gao Chen and Imre Z. Ruzsa, On the irrationality of certain series, Periodica Mathematica Hungarica, Vol. 38, No. 1 (1999), pp. 31-37.
Paul Erdős, Sur l'irrationalité d'une certaine série, C. R. Acad. Sci. Paris, Sér. 1, Vol. 292 (1981), pp. 765-768.
Terence Tao, Erdős problem database, see no. 259.

Cf. A005117, A008683, A333182, A346173, A371135.

m:= 510;
S:= func< n | (&+[j/2^j: j in [1..3*m] | IsSquarefree(j)]) >;
Prune(Reverse(IntegerToSequence(Floor(S(m)*10^( Floor(m/2)) )))); // G. C. Greubel, Sep 06 2025

RealDigits[Sum[n/2^n, {n, Select[Range[1000], SquareFreeQ]}], 10, 120][[1]]

numerical_approx( sum(k/2^k for k in range(1,1000) if abs(moebius(k)) == 1 ) , digits=265 ) # G. C. Greubel, Sep 06 2025

Equals Sum_{k>=1} A005117(k) / 2^A005117(k).

Equals Sum_{k>=1} k * mu(k)^2 / 2^k.

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A332905 Decimal expansion of Sum_{p prime, k>=1} 1 / 2^(p^k).

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A371134 Decimal expansion of Sum_{squarefree k>=1} k / 2^k.

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