A051006 -id:A051006 - cp's OEIS frontend

A132800 Decimal expansion of Sum_{n >= 1} 1/3^prime(n).

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

Offset: 0

Cino Hilliard, Nov 17 2007

Equivalently, the real number in (0,1) having the characteristic function of the primes, A010051, as its base-3 expansion. - M. F. Hasler, Jul 04 2017.

			0.15272690272572247152817541874425912430342364271463298508628837536732...

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Cf. A000720, A051006 (analog for base 2), A132797 (analog for base 5), A010051 (characteristic function of the primes), A057901, A132806 (base 4).

RealDigits[Sum[1/3^Prime[k], {k, 100}], 10, 100][[1]] (* Vincenzo Librandi, Jul 05 2017 *)

/* Sum of 1/m^p for primes p */ sumnp(n,m) = { local(s=0,a,j); for(x=1,n, s+=1./m^prime(x); ); a=Vec(Str(s)); for(j=3,100, print1(eval(a[j])",") ) }

suminf(n=1,1/3^prime(n)) \\ Then: digits(%\.1^default(realprecision))[1..-3] to remove the last 2 digits. N.B.: Functions sumpos() and sumnum() yield much less accurate results. - M. F. Hasler, Jul 04 2017

From Amiram Eldar, Aug 11 2020: (Start)
Equals Sum_{k>=1} 1/A057901(k).
Equals 2 * Sum_{k>=1} pi(k)/3^(k+1), where pi(k) = A000720(k). (End)

Offset corrected R. J. Mathar, Jan 26 2009

Edited by M. F. Hasler, Jul 04 2017

A092858 "Sum" of the sequences of primes and the triangular numbers (A000217).

Original entry on oeis.org

5, 6, 7, 10, 11, 13, 15, 17, 19, 21, 23, 28, 29, 31, 36, 37, 41, 43, 45, 47, 53, 55, 59, 61, 66, 67, 71, 73, 78, 79, 83, 89, 91, 97, 101, 103, 105, 107, 109, 113, 120, 127, 131, 136, 137, 139, 149, 151, 153, 157, 163, 167, 171, 173, 179, 181, 190, 191, 193, 197, 199

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

If two monotonic sequences are mapped into the real codomain of (0,1) as it is defined in A051006, then the fractional part of the sum of the two reals can be mapped back into a sequence as defined in A092855, yielding the "sum" of the two sequences.

Ferenc Adorjan, Binary mapping of monotonic sequences and the Aronson function

Cf. A092855, A051006, A092857, A092859, A092860, A092861, A092862, A092863, A092874.

{ssum(a,b)= /*Returns the "sum" monotonic sequences a and b */ return(mtinv(mt(a)+mt(b))) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ }

A092859 "Difference" of the sequences of triangular numbers (A000217) and the primes (cf. A092858).

Original entry on oeis.org

3, 4, 5, 7, 12, 13, 16, 18, 19, 22, 23, 30, 31, 38, 39, 40, 42, 43, 46, 48, 49, 50, 51, 52, 53, 56, 57, 58, 60, 61, 68, 69, 70, 72, 73, 80, 81, 82, 84, 85, 86, 87, 88, 89, 92, 93, 94, 95, 96, 98, 99, 100, 102, 103, 106, 108, 110, 111, 112, 113, 121, 122, 123, 124, 125, 126

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

Here the complement of the sequence of primes (1 and the composites) is "added" to the sequence of triangulars, according to the definition outlined in A092858.

Ferenc Adorjan, Binary mapping of monotonic sequences and the Aronson function

Cf. A092855, A051006, A092857, A092858, A092860, A092861, A092862, A092863, A092874.

{sdif(a,b)= /*Returns the "difference" of monotonic sequences a and b */ return(mtinv(mt(a)+mt(compl(b)))) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ } {compl(v)=/* Returns the complement of v monotonic positive sequence */ local(n,p=0,vv=[]);n=matsize(v)[2];for(i=1,n, for(j=p+1,v[i]-1,vv=concat (vv,j));p=v[i]);return(vv)}

A092860 "3 times the prime sequence".

Original entry on oeis.org

3, 4, 5, 6, 7, 10, 11, 12, 13, 16, 17, 18, 19, 22, 23, 28, 29, 30, 31, 36, 37, 40, 41, 42, 43, 46, 47, 52, 53, 58, 59, 60, 61, 66, 67, 70, 71, 72, 73, 78, 79, 82, 83, 88, 89, 96, 97, 100, 101, 102, 103, 106, 107, 108, 109, 112, 113, 126, 127, 130, 131, 136, 137, 138, 139

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

By iterating the addition to itself a monotonic sequence, according to the definition given in A092858, we can multiply the monotonic sequences by natural numbers.

Note, that it is easy to see that for an i natural and a v monotonic sequence, i(x)compl(v)=compl(i(x)v); where the "(x)" mark stands for the "integer multiplication of a sequence" and the function "compl" produces the complement of a positive monotonic sequence.

Ferenc Adorjan, Binary mapping of monotonic sequences and the Aronson function

Cf. A092855, A051006, A092857, A092858, A092859, A092861, A092862, A092863, A092874.

{imulv(i,v)= /*Returns "i (x) v" monotonic sequence */ return(mtinv(i*mt(v))) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ }

A092861 "Product" of the sequence of primes and the "evil" numbers (A001969).

Original entry on oeis.org

4, 7, 9, 12, 14, 15, 18, 19, 21, 25, 26, 33, 35, 36, 37, 40, 41, 42, 44, 47, 48, 50, 54, 55, 58, 59, 60, 64, 65, 66, 69, 72, 77, 78, 79, 80, 84, 86, 87, 88, 89, 90, 91, 97, 99, 100, 105, 106, 107, 108, 110, 111, 112, 114, 115, 116, 117, 118, 120, 122, 123, 125, 127, 128

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

If two monotonic sequences are mapped into the real section of (0,1), as it is defined in A051006 and the product of the two reals mapped back into the set of monotonic sequences as defined in A092855, then we have the "product" of the two sequences.

Ferenc Adorjan, Binary mapping of monotonic sequences and the Aronson function

Cf. A092855, A051006, A092857, A092858, A092859, A092860, A092862, A092863, A092874.

{prod(a,b)= /*Returns the "product" of monotonic sequences a and b */ return(mtinv(mt(a)*mt(b))) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ }

A092862 "Square" of the prime sequence.

Original entry on oeis.org

3, 5, 6, 14, 16, 17, 19, 21, 22, 25, 27, 31, 32, 34, 36, 37, 41, 42, 44, 45, 48, 49, 52, 54, 57, 59, 60, 62, 64, 65, 69, 74, 75, 78, 81, 88, 90, 91, 92, 94, 97, 98, 100, 103, 104, 108, 109, 114, 118, 119, 121, 123, 124, 125, 127, 128, 129, 130, 131, 133, 135, 136, 137

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

By following the definition outlined in A092861, one can multiply a monotonic sequence by itself, thus squaring it.

Ferenc Adorjan, Binary mapping of monotonic sequences and the Aronson function

Cf. A092855, A051006, A092857, A092858, A092859, A092860, A092861, A092863, A092874.

{pow(a,n)= /*Returns the "n-th power" of monotonic sequence a */ return(mtinv(mt(a)^n)) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ }

A092863 Prime sequence to the power Pi.

Original entry on oeis.org

4, 7, 10, 16, 18, 20, 22, 23, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 40, 42, 46, 51, 57, 60, 65, 66, 67, 68, 69, 70, 72, 73, 74, 77, 78, 80, 81, 82, 84, 85, 89, 91, 92, 93, 94, 95, 99, 101, 103, 107, 108, 110, 111, 112, 115, 117, 122, 123, 124, 125, 127, 128, 129, 130

Offset: 1

Ferenc Adorjan (fadorjan(AT)freemail.hu)

F. Adorjan, Binary mapping of monotonic sequences and the Aronson function

Cf. A092855, A051006, A092857, A092858, A092859, A092860, A092861, A092862, A092874.

{prow(a,r)= /*Returns the "r-th power" of monotonic sequence a */ return(mtinv(mt(a)^r)) /* the functions mt(a) and mtinv(r) are defined in A051006 and A092855, respectively */ }

A119523 Decimal expansion of 2^-1 + 2^-2 + 2^-4 + 2^-6 + 2^-10 + ..., where the exponents are 1 less than the primes.

Original entry on oeis.org

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

Offset: 0

Roger L. Bagula, May 27 2006

Decimal expansion of Sum_{k >= 1} A010051(k)/2^(k-1).
The primes have a larger measure than the composites as they dominate the lower integers.
The binary JIS function (as defined in A113829) for this constant (that we may call the van der Waerden-Ulam constant W) is given by the first differences of A000720, A000720(n+1) - A000720(n) = A010051(n+1) = JIS[W,2]. - Artur Jasinski, Jun 02 2008

			0.829365...

S. M. Ulam, Problems in Modern Mathematics, John Wiley and Sons, New York, 1960, page 54.

Cf. A000720, A010051, A061286, A113829, A119524 (measure of composites).

b = 0; Do[k = PrimePi[n + 1] - PrimePi[n]; b = b + k/2^n, {n, 1, 200}]; First[RealDigits[N[b, 200]]] (* Artur Jasinski, Jun 02 2008 *)

s=0;forprime(p=2,1000,s+=1.>>p);2*s \\ Charles R Greathouse IV, Apr 05 2012

Equals 2*A051006 = 1/2 + 1/4 + 1/16 + 1/64 + 1/1024 +1/4096 + 1/65536 + ... (see A061286).
Equals Sum_{k>=1} pi(k)/2^k, where pi(k) = A000720(k). - Amiram Eldar, Aug 11 2020
Equals 1 - A119524. - Antonio Graciá Llorente, Jan 14 2024

More terms from Peter Pein (petsie(AT)dordos.net), May 31 2006
Edited by N. J. A. Sloane, Nov 17 2006
Use of PrimePi in the first comment line corrected by R. J. Mathar, Oct 30 2010, Alonso Del Arte, Apr 05 2012
a(99) corrected by Sean A. Irvine, Jun 09 2024

A132797 Decimal expansion of Sum_{n >= 1} 1/5^prime(n).

Original entry on oeis.org

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

Offset: 0

Cino Hilliard, Nov 17 2007

Equivalently, the real number in (0,1) having the characteristic function of the primes, A010051, as its base-5 expansion. - M. F. Hasler, Jul 04 2017

			0.0483328213005632326916634712515665965227023410340158272294967746839279...

Cf. A000720, A051006 (analog for base 2), A057902, A132800 (analog for base 3), A132806 (analog for base 4), A010051 (characteristic function of the primes), A132817 (base 6).

/* Sum of 1/m^p for primes p */ sumnp(n,m) = { local(s=0,a,j); for(x=1,n, s+=1./m^prime(x); ); a=Vec(Str(s)); for(j=3,n, print1(eval(a[j])",") ) }

suminf(n=1, 1/5^prime(n)) \\ Then: digits(%\.1^default(realprecision))[1..-3] to remove the last 2 digits. N.B.: Functions sumpos() and sumnum() yield much less accurate results. - M. F. Hasler, Jul 04 2017

From Amiram Eldar, Aug 11 2020: (Start)
Equals Sum_{k>=1} 1/A057902(k).
Equals 4 * Sum_{k>=1} pi(k)/5^(k+1), where pi(k) = A000720(k). (End)

Offset corrected R. J. Mathar, Jan 26 2009

Edited by M. F. Hasler, Jul 04 2017

A132806 Decimal expansion of Sum_{n >= 1} 1/4^prime(n).

Original entry on oeis.org

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

Offset: 0

Cino Hilliard, Nov 17 2007

Equivalently, the real number in (0,1) having the characteristic function of the primes, A010051, as its base-4 expansion. - M. F. Hasler, Jul 04 2017

			0.079162851037850149671771117962208184613038569751878...

Cf. A000720, A051006 (analog for base 2), A132800 (analog for base 3), A132797 (analog for base 5), A010051 (characteristic function of the primes), A000040 (the primes).

/* Sum of 1/m^p for primes p */ sumnp(n,m) = { local(s=0,a,j); for(x=1,n, s+=1./m^prime(x); ); a=Vec(Str(s)); for(j=3,n, print1(eval(a[j])",") ) }

suminf(n=1, 1/4^prime(n)) \\ Then: digits(%\.1^default(realprecision))[1..-3] to remove the last 2 digits. N.B.: Functions sumpos() and sumnum() yield much less accurate results. - M. F. Hasler, Jul 04 2017

Equals 3 * Sum_{k>=1} pi(k)/4^(k+1), where pi(k) = A000720(k). - Amiram Eldar, Aug 11 2020

Offset corrected R. J. Mathar, Jan 26 2009

Edited by M. F. Hasler, Jul 04 2017

cp's OEIS Frontend

A132800 Decimal expansion of Sum_{n >= 1} 1/3^prime(n).

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A092858 "Sum" of the sequences of primes and the triangular numbers (A000217).

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A092859 "Difference" of the sequences of triangular numbers (A000217) and the primes (cf. A092858).

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A092860 "3 times the prime sequence".

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A092861 "Product" of the sequence of primes and the "evil" numbers (A001969).

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A092862 "Square" of the prime sequence.

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A092863 Prime sequence to the power Pi.

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A119523 Decimal expansion of 2^-1 + 2^-2 + 2^-4 + 2^-6 + 2^-10 + ..., where the exponents are 1 less than the primes.

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