A333242 -id:A333242 - cp's OEIS frontend

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

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

A262275 Prime numbers with an even number of steps in their prime index chain.

Original entry on oeis.org

3, 11, 17, 41, 67, 83, 109, 127, 157, 191, 211, 241, 277, 283, 353, 367, 401, 461, 509, 547, 563, 587, 617, 739, 773, 797, 859, 877, 967, 991, 1031, 1063, 1087, 1171, 1201, 1217, 1409, 1433, 1447, 1471, 1499, 1597, 1621, 1669, 1723, 1741, 1823, 1913, 2027, 2063, 2081, 2099, 2221, 2269, 2341, 2351

Offset: 1

Zak Seidov and Robert G. Wilson v, Sep 17 2015

nonn

Old (incorrect) name was: Primes not appearing in A121543.

Number of terms less than 10^n: 1, 6, 30, 165, 1024, ... .

			11 is a term: 11 -> 5 -> 3 -> 2 -> 1, four (an even number of) steps "->" = pi = A000720.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1025 terms from Zak Seidov and Robert G. Wilson v)
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, Application of the Inclusion-Exclusion Principle to Prime Number Subsequences, arXiv:2402.13214 [math.GM], 2024.

Cf. A000040, A000720, A078442, A121543, A333242 (complement in primes).

b:= proc(n) option remember;
       `if`(isprime(n), 1+b(numtheory[pi](n)), 0)
    end:
a:= proc(n) option remember; local p; p:= a(n-1);
      do p:= nextprime(p);
         if b(p)::even then break fi
      od; p
    end: a(1):=3:
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 15 2020

fQ[n_] := If[ !PrimeQ[n] || (PrimeQ[n] && FreeQ[lst, PrimePi[n]]), AppendTo[lst, n]]; k = 2; lst = {1}; While[k < 2401, fQ@ k; k++]; Select[lst, PrimeQ]

b(n)={my(k=0); while(isprime(n), k++; n=primepi(n)); k};
apply(prime, select(n->b(n)%2, [1..500])) \\ Michel Marcus, Jan 03 2022; after A333242

From Alois P. Heinz, Mar 15 2020: (Start)
{ p in primes : A078442(p) mod 2 = 0 }.
a(n) = prime(A333242(n)). (End)

New name from Alois P. Heinz, Mar 15 2020

A333243 Prime numbers with prime indices in A262275.

Original entry on oeis.org

5, 31, 59, 179, 331, 431, 599, 709, 919, 1153, 1297, 1523, 1787, 1847, 2381, 2477, 2749, 3259, 3637, 3943, 4091, 4273, 4549, 5623, 5869, 6113, 6661, 6823, 7607, 7841, 8221, 8527, 8719, 9461, 9739, 9859, 11743, 11953, 12097, 12301, 12547, 13469, 13709, 14177

Offset: 1

Michael P. May, Mar 12 2020

nonn

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

			a(1) = prime(A262275(1)) = prime(3) = 5.

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, A333244.

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>1 and h::odd)(b(p)) then break fi
      od; p
    end:
seq(a(n), n=1..50);  # 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[#>1 && OddQ[#]&[b[p]], Break[]]]; p];
Array[a, 50] (* 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(x)), select(n->b(n)%2, [1..500])) \\ Michel Marcus, Nov 18 2022

a(n) = prime(A262275(n)).

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)).

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

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

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)).

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

cp's OEIS Frontend

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

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A262275 Prime numbers with an even number of steps in their prime index chain.

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A333243 Prime numbers with prime indices in A262275.

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A333244 Prime numbers with prime indices in A333243.

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

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

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A358179 Prime numbers with prime indices in A333244.

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