A077648 -id:A077648 - cp's OEIS frontend

A376129 Run lengths of the most significant decimal digit in the primes (A077648).

1, 1, 1, 1, 4, 2, 2, 3, 2, 2, 3, 2, 1, 21, 16, 16, 17, 14, 16, 14, 15, 14, 135, 127, 120, 119, 114, 117, 107, 110, 112, 1033, 983, 958, 930, 924, 878, 902, 876, 879, 8392, 8013, 7863, 7678, 7560, 7445, 7408, 7323, 7224, 70435, 67883, 66330, 65367, 64336, 63799, 63129, 62712, 62090

Offset: 1

Stuart Coe, Sep 11 2024

			The primes and the run lengths of their initial digits begin
  primes   2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, ...
  runs                 \------------/  \----/  \----/
  lengths  1  1  1  1         4          2       2     ...

Cf. A000720, A037124, A077648.

from sympy import primepi
def A376129(n):
    if n<5: return 1
    def f(m): return (lambda x:primepi(10**x[0]*(x[1]+1)))(divmod(m,9))
    return int(f(n+5)-f(n+4)) # Chai Wah Wu, Oct 16 2024

a(n) = A000720(A037124(n+6)) - A000720(A037124(n+5)) for n >= 5. - Pontus von Brömssen, Oct 07 2024

A138840 Concatenation of initial and final digits of n-th prime.

Original entry on oeis.org

22, 33, 55, 77, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 11, 13, 17, 19, 13, 17, 11, 17, 19, 19, 11, 17, 13, 17, 13, 19, 11, 11, 13, 17, 19, 21, 23, 27, 29, 23, 29, 21, 21, 27, 23, 29, 21, 27, 21, 23, 23, 37, 31, 33, 37, 31, 37, 37

Offset: 1

Omar E. Pol, Apr 01 2008

There are only 38 distinct terms in this sequence, all of them odd with the exception of 22. 55 is the only term divisible by 5. 22 and 55 each appear only once. The other terms, each of which appears multiple times, are the odd two-digit numbers not divisible by 5. - Harvey P. Dale, May 15 2012

a(n) is the concatenation of A077648(n) and A007652(n), hence all terms of this sequence have two digits in the same way as A073729. - Omar E. Pol, Mar 23 2018

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A000040, A007652, A073729, A077648, A138841, A138842, A138843, A138844.

Cf. A137589 (same except for first four terms).

a:= n-> (p-> parse(cat(p[1], p[-1])))(""||(ithprime(n))):
seq(a(n), n=1..92);  # Alois P. Heinz, Nov 23 2023

cifd[n_]:=Module[{il=IntegerLength[n],idn=IntegerDigits[n]},Which[ il==1, 10n+n, il==2,n,il>2,FromDigits[Join[{First[idn],Last[idn]}]]]]; cifd/@ Prime[ Range[70]] (* Harvey P. Dale, May 15 2012 *)

a(n) = my(d=digits(prime(n))); fromdigits(concat(d[1], d[#d])); \\ Michel Marcus, Mar 23 2018

A135617 a(n) is the initial digit of n-th even perfect number.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Mar 01 2008

a(n) is also the initial digit of n-th perfect number A000396(n) if there are no odd perfect numbers.

			a(5) = 3 because the 5th even perfect number is 33550336 and the initial digit of 33550336 is 3.

Omar E. Pol, Los números perfectos, (in Spanish).

Cf. A000030, A000396, A077648, A094540, A133033.

lst = {* the list of terms in A000043 *}; f[n_] := Block[{pn = (2^n - 1) (2^(n - 1))}, Quotient[pn, 10^Floor[ Log[10, pn]] ]]; f@# & /@ lst (* Robert G. Wilson v, Apr 01 2008 *)

More terms from Robert G. Wilson v, Apr 01 2008
Definition clarified by Omar E. Pol, Apr 14 2018
a(40)-a(47) from Ivan Panchenko, Apr 16 2018
a(48) from Amiram Eldar, Oct 16 2024

A135613 Initial digit of Mersenne primes A000668.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Mar 01 2008

			a(4) = 1 because the 4th Mersenne prime A000668(4) is 127 and the initial digit of 127 is 1.

Landon Curt Noll, Mersenne Prime Digits.

Cf. A000030, A000668, A077648, A080172.

lst = {* the list of terms in A000043 *}; f[n_] := Block[{pn = 2^n - 1}, Quotient[pn, 10^Floor[ Log[10, pn]] ]]; f@# & /@ lst (* Robert G. Wilson v, Apr 01 2008 *)
IntegerDigits[#][[1]]&/@(2^#-1&/@MersennePrimeExponent[Range[47]]) (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 04 2019 *)

a(n) = A000030(A000668(n)). - Omar E. Pol, Jul 04 2019

More terms from Robert G. Wilson v, Apr 01 2008
a(40)-a(44) from David Radcliffe, Jan 21 2016
a(45)-a(47) from Ivan Panchenko, Apr 11 2018
a(48) from Amiram Eldar, Oct 16 2024

A138125 Final digit of n-th even superperfect number A061652(n).

Original entry on oeis.org

2, 4, 6, 4, 6, 6, 4, 4, 6, 6, 4, 4, 6, 4, 4, 4, 6, 6, 6, 4, 6, 6, 6, 6, 6, 6, 6, 4, 4, 6, 4, 4, 6, 4, 6, 6, 6, 6, 6, 4, 4, 4, 6, 6, 4, 6, 6

Offset: 1

Omar E. Pol, Apr 01 2008, corrected Apr 03 2008

Also, final digit of n-th superperfect number A019279(n), if there are no odd superperfect numbers.

			a(5)=6 because the 5th even superperfect number A061652(5) is 4096 and the final digit of 4096 is 6.
a(34)=4 because the final digit of 34th Mersenne prime is 7. a(39)=6 because the final digit of 39th Mersenne prime is 1.
.............................................................
............... SHORT TABLE OF FINAL DIGITS .................
.............................................................
Final digit of ..... Final digit of Even ..... Final digit of
Mersenne prime ..... Superperfect number ..... Perfect number
A000668 ............ A061652 ................. A000396........
(3) ................ (2) ..................... (6) ........... (For n=1, only)
(7) ................ (4) ..................... (8) ...........
(1) ................ (6) ..................... (6) ...........

Landon Curt Noll, Mersenne Prime Digits.

Cf. A000030, A000396, A000668, A077648, A019279, A061652, A135613, A135617, A138125.

Mod[#,10]&/@(2^(MersennePrimeExponent[Range[47]]-1)) (* Harvey P. Dale, Feb 23 2023 *)

a(1)=2. For n>1, if final digit of n-th Mersenne prime A000668(n) is equal to 1 then a(n)=6, otherwise a(n)=4.

a(40)-a(47) from Jinyuan Wang, Mar 14 2020

A166499 Concatenation of the rightmost digit of the n-th prime and the leftmost digit of the (n+1)th prime.

Original entry on oeis.org

23, 35, 57, 71, 11, 31, 71, 92, 32, 93, 13, 74, 14, 34, 75, 35, 96, 16, 77, 17, 37, 98, 38, 99, 71, 11, 31, 71, 91, 31, 71, 11, 71, 91, 91, 11, 71, 31, 71, 31, 91, 11, 11, 31, 71, 92, 12, 32, 72, 92, 32, 92, 12, 12, 72, 32, 92, 12, 72, 12, 32, 33, 73, 13, 33, 73, 13, 73, 73

Offset: 1

Zak Seidov, Oct 15 2009

This is the comma transform of the primes (see A367360).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A007652, A077648, A367360.

a:= n-> parse(cat(""||(ithprime(n))[-1],""||(ithprime(n+1))[1])):
seq(a(n), n=1..99);  # Alois P. Heinz, Nov 22 2023

With[{nmax=100},Map[10Mod[#[[1]],10]+IntegerDigits[#[[2]]][[1]]&,Partition[Prime[Range[nmax+1]],2,1]]] (* Paolo Xausa, Nov 24 2023 *)

a(n) = 10 * A007652(n) + A077648(n+1). - Alois P. Heinz, Nov 23 2023

A138124 Initial digit of n-th even superperfect number A061652(n).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol and Robert G. Wilson v, Apr 01 2008

Also, initial digit of n-th superperfect number A019279(n), if there are no odd superperfect numbers.

			a(5)=4 because the 5th even superperfect number A061652(5) is 4096 and the initial digit of 4096 is 4.

Cf. A000030, A077648, A019279, A061652, A135613, A135617, A138125.

lst = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917}; f[n_] := Block[{pn = 2^(n - 1)}, Quotient[pn, 10^Floor[Log[10, pn]]]]; f@# & /@ (* Robert G. Wilson v, Apr 01 2008 *)

a(13)-a(39) from Robert G. Wilson v, Apr 01 2008

A093339 Middle digits of primes with an odd number of digits.

Original entry on oeis.org

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

Offset: 2

Cino Hilliard, Apr 25 2004

Harvey P. Dale, Table of n, a(n) for n = 2..1000

Cf. A077648, A007652.

mdp[n_]:=Module[{idn=IntegerDigits[n],len=IntegerLength[n]},If[EvenQ[len], -1,idn[[(len-1)/2+1]]]]; Select[mdp/@Prime[Range[500]],NonNegative] (* Harvey P. Dale, Feb 26 2012 *)

midd(n) = { forprime(x=2,n, s = Str(x); ln = length(s); if(ln<2,p=1,p = (ln-1)/2+1); if(ln%2, md = eval(mid(s,p,1)); print1(md",") ) ) } \ Get a substring of length n from string str starting at position s in str. mid(str,s,n) = { v =""; tmp = Vec(str); ln=length(tmp); for(x=s,s+n-1, v=concat(v,tmp[x]); ); return(v) }

A093336 Second digit of prime(n).

Original entry on oeis.org

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

Offset: 5

Cino Hilliard, Apr 25 2004

Matthew House, Table of n, a(n) for n = 5..10000

Cf. A077648, A007652.

IntegerDigits[#][[2]]&/@Prime[Range[5,110]] (* Harvey P. Dale, Dec 22 2013 *)

second(n) = { forprime(x=11,n, sd = mid(Str(x),2,1); print1(sd",") ) } \ Get a substring of length n from string str starting at position s in str. mid(str,s,n) = { v =""; tmp = Vec(str); ln=length(tmp); for(x=s,s+n-1, v=concat(v,tmp[x]); ); return(v) }

Offset corrected by Matthew House, Aug 09 2015

A120843 Initial digit of the (10^n)-th prime.

Original entry on oeis.org

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

Offset: 0

Cino Hilliard, Aug 18 2006

The algorithm in the PARI program approximates the (10^n)-th prime to an accuracy of roughly n/2 + 1 digits. As a result, we are almost certain to get the initial digit correctly. It remains to prove this. Since the Riemann approximation of pi(x) is used as a boundary in the exponential bisection routine, it would seem that a proof is possible in view of the fact that bisection almost always guarantees convergence. "Almost" is an appropriate term here, as will be demonstrated when we let the initial parameter r2 = 1. For example, we can toggle print(dx) to check the convergence. For primex(1e116), we get 9.999999999999999999999999970 E115.

			The (10^3)-th prime is 7919, so a(3)=7.

Robert G. Wilson v, Table of n, a(n) for n = 0..1000

Cf. A000030, A006988, A077648.

f[n_] := RealDigits[ n (Log[n] + Log[Log[n]] - 1 + (Log[Log[n]] - 2)/Log[n] - (Log[Log[n]]^2 - 6 Log[Log[n]] + 11)/(2 Log[n]^2)), 10, 10][[1, 1]]; f[1] = f[10] = 2; f[100] = 5; Array[ f[10^#] &, 105, 0] (* Robert G. Wilson v, Jan 15 2017 *)

g(n) = print1(2", "); for(x=1, n, y=Vec(Str(primex(10^x))); print1(y[1]", "))
primex(n) = /* Efficient Algorithm to accurately approximate the n-th prime */ { local(x, px, r1, r2, r, p10, b, e); b=10; /*Select base*/ p10=log(n)/log(10); /*p10=pow of 10 n is to adjust in b^p10*/ if(Rg(b^p10*log(b^(p10+1)))< b^p10, m=p10+1, m=p10); r1 = 0; r2 = 2.718281828; /*Real kicker. if 1, it fails at 1e117*/ for(x=1, 360, r=(r1+r2)/2; px = Rg(b^p10*log(b^(m+r))); if(px <= b^p10, r1=r, r2=r); r=(r1+r2)/2.; ); (b^p10*log(b^(m+r))+.5); }
Rg(x) = /* Gram's Riemann's Approx of Pi(x) */{ local(n=1, L, s=1, r); L=r=log(x); while(s<10^100*r, s=s+r/zeta(n+1)/n; n=n+1; r=r*L/n); (s) }

a(n) = most significant digit of A006988(n). - Robert G. Wilson v, Jan 17 2017

a(n) = A000030(A006988(n)). - Michel Marcus, Jan 18 2017

cp's OEIS Frontend

A376129 Run lengths of the most significant decimal digit in the primes (A077648).

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A138840 Concatenation of initial and final digits of n-th prime.

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PARI

A135617 a(n) is the initial digit of n-th even perfect number.

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A135613 Initial digit of Mersenne primes A000668.

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A138125 Final digit of n-th even superperfect number A061652(n).

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A166499 Concatenation of the rightmost digit of the n-th prime and the leftmost digit of the (n+1)th prime.

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A138124 Initial digit of n-th even superperfect number A061652(n).

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A093339 Middle digits of primes with an odd number of digits.

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