Aldo Roberto Pessolano - cp's OEIS frontend

A355420 Integers whose third power is a digital permutation of a term in A007908.

1, 2326, 308344, 416308, 22330489, 23584549, 25262887, 100369113, 103697628, 112085871, 117764571, 123236271, 128235558, 480765411, 487901778, 492021537, 498423726, 507761406, 520620501, 552317646, 622410993, 2231515936, 2245722316, 2259865441, 2277355234

Offset: 1

Marco Ripà and Aldo Roberto Pessolano, Jul 01 2022

			2326 is a term since 2326^3 = 12584301976 is a digital permutation of 12345678910 = A007908(10).
308344 is a term since 308344^3 = 29316121031171584 is a permutation of the digits of A007908(13).

Marco Ripà, On some conjectures concerning perfect powers, arXiv:2205.10163 [math.GM], 2022.

Cf. A007908, A033307, A353025.

A307385 Decimal expansion of the constant S_2* = Sum_{j>=1} prime((2j + 1) - 1)!/prime((2j + 2) - 1)!.

Original entry on oeis.org

0, 4, 5, 2, 9, 4, 3, 4, 8, 8, 5, 0

Offset: 0

Marco Ripà and Aldo Roberto Pessolano, Apr 06 2019

The constant S_2* is related to the prime gaps, since twin primes produce the largest terms of the sum compared with neighboring terms.

On Apr 06 2019, the first 4200000000 prime numbers were used in order to calculate S_1* and S_2* and using Rosser's theorem we get: 0.04529434885014 < S_1* + S_2* < 0.04529434885035.

			S_2* = 0.045294348850...

Wikipedia, Rosser's theorem

Cf. A000040, A306658 (S_1) A306700 (S_2), A306744 (S_1 + S_2), A307383 (S_1* + S_2*), A307384 (S_1*).

S_2* = Sum_{j>=1} prime((2*j + 1) - 1)!/prime((2*j + 2) - 1)! = Sum_{j>=1} 1/(Product{k=prime(2*j + 1), prime((2*j + 2) - 1)} k) = 1/(5*6) + 1/(11*12) + 1/(17*18) + 1/(23*24*25*26*27*28) +...

A307384 Decimal expansion of the constant S_1* = Sum_{j>=1} prime((2j) - 1)!/prime((2j + 1) - 1)!.

Original entry on oeis.org

0, 8, 5, 1, 6, 1, 9, 1, 0, 9, 8, 5

Offset: 0

Marco Ripà and Aldo Roberto Pessolano, Apr 06 2019

Together with the constant S_2* and S_1* + S_2* (see A307383), S_1* involves the prime gaps, since twin primes produce the heaviest terms of the summation in comparison to their next and previous addend.

On Apr 06 2019, the first 4200000000 prime numbers were used and using Rosser's theorem we get: 0.08516191098523 < S_1* < 0.08516191098543.

			S_1* = 0.085161910985...

Wikipedia, Rosser's theorem

Cf. A000040, A306658 (S_1) A306700 (S_2), A306744 (S_1 + S_2), A307383 (S_1* + S_2*).

S_1* = Sum_{j>=1} prime(2*j - 1)!/prime((2*j + 1) - 1)! = Sum_{j>=1} 1/(Product{k=prime(2*j), prime(2*j + 1)} k) = 1/(3*4) + 1/(7*8*9*10) + 1/(13*14*15*16) + 1/(19*20*21*22) +...

A307383 Decimal expansion of the constant S_1* + S_2* = Sum_{j>=1} prime((j + 1) - 1)!/prime((j + 2) - 1)!.

Original entry on oeis.org

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

Offset: 0

Marco Ripà and Aldo Roberto Pessolano, Apr 06 2019

The constant S_1* + S_2* is related to the prime gaps, since twin primes produce the largest terms of the sum compared with neighboring terms.

On Apr 06 2019, the first 4200000000 prime numbers were used in order to calculate S_1* and S_2* and using Rosser's theorem we get: 0.13045626983537 < S_1* + S_2* < 0.13045626983578.

			S_1* + S_2* = 0.130456269835...

Cf. A000040, A306658 (S_1) A306700 (S_2), A306744 (S_1 + S_2).

S_1* + S_2* = Sum_{j>=1} prime((j + 1) - 1)!/prime((j + 2) - 1)! = Sum_{j>=1} 1/(Product{k=prime(j + 1), prime((j + 2) - 1)} k) = 1/(3*4) + 1/(5*6) + 1/(7*8*9*10) + 1/(11*12) + ...

A306780 Decimal expansion of the constant S_1 - S_2 = Sum_{j>=1} (-1)^(j+1)*(prime(j)!/prime(j + 1)!).

Original entry on oeis.org

3, 1, 5, 8, 8, 8, 8, 1, 9, 3, 5, 0

Offset: 0

Marco Ripà and Aldo Roberto Pessolano, Mar 09 2019

The constant S_1 - S_2 is related to the prime gaps, since twin primes produce the largest terms of the algebraic sum compared with neighboring terms.

			S_1 - S_2 = 0.315888819350...

Cf. A000040, A306658 (S_1), A306700 (S_2), A306744 (S_1 + S_2).

sumalt(j=1, (-1)^(j+1)*(prime(j)!/prime(j + 1)!)) \\ Michel Marcus, Apr 02 2019  \\ Needs default(realprecision, 10^4)  Jinyuan Wang, May 19 2019

S_1 - S_2 = Sum_{j>=1} (-1)^(j+1)*(prime(j)!/prime(j + 1)!) = Sum_{j>=2} (-1)^j/(Product{k=prime(j - 1) + 1, prime(j)} k) = 1/3 - 1/(4*5) + 1/(6*7) - 1/(8*9*10*11) + ...

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User: Aldo Roberto Pessolano

A355420 Integers whose third power is a digital permutation of a term in A007908.

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A307385 Decimal expansion of the constant S_2* = Sum_{j>=1} prime((2*j + 1) - 1)!/prime((2*j + 2) - 1)!.

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A307384 Decimal expansion of the constant S_1* = Sum_{j>=1} prime((2*j) - 1)!/prime((2*j + 1) - 1)!.

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A307383 Decimal expansion of the constant S_1* + S_2* = Sum_{j>=1} prime((j + 1) - 1)!/prime((j + 2) - 1)!.

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A306780 Decimal expansion of the constant S_1 - S_2 = Sum_{j>=1} (-1)^(j+1)*(prime(j)!/prime(j + 1)!).

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A307385 Decimal expansion of the constant S_2* = Sum_{j>=1} prime((2j + 1) - 1)!/prime((2j + 2) - 1)!.

A307384 Decimal expansion of the constant S_1* = Sum_{j>=1} prime((2j) - 1)!/prime((2j + 1) - 1)!.