A114850 -id:A114850 - cp's OEIS frontend

A140052 Indices m such that A114850(m)+A114850(k) is prime for some k

6, 9, 9, 19, 20, 25, 33, 38, 40, 59, 69, 76, 99, 111, 126, 141, 147, 167, 188, 202, 211, 211, 220, 238, 263, 264, 279, 284, 297, 329, 336, 354, 407, 407, 407, 410, 426, 540, 568, 652, 683, 696, 769, 780, 948, 951

Offset: 1

Jonathan Vos Post, May 03 2008

The associated primes b(n), which grow too quickly for many to be given as a sequence themselves, are {primes of the form A114850(a) + A114850(b)} = {primes of the form A114850(a) + A114850(b)} and begin as follows. b(1) = 437893890380859631 = 256 + 437893890380859375 = 4^4 + 15^15 = semiprime(1)^semiprime(1) + semiprime(6)^semiprime(6).
b(2) = 88817841970012523233890533447265881 = 256 + 88817841970012523233890533447265625 = 4^4 + 25^25 = semiprime(1)^semiprime(1) + semiprime(9)^semiprime(9).
b(3) = 46656 + 88817841970012523233890533447265625 = 6^6 + 24^25 = semiprime(2)^semiprime(2) + semiprime(9)^semiprime(9). This is to A068145 "Primes of the form a^a + b^b" as A001358 semiprimes is to A000040 primes; and as A114850 "(n-th semiprime)^(n-th semiprime)" is to A051674 "(n-th prime)^(n-th prime)."
M. F. Hasler gave the present definition which allows us to list merely the indices, which in the 3 examples above, are [6, 1],[9, 1],[9, 2]. The first 13 [m,k] value pairs are (as found by M. F. Hasler as an extension) are [6, 1], [9, 1], [9, 2], [19, 5], [20, 8], [25, 7], [33, 11], [38, 6], [40, 33], [59, 14], [69, 62], [76, 57], [99, 22]. Hence our sequence begins a(1) = 6, a(2) = 9, a(3) = 9. For the sequence of corresponding k values {1, 1, 2, 5, 8, ...}, see A140053.

			a(1) = 6 because semiprime(6)^semiprime(6) + semiprime(1)^semiprime(1) = 15^15 + 4^4 = 437893890380859375 + 256 = 437893890380859631 is prime.

Cf. A000040, A001358, A051674, A068145, A114850, A137701, A140053.

? t=0;A001358=vector(100,i,until(bigomega(t++)==2,);t); ? for(i=1,#A001358, for(j=1,i-1, ispseudoprime(A001358[i]^A001358[i]+A001358[j]^A001358[j]) | next; print1([i,j]",")))

A001358(a(n))^A001358(a(n)) + A001358(A140053(n))^A001358(A140053(n)) is prime.

a(14)-a(46) from Donovan Johnson, Nov 11 2008

A140053 Indices k such that A114850(m)+A114850(k) is prime for some m>k.

Original entry on oeis.org

1, 1, 2, 5, 8, 7, 11, 6, 33, 14, 62, 57, 22, 7, 86, 61, 28, 70, 66, 134, 77, 131, 107, 58, 161, 252, 240, 52, 155, 32, 152, 322, 167, 200, 284, 258, 28, 173, 95, 563, 369, 57, 58, 126, 113, 369

Offset: 1

Jonathan Vos Post, May 03 2008

The associated primes b(n), which grow too quickly for many to be given as a sequence themselves, are {primes of the form A114850(a) + A114850(b)} = {primes of the form A114850(a) + A114850(b)} and begin as follows. b(1) = 437893890380859631 = 256 + 437893890380859375 = 4^4 + 15^15 = semiprime(1)^semiprime(1) + semiprime(6)^semiprime(6).
b(2) = 88817841970012523233890533447265881 = 256 + 88817841970012523233890533447265625 = 4^4 + 25^25 = semiprime(1)^semiprime(1) + semiprime(9)^semiprime(9).
b(3) = 46656 + 88817841970012523233890533447265625 = 6^6 + 24^25 = semiprime(2)^semiprime(2) + semiprime(9)^semiprime(9). This is to A068145 "Primes of the form a^a + b^b" as A001358 semiprimes is to A000040 primes; and as A114850 "(n-th semiprime)^(n-th semiprime)" is to A051674 "(n-th prime)^(n-th prime)."
M. F. Hasler gave the present definition which allows us to list merely the indices, which in the 3 examples above, are [6, 1],[9, 1],[9, 2]. The first 13 [m,k] value pairs are (as found by M. F. Hasler as an extension) are [6, 1], [9, 1], [9, 2], [19, 5], [20, 8], [25, 7], [33, 11], [38, 6], [40, 33], [59, 14], [69, 62], [76, 57], [99, 22]. Hence our sequence begins a(1) = 6, a(2) = 9, a(3) = 9. For the sequence of corresponding k values {6, 9, 9, 19, 20, ...}, see A140052.

			a(1) = 1 because semiprime(6)^semiprime(6) + semiprime(1)^semiprime(1) = 15^15 + 4^4 = 437893890380859375 + 256 = 437893890380859631 is prime.

Cf. A000040, A001358, A051674, A068145, A114850, A137701, A140052.

? t=0;A001358=vector(100,i,until(bigomega(t++)==2,);t); ? for(i=1,#A001358, for(j=1,i-1, ispseudoprime(A001358[i]^A001358[i]+A001358[j]^A001358[j]) | next; print1([i,j]",")))

A001358(a(n))^A001358(a(n)) + A001358(A140052(n))^A001358(A140052(n)) is prime.

a(14)-a(46) from Donovan Johnson, Nov 11 2008

A137701 a(n) = semiprime(n)^prime(n).

Original entry on oeis.org

16, 216, 59049, 10000000, 4049565169664, 1946195068359375, 30041942495081691894741, 32064977213018365645815808, 142108547152020037174224853515625, 108199957741720996894562347292921981566976, 118558347188026655500106547231096910504441858017

Offset: 1

Jonathan Vos Post, Apr 27 2008

			a(1) = Semiprime(1)^prime(1) = 4^2 = 16.
a(2) = Semiprime(2)^prime(2) = 6^3 = 216.
a(3) = Semiprime(3)^prime(3) = 9^5 = 59049.
a(4) = Semiprime(4)^prime(4) = 10^7 = 10000000.

Cf. A000040, A001358, A051674, A114850, A111096.

a(n) = A001358(n)^A000040(n).

A114967 (n-th 3-almost prime)^(n-th 3-almost prime).

Original entry on oeis.org

16777216, 8916100448256, 39346408075296537575424, 104857600000000000000000000, 443426488243037769948249630619149892803, 33145523113253374862572728253364605812736

Offset: 1

Jonathan Vos Post, Feb 21 2006

3-almost prime analog of A051674. A114850 is semiprime analog of A051674.

			a(1) = A014612(1)^A014612(1) = 8^8 = 16777216 = 2^24.
a(2) = A014612(2)^A014612(2) = 12^12 = 8916100448256 = 2^24 * 3^12.
a(3) = A014612(3)^A014612(3) = 18^18 = 39346408075296537575424 = 2^18 * 3^36.

Harvey P. Dale, Table of n, a(n) for n = 1..92

Cf. A001358, A014612, A051674, A113877, A114850.

#^#&/@Select[Range[40],PrimeOmega[#]==3&] (* Harvey P. Dale, Apr 25 2015 *)

a(n) = A014612(n)^A014612(n).

A118055 Numerator of Sum_{i=1..n} 1/(s(i)^s(i)) where s(i) = i-th semiprime.

Original entry on oeis.org

1, 733, 389546509, 15216660895232989, 165124648173861912289213141201, 516014525543318775927975356319557, 11473924061057077116469420939475877122177

Offset: 1

Jonathan Vos Post, Apr 11 2006

Semiprime analog of A117579. Fractions are 1/256, 733/186624, 389546509/99179645184, 15216660895232989/3874204890000000000, 165124648173861912289213141201/42041202325478752505760000000000, 516014525543318775927975356319557/131378757267121101580500000000000000, 11473924061057077116469420939475877122177 / 2921293509192991260690562210500000000000000, 239106294995420151295311285049507497083520504633431021289373163777 / 6087713879404511830817263262876196035025072.

			a(2) = 733 because (1/semiprime(1)^semiprime(1)) + (1/semiprime(2)^semiprime(2))
= (1/256) + (1/46656) = 733/186624.

Denominators = A118055. Cf. A001358, A051674, A114850, A117579.

Numerator[Accumulate[1/#^#&/@Select[Range[25],PrimeOmega[#]==2&]]] (* Harvey P. Dale, Aug 09 2012 *)

a(n) = Numerator of Sum_{i=1..n} 1/(semiprime(i)^semiprime(i)).
a(n) = Numerator of Sum_{i=1..n} 1/(A001358(i)^A001358(i)).
a(n) = Numerator of Sum_{i=1..n} 1/A114850(n).

A118056 Denominator of Sum_{i=1..n} 1/(s(i)^s(i)) where s(i) = i-th semiprime.

Original entry on oeis.org

256, 186624, 99179645184, 3874204890000000000, 42041202325478752505760000000000, 131378757267121101580500000000000000, 2921293509192991260690562210500000000000000, 60877138794045118308172632628761960350250724033554048000000000000000

Offset: 1

Jonathan Vos Post, Apr 11 2006

Semiprime analog of A076265. Fractions are 1/256, 733/186624, 389546509/99179645184, 15216660895232989/3874204890000000000, 165124648173861912289213141201/42041202325478752505760000000000, 516014525543318775927975356319557/131378757267121101580500000000000000, 11473924061057077116469420939475877122177 / 2921293509192991260690562210500000000000000, 239106294995420151295311285049507497083520504633431021289373163777 / 60877138794045118308172632628761960350250724033554048000000000000000.

			a(2) = 186624 because (1/semiprime(1)^semiprime(1)) + (1/semiprime(2)^semiprime(2))= (1/256) + (1/46656) = 733/186624.

Numerators = A118055. Cf. A001358, A051674, A114850, A117579.

Denominator[Accumulate[1/#^#&/@Select[Range[30],PrimeOmega[#]==2&]]] (* Harvey P. Dale, Feb 15 2012 *)

a(n) = Denominator of Sum_{i=1..n} 1/(semiprime(i)^semiprime(i)).
a(n) = Denominator of Sum_{i=1..n} 1/(A001358(i)^A001358(i)).
a(n) = Denominator of Sum_{i=1..n} 1/A114850(n).

Corrected by Harvey P. Dale, Feb 15 2012

A114993 (n-th 4-almost prime)^(n-th 4-almost prime).

Original entry on oeis.org

18446744073709551616, 1333735776850284124449081472843776, 106387358923716524807713475752456393740167855629859291136, 12089258196146291747061760000000000000000000000000000000000000000

Offset: 1

Jonathan Vos Post, Feb 22 2006

4-almost prime analog of A051674. A114850 is semiprime analog of A051674. a(5) = 56^56 has 94 digits.

			a(1) = A014613(1)^A014613(1) = 16^16 = 18446744073709551616 = 2^64.
a(2) = A014613(2)^A014613(2) = 24^24 = 1333735776850284124449081472843776 = 2^72 * 3^24.
a(3) = A014613(3)^A014613(3) = 36^36 = 2^72 * 3^72.

Cf. A001358, A014613, A051674, A113877, A114850.

a(n) = A014613(n)^A014613(n).

A118062 Numerator of Sum_{i=1..n} 1/(t(i)^t(i)) where t(i) = i-th 3-almost prime.

Original entry on oeis.org

1, 265721, 75047458863267833, 938093235790847912650094635296999121, 2771420766426289313598405374054613260285749630619149892803, 83546357082134777747819786589906868700938637689935705237433756853637190925073724793683

Offset: 1

Jonathan Vos Post, Apr 11 2006

3-almost prime analog of A117579. Semiprime analog of A117579 is A118056. Fractions are 1/16777216, 265721/4458050224128, 75047458863267833/1259085058409489202413568, 938093235790847912650094635296999121 / 15738563230118615030169600000000000000000000, 2771420766426289313598405374054613260285749630619149892803 / 46496637333593157266125580467610571799579852800000000000000000000.

			a(2) = 265721 because (1/A014612(1)^A014612(1)) + (1/A014612(2)^A014612(2))= (1/(8^8)) + (1/(12^12)) = (1/16777216) + (1/8916100448256) = 265721/4458050224128.

Denominators = A118063. Cf. A001358, A014612, A051674, A114850, A114967, A117579, A118056.

a(n) = Numerator of Sum_{i=1..n} 1/(3almostprime(i)^3almostprime(i)).
a(n) = Numerator of Sum_{i=1..n} 1/(A014612(i)^A014612(i)).
a(n) = Numerator of Sum_{i=1..n} 1/A114967(n).

A118063 Denominator of Sum_{i=1..n} 1/(t(i)^t(i)) where t(i) = i-th 3-almost prime.

Original entry on oeis.org

16777216, 4458050224128, 1259085058409489202413568, 15738563230118615030169600000000000000000000, 46496637333593157266125580467610571799579852800000000000000000000

Offset: 1

Jonathan Vos Post, Apr 11 2006

3-almost prime analog of A076265. (Semiprime analog of A076265 is A118056.) Fractions are 1/16777216, 265721/4458050224128, 75047458863267833/1259085058409489202413568, 938093235790847912650094635296999121 / 15738563230118615030169600000000000000000000, 2771420766426289313598405374054613260285749630619149892803 / 46496637333593157266125580467610571799579852800000000000000000000.

			a(2) = 4458050224128 because (1/A014612(1)^A014612(1)) + (1/A014612(2)^A014612(2))= (1/(8^8)) + (1/(12^12)) = (1/16777216) + (1/8916100448256) = 265721/4458050224128.

Numerators = A118062. Cf. A001358, A014612, A051674, A114850, A114967, A117579, A118056.

Accumulate[1/#^#&/@Select[Range[30],PrimeOmega[#]==3&]]//Denominator (* Harvey P. Dale, Apr 05 2020 *)

a(n) = Denominator of Sum_{i=1..n} 1/(3almostprime(i)^3almostprime(i)).
a(n) = Denominator of Sum_{i=1..n} 1/(A014612(i)^A014612(i)).
a(n) = Denominator of Sum_{i=1..n} 1/A114967(n).

cp's OEIS Frontend

A140052 Indices m such that A114850(m)+A114850(k) is prime for some k

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A140053 Indices k such that A114850(m)+A114850(k) is prime for some m>k.

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A137701 a(n) = semiprime(n)^prime(n).

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A114967 (n-th 3-almost prime)^(n-th 3-almost prime).

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A118055 Numerator of Sum_{i=1..n} 1/(s(i)^s(i)) where s(i) = i-th semiprime.

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A118056 Denominator of Sum_{i=1..n} 1/(s(i)^s(i)) where s(i) = i-th semiprime.

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A114993 (n-th 4-almost prime)^(n-th 4-almost prime).

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A118062 Numerator of Sum_{i=1..n} 1/(t(i)^t(i)) where t(i) = i-th 3-almost prime.

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A118063 Denominator of Sum_{i=1..n} 1/(t(i)^t(i)) where t(i) = i-th 3-almost prime.

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