Michael P. May - cp's OEIS frontend

A380334 Decimal expansion of the smallest possible average gap for the prime number subsequence P" = A262275.

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

Offset: 1

Michael P. May, Jan 21 2025

			9.45193981344780631141075022470840...

Cf. A262275, A338460.

f[x_] := Log[x!]*{Log[x!]+1}-x*{x-1};
root = FindRoot[f[x]  == 0, {x, 3},WorkingPrecision -> 100];
y=root;
z=x/. y;
RealDigits[Log[z!]*{Log[z!]+1}]

my(y=solve(x=3, 4, lngamma(x+1)-x+1)); y*(y-1) \\ Michel Marcus, Jan 28 2025

Equals x*(x-1) where x = A338460.

A373497 If n is prime, a(n) = 1 if the number of steps in its prime index chain is odd, a(n) = -1 if the number of steps is even, and a(n) = 0 is n is composite or 1.

Original entry on oeis.org

0, 1, -1, 0, 1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0

Offset: 1

Michael P. May, Jun 06 2024

sign

Cf. A333242, A262275, A018252.

b(n)={my(k=0); while(isprime(n), k++; n=primepi(n)); k}; \\ A078442
a(n) = if ((n==1) || !isprime(n), return(0)); if (b(n)%2, 1, -1); \\ Michel Marcus, Jun 11 2024

a(n) = 1 iff n is in A333242.
a(n) = -1 iff n is in A262275.
a(n) = 0 iff n is in A018252.

More terms from Michel Marcus, Jun 11 2024

A373338 Characteristic function of A333242: a(n) = 1 if n is a term of A333242.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 1

Michael P. May, Jun 01 2024

nonn

This sequence is the result of applying the N-sieve to generate the prime number subsequence A333242 where 1 indicates a prime number chosen to be included in sequence A333242 and 0 indicates the prime numbers and composites not in A333242.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Michael P. May, Properties of Higher-Order Prime Number Sequences, Missouri J. Math. Sci. (2020) Vol. 32, No. 2, 158-170; and arXiv version, arXiv:2108.04662 [math.NT], 2021.
Index entries for characteristic functions
Index entries for sequences related to prime indices in the factorization of n

Cf. A333242, A262275, A348677, A078442.

Select[Prime@ Range@ 75, EvenQ@ Length@ NestWhileList[ PrimePi, #, PrimeQ] &]

A078442(n) = my(k=0); while(isprime(n), k++; n=primepi(n)); k;
a(n) = A078442(n) % 2; \\ Michel Marcus, Jun 15 2024

a(n) = A078442(n) mod 2

A358274 a(n) is the prime before A262275(n).

Original entry on oeis.org

2, 7, 13, 37, 61, 79, 107, 113, 151, 181, 199, 239, 271, 281, 349, 359, 397, 457, 503, 541, 557, 577, 613, 733, 769, 787, 857, 863, 953, 983, 1021, 1061, 1069, 1163, 1193, 1213, 1399, 1429, 1439, 1459, 1493, 1583, 1619, 1667, 1721, 1733, 1811, 1907, 2017, 2053

Offset: 1

Michael P. May, Nov 11 2022

nonn

The sum of the individual gaps formed by the subtraction of the next lower prime number from each prime in A262275 approximates the prime counting function at very large n.

			a(3) = A262275(3) - A348677(3) = 17 - 4 = 13.

Michael P. May, "Relationship Between the Prime Counting Function and a Unique Prime Number Sequence", accepted for publication in the March 2023 edition of the Missouri Journal of Mathematical Sciences.

Cf. A151799, A262275, A348677.

b(n) = {my(k=0); while(isprime(n), k++; n=primepi(n)); k};
apply(x->precprime(x-1), apply(prime, select(n->b(n)%2, [1..500]))) \\ Michel Marcus, Nov 18 2022

a(n) = A262275(n) - A348677(n).

a(n) = A151799(A262275(n)).

A358179 Prime numbers with prime indices in A333244.

Original entry on oeis.org

31, 709, 1787, 8527, 19577, 27457, 42043, 52711, 72727, 96797, 112129, 137077, 167449, 173867, 239489, 250751, 285191, 352007, 401519, 443419, 464939, 490643, 527623, 683873, 718807, 755387, 839483, 864013, 985151, 1021271, 1080923, 1128889, 1159901, 1278779, 1323503, 1342907, 1656649, 1693031

Offset: 1

Michael P. May, Nov 11 2022

nonn

This sequence can also be generated by the N-sieve.

			a(1) = prime(A333244(1)) = prime(11) = 31.

Michael P. May, Properties of Higher-Order Prime Number Sequences, Missouri J. Math. Sci. (2020) Vol. 32, No. 2, 158-170; and arXiv version, arXiv:2108.04662 [math.NT], 2021.

Cf. A078442, A262275, A333242, A333243, A333244.

b[n_] := b[n] = If[PrimeQ[n], 1+b[PrimePi[n]], 0];
a[n_] := a[n] = Module[{p}, p = If[n==1, 1, a[n-1]]; While[True, p = NextPrime[p]; If[#>3 && OddQ[#]&[b[p]], Break[]]]; p];
Array[a, 50]

b(n)={my(k=0); while(isprime(n), k++; n=primepi(n)); k};
apply(x->prime(prime(prime(prime(x)))), select(n->b(n)%2, [1..500])) \\ Michel Marcus, Nov 18 2022

a(n) = prime(A333244(n)).
a(n) = A049090(A333242(n)).
a(n) = A038580(A262275(n)).
a(n) = A006450(A333243(n)).

A348677 a(n) is the difference between A262275(n) and the next lower prime.

Original entry on oeis.org

1, 4, 4, 4, 6, 4, 2, 14, 6, 10, 12, 2, 6, 2, 4, 8, 4, 4, 6, 6, 6, 10, 4, 6, 4, 10, 2, 14, 14, 8, 10, 2, 18, 8, 8, 4, 10, 4, 8, 12, 6, 14, 2, 2, 2, 8, 12, 6, 10, 10, 12, 10, 8, 2, 2, 4, 6, 6, 16, 14, 6, 6, 2, 10, 6, 2, 8, 6, 20, 2, 8, 28, 6, 16, 2, 6, 2, 10, 6, 22, 4, 6, 4, 14, 6, 2

Offset: 1

Michael P. May, Oct 30 2021

nonn

This sequence can be used as an alternate method of approximating the prime-counting function pi(n).

			For n = 3, a(3) = 17 - 13 = 4.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Michael P. May, Properties of Higher-Order Prime Number Sequences, Missouri J. Math. Sci. (2020) Vol. 32, No. 2, 158-170; and arXiv version, arXiv:2108.04662 [math.NT], 2021.
Michael P. May, Approximating the Prime Counting Function via an Operation on a Unique Prime Number Subsequence, arXiv:2112.08941 [math.GM], 2021.
Michael P. May, Relationship Between the Prime-Counting Function and a Unique Prime Number Sequence, Missouri J. Math. Sci. (2023), Vol. 35, No. 1, 105-116.

Cf. A151799, A262275, A333242.

b:= proc(n) option remember;
       `if`(isprime(n), 1+b(numtheory[pi](n)), 0)
    end:
g:= proc(n) option remember; local p; p:= g(n-1);
      do p:= nextprime(p);
         if b(p)::even then break fi
      od; p
    end: g(1):=3:
a:= n-> (t-> t-prevprime(t))(g(n)):
seq(a(n), n=1..86);  # Alois P. Heinz, Jan 06 2022

fQ[n_]:=If[!PrimeQ[n]||(PrimeQ[n]&&FreeQ[lst,PrimePi[n]]),AppendTo[lst,n]];k=2;lst={1};While[k<10000000,fQ@k;k++];tab1=Select[lst,PrimeQ]
lowerP[n_]:=Module[{m}, m=n;While[!PrimeQ[m-1],m--]; m-1]
tab2=lowerP/@tab1
tab3=tab1-tab2

a(n) = p_p'(n) - p_(p'(n) - 1), where p' is a prime number in the sequence A333242, p_p' is a prime number with index in A333242 (forms the prime number sequence A262275), and p_(p'(n)-1) is a prime number which is the next lower prime than those in A262275.
a(n) = A001223(A000720(A262275(n)) - 1).
a(n) = A262275(n) - A151799(A262275(n)). - Alois P. Heinz, Jan 06 2022

A342574 Remainder of e when the Dirichlet series for e is converted to an infinite product (negated).

Original entry on oeis.org

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

Offset: 0

Michael P. May, Mar 27 2021

Remainder term when the Dirichlet series representation of e is converted to an infinite product via the Euler Product Formula method. The remainder can be calculated via the infinite series found at the top of page 2 of "A Non-Sieving Application ..." (see links) or by the much simpler method A001113*A282529-2.

			-0.925358356236040633370884166370763828049565...

Michael P. May, A Non-Sieving Application of the Euler Zeta Function, arXiv:1510.01549 [math.NT], 2015.

Cf. A001113 (e), A282529.

Equals A001113*A282529-2.

More digits from Alois P. Heinz, Mar 31 2021

A341826 Decimal expansion of the largest real root of Gamma(x+1) = e.

Original entry on oeis.org

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

Offset: 1

Michael P. May, Feb 20 2021

Decimal expansion of the root of Gamma(x+1)/log(Gamma(x+1)) = Gamma(x+1).

			2.3124408825391603702...

Cf. A001113.

RealDigits[x /. FindRoot[Gamma[x+1] - E, {x, 2}, WorkingPrecision -> 120], 10, 100][[1]] (* Amiram Eldar, May 29 2021 *)

solve(x=2, 3, gamma(x+1)/log(gamma(x+1))-gamma(x+1)) \\ Michel Marcus, Feb 21 2021

solve(x=2, 3, gamma(x+1) - exp(1)) \\ Michel Marcus, Feb 21 2021

{x| Gamma(x+1)/log(Gamma(x+1))=Gamma(x+1)}.

A338460 Decimal expansion of the largest real root of e^(x-1) = Gamma(x+1).

Original entry on oeis.org

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

Offset: 1

Michael P. May, Jan 31 2021

Decimal expansion of the constant value for which the average and the minimum prime gaps are equal for the prime number sequences of higher order P', P'', P''', and P'''' as represented by A333242, A262275, A333243 and A333244.

x-1 is the smallest average gap size for the set of all prime numbers P. - Michael P. May, Jan 26 2025

			3.61479370319252544738653662560345463353151659694750...

Cf. A333242, A262275, A333243, A333244.

Cf. A001113, A078335, A336763.

Digits:= 155:
fsolve(exp(x-1)=GAMMA(x+1), x=3..4);  # Alois P. Heinz, Feb 01 2021

RealDigits[x /. FindRoot[LogGamma[x + 1] - x + 1, {x, 3}, WorkingPrecision -> 110], 10, 105][[1]] (* Amiram Eldar, Feb 01 2021 *)

solve(x=3,4,lngamma(x+1)-x+1) \\ Hugo Pfoertner, Feb 01 2021

x| (log(x!))^n * (log(x!) + 1) = x * (x-1)^n, for n >= 0

A333244 Prime numbers with prime indices in A333243.

Original entry on oeis.org

11, 127, 277, 1063, 2221, 3001, 4397, 5381, 7193, 9319, 10631, 12763, 15299, 15823, 21179, 22093, 24859, 30133, 33967, 37217, 38833, 40819, 43651, 55351, 57943, 60647, 66851, 68639, 77431, 80071, 84347, 87803, 90023, 98519, 101701, 103069, 125113, 127643

Offset: 1

Michael P. May, Mar 12 2020

nonn

This sequence can also be generated by the N-sieve.

			a(1) = prime(A333243(1)) = prime(5) = 11.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael P. May, On the Properties of Special Prime Number Subsequences, arXiv:1608.08082 [math.GM], 2016-2020.
Michael P. May, Properties of Higher-Order Prime Number Sequences, Missouri J. Math. Sci. (2020) Vol. 32, No. 2, 158-170; and arXiv version, arXiv:2108.04662 [math.NT], 2021.

Cf. A078442, A262275, A333242, A333243.

b:= proc(n) option remember;
      `if`(isprime(n), 1+b(numtheory[pi](n)), 0)
    end:
a:= proc(n) option remember; local p;
      p:= `if`(n=1, 1, a(n-1));
      do p:= nextprime(p);
        if (h-> h>2 and h::even)(b(p)) then break fi
      od; p
    end:
seq(a(n), n=1..42);  # Alois P. Heinz, Mar 15 2020

b[n_] := b[n] = If[PrimeQ[n], 1+b[PrimePi[n]], 0];
a[n_] := a[n] = Module[{p}, p = If[n==1, 1, a[n-1]]; While[True, p = NextPrime[p]; If[#>2 && EvenQ[#]&[b[p]], Break[]]]; p];
Array[a, 42] (* Jean-François Alcover, Nov 16 2020, after Alois P. Heinz *)

b(n)={my(k=0); while(isprime(n), k++; n=primepi(n)); k};
apply(x->prime(prime(prime(x))), select(n->b(n)%2, [1..500])) \\ Michel Marcus, Nov 18 2022

a(n) = prime(A333243(n)).

cp's OEIS Frontend

User: Michael P. May

A380334 Decimal expansion of the smallest possible average gap for the prime number subsequence P" = A262275.

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A373497 If n is prime, a(n) = 1 if the number of steps in its prime index chain is odd, a(n) = -1 if the number of steps is even, and a(n) = 0 is n is composite or 1.

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A373338 Characteristic function of A333242: a(n) = 1 if n is a term of A333242.

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A358274 a(n) is the prime before A262275(n).

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A358179 Prime numbers with prime indices in A333244.

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A348677 a(n) is the difference between A262275(n) and the next lower prime.

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A342574 Remainder of e when the Dirichlet series for e is converted to an infinite product (negated).

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A341826 Decimal expansion of the largest real root of Gamma(x+1) = e.

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