A112037 -id:A112037 - cp's OEIS frontend

A114461 Position in A112037 where the n-th prime appears.

1, 2, 3, 5, 4, 7, 10, 16, 6, 8, 22, 12, 9, 14, 20, 11, 41, 26, 19, 35, 21, 23, 13, 15, 28, 37, 38, 39, 53, 17, 32, 18, 45, 33, 56, 49, 68, 40, 92, 24, 25, 52, 27, 43, 123, 44, 87, 98, 177, 100, 29, 30, 63, 31, 67, 69, 88, 71, 54, 34, 73, 36, 57, 76, 77, 252, 80, 118, 85, 121, 165

Offset: 1

Robert G. Wilson v, Nov 29 2005

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A112037, A112038.

lst = {}; r[n_] := (len = Length@lst; lst = Flatten@ Join[lst, Select[First /@ FactorInteger[Prime@n - 1], ! MemberQ[lst, # ] &]]; If[l < Length@lst, 1, 0]); Do[ r[n], {n, 260}]; Table[Position[lst, Prime[n]], {n, 71}] // Flatten

A112038 The p values which produce new terms in A112037.

Original entry on oeis.org

3, 7, 11, 23, 29, 47, 53, 59, 83, 103, 107, 149, 167, 173, 179, 191, 227, 263, 269, 283, 293, 311, 317, 347, 359, 367, 383, 389, 467, 479, 503, 509, 557, 563, 569, 587, 607, 619, 643, 653, 709, 719, 773, 797, 823, 839, 863, 887, 907, 983, 1019, 1087, 1091

Offset: 1

Michel Dauchez (mdzdm(AT)yahoo.fr), Nov 28 2005

The p values which do not produce new terms in A112037 are given by A061303. - Ray Chandler, Nov 30 2005

Cf. A112037, A114461.

lst = {}; r[n_] := (len = Length@lst; lst = Flatten@ Join[lst, Select[First /@ FactorInteger[Prime@n - 1], ! MemberQ[lst, # ] &]]; If[l < Length@lst, 1, 0]); Prime /@ Select[Range@185, r[ # ] == 1 &] (* Robert G. Wilson v *)

Better description from Jack Brennen, Nov 28 2005

Extended by Ray Chandler and Robert G. Wilson v, Nov 30 2005

A038111 Denominator of density of integers with smallest prime factor prime(n).

Original entry on oeis.org

2, 6, 15, 105, 385, 1001, 17017, 323323, 7436429, 19605131, 86822723, 3212440751, 131710070791, 5663533044013, 266186053068611, 613385252723321, 2783825377744303, 5855632691117327, 392327390304860909, 27855244711645124539, 2033432863950094091347, 160641196252057433216413

Offset: 1

Wouter Meeussen

Denominator of (Product_{k=1..n-1} (1 - 1/prime(k)))/prime(n). - Vladimir Shevelev, Jan 09 2015

a(n)/a(n-1) = prime(n)/q(n) where q(n) is 1 or a prime for all n < 1000. What are the first indices for which q(n) is composite? - M. F. Hasler, Dec 04 2018

			From _M. F. Hasler_, Dec 03 2018: (Start)
The density of the even numbers is 1/2, thus a(1) = 2.
The density of the numbers divisible by 3 but not by 2 is 1/6, thus a(2) = 6.
The density of multiples of 5 not divisible by 2 or 3 is 2/30, thus a(3) = 15. (End)

Robert Israel, Table of n, a(n) for n = 1..277
Fred Kline and Gerry Myerson, Identity for frequency of integers with smallest prime(n) divisor, Mathematics Stack Exchange, Jul 2014.
Vladimir Shevelev, Generalized Newman phenomena and digit conjectures on primes, Int'l J. Math. and Math. Sci. (2008) Art. ID 908045, 1-12. See Eq. (5.8).

Cf. A038110, A060753, A112037.

N:= 100: # for the first N terms
Q:= 1: p:= 1:
for n from 1 to N do
  p:= nextprime(p);
  A[n]:= denom(Q/p);
  Q:= Q * (1 - 1/p);
end:
seq(A[n],n=1..N); # Robert Israel, Jul 14 2014

Denominator@Table[ Product[ 1-1/Prime[ k ], {k, n-1} ]/Prime[ n ], {n, 1, 64} ]
(* Wouter Meeussen *)
Denominator@
Table[EulerPhi[Exp[Sum[MangoldtLambda[m], {m, 1, Prime[n] - 1}]]]/
Exp[Sum[MangoldtLambda[m], {m, 1, Prime[n]}]], {n, 1, 21}]
(* Fred Daniel Kline, Jul 14 2014 *)

apply( A038111(n)=denominator(prod(k=1,n-1,1-1/prime(k)))*prime(n), [1..30]) \\ M. F. Hasler, Dec 03 2018

a(n) = denominator of phi(e^(psi(p_n-1)))/e^(psi(p_n)), where psi(.) is the second Chebyshev function and phi(.) is Euler's totient function. - Fred Daniel Kline, Jul 17 2014
a(n) = prime(n)*A060753(n). - Vladimir Shevelev, Jan 10 2015
a(n) = a(n-1)*prime(n)/q(n), where q(n) = 1 except for q({3, 5, 6, 10, 11, 16, 17, 18, ...}) = (2, 3, 5, 11, 7, 23, 13, 29, ...), cf. A112037. - M. F. Hasler, Dec 03 2018

Name edited by M. F. Hasler, Dec 03 2018

A071350 Distinct values of A058250; these terms appear first at subscripts listed in A071349.

Original entry on oeis.org

1, 2, 6, 30, 330, 2310, 53130, 690690, 20030010, 821230410, 13960916970, 739928599410, 27377358178170, 2272320728788110, 97709791337888730, 8696171429072096970, 165227257152369842430, 18670680058217792194590

Offset: 1

Labos Elemer, May 21 2002

Michael De Vlieger, Table of n, a(n) for n = 1..341

Cf. A058250, A002110, A000010, A005867, A071349, A112037.

Prepend[FoldList[Times,DeleteDuplicates[Rest[Flatten[FactorInteger[#][[All, 1]]&/@(Prime[Range[100]]-1)]]]],1] (* Jamie Morken, Apr 27 2021 after Harvey P. Dale at A112037, May 26 2019 *)

f(n) = my(pr=prod(k=1, n, prime(k))); gcd(pr, eulerphi(pr)); \\ A058250
lista(nn) = Set(vector(nn, k, f(k))); \\ Michel Marcus, Apr 27 2021

a(n) = a(n-1) * A112037(n), n >= 2. - David A. Corneth, Apr 27 2021

A236388 Primes in order of first appearance among the prime factors of p+1 where p is a prime.

Original entry on oeis.org

3, 2, 7, 5, 19, 11, 31, 17, 37, 13, 23, 79, 41, 29, 97, 53, 43, 139, 47, 71, 157, 83, 59, 199, 67, 211, 229, 61, 131, 271, 137, 307, 103, 107, 109, 331, 337, 113, 173, 367, 379, 127, 197, 439, 227, 163, 499, 101, 73, 263, 547, 281, 577, 293, 601, 607

Offset: 1

Joseph L. Pe, Jan 24 2014

The first p+1 (p prime) is 2+1=3, so 3 is the first term of the sequence. The next is 3+1=4=2*2, and the prime 2 appears next, so it is the second term of the sequence. The next p+1 = 5+1 = 6 gives no new prime factor; neither do 7+1 = 8 and 11+1 = 12. 13+1 = 14 = 2*7 gives the new prime factor 7, so 7 is the third term of the sequence.

If "p+1" is changed to "p-1" we get A112037. - N. J. A. Sloane, Jan 24 2014

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

Cf. A112037.

DeleteDuplicates[ First /@ Flatten[FactorInteger[1 + Prime@Range@200], 1]] (* Giovanni Resta, Jan 24 2014 *)

A307805 a(n) = first position of prime(n) in A023503.

Original entry on oeis.org

2, 4, 5, 10, 9, 16, 27, 43, 15, 17, 64, 35, 23, 40, 61, 28, 127, 73, 57, 104, 62, 66, 39, 41, 77, 111, 114, 117, 182, 49, 97, 56, 143, 102, 196, 155, 248, 119, 346, 69, 72, 181, 76, 137, 497, 139, 318, 388, 721, 401, 91, 92, 229, 96, 243, 249, 325, 258, 186, 103

Offset: 1

Michael De Vlieger, Apr 29 2019

nonn

A023503(n) = A006530(A006093(n)).
Apparent permutation of A071349(n) apart from A071349(1) = 1.
Let i = a(n). Sorting prime(n) in order of increasing i yields A112037 = {2, 3, 5, 11, 7, 23, 13, 29, 41, ...}. The product of the first j terms of A112037 = A071350(j).

			a(1) = 2 since prime(1) = gpf(prime(2) - 1), i.e., 2 = gpf(2).
a(2) = 4 since prime(2) = gpf(prime(4) - 1), i.e., 3 = gpf(6).
a(3) = 5 since prime(3) = gpf(prime(5) - 1), i.e., 5 = gpf(10).
a(4) = 10 since prime(4) = gpf(prime(10) - 1), i.e., 7 = gpf(28).

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A006093, A006530, A023503, A071349, A112037.

With[{s = Array[FactorInteger[Prime@ # - 1][[-1, 1]] &, 1000]}, Reap[Do[If[FreeQ[s, #], Break[], Sow@ FirstPosition[s, #][[1]]] &@ Prime@ i, {i, Length@ s}]][[-1, -1]]]

{ a = vector(60); pr = primes(#a); u = 1; n = 1;
forprime (p=3, oo, n++; f=factor(p-1); g=setsearch(pr, f[#f~,1]);
if (g && !a[g], a[g]=n; while (a[u], print1 (a[u]", "); u++; if (u>#a, break (2))))) } \\ Rémy Sigrist, May 28 2019

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A114461 Position in A112037 where the n-th prime appears.

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A112038 The p values which produce new terms in A112037.

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A038111 Denominator of density of integers with smallest prime factor prime(n).

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A071350 Distinct values of A058250; these terms appear first at subscripts listed in A071349.

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A236388 Primes in order of first appearance among the prime factors of p+1 where p is a prime.

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A307805 a(n) = first position of prime(n) in A023503.

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