A014545 -id:A014545 - cp's OEIS frontend

A376416 a(n) = A276085(A006862(n)), where A276085 is the primorial base log-function, and A006862 is the Euclid numbers, one more than primorials.

1, 2, 30, 6469693230, 7799922041683461553249199106329813876687996789903550945093032474868511536164700810

Offset: 0

Antti Karttunen, Nov 17 2024

nonn

Numbers k such that when we apply primorial base exp function (A276086) twice to them, the results are squarefree even semiprimes, A100484 after its initial 4. See comments in A377871.
a(5)..a(8) have 976, 209, 111, 12051 decimal digits.
a(n) is a primorial for those n that are in A014545, that is, when A006862(n) is one of the primorial primes, A018239.

Cf. A002110, A006862, A014545, A018239, A100484, A276085, A276086, A276087, A377871.

A276085(n) = { my(f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= prime(i); i++); s += f[k, 2]*pr); (s); };
A376416(n) = A276085(1+prod(i=1,n,prime(i)));

a(n) = A276085(1+A002110(n)) = A276085(A276085(A100484(1+n))).

For n >= 1, A276087(a(n)) = A100484(1+n).

A088411 A088257 indexed by A002110.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 11, 13, 24, 66, 68, 75, 167, 171, 172, 287, 310, 352, 384, 457, 564, 590, 616, 620, 643, 849, 1391, 1552, 1613, 1849, 2122, 2647, 2673, 4413, 13494, 31260, 33237, 67132, 85586, 234725

Offset: 1

Ray Chandler, Sep 29 2003

Union of A057704 and A014545. - Jeppe Stig Nielsen, Aug 01 2019

			3 is in the sequence because primorial p_3# = 2 * 3 * 5 = 30 has two prime neighbors 29 and 31.
4 is in the sequence because primorial p_4# = 2 * 3 * 5 * 7 = 210 has one prime neighbor 211; 209 = 11 * 19.
7 is not in the sequence because the product of the smallest 7 primes has two composite neighbors.

Cf. A002110, A088257, A014545, A057704.

A:= NULL:
P:= 1: p:= 1;
for n from 1 to 700 do
  p:= nextprime(p);
  P:= P*p;
  if isprime(P+1) or isprime(P-1) then A:= A, n fi
od:
A; # Robert Israel, Aug 03 2016

Select[Range[0, 600], Total@ Boole@ PrimeQ@ {# - 1, # + 1} > 0 &@ Apply[Times, Prime@ Range@ #] &] (* Michael De Vlieger, Aug 03 2016 *)

is(k)=pr=prod(j=1,k,prime(j));ispseudoprime(pr-1)||ispseudoprime(pr+1) \\ Jeppe Stig Nielsen, Aug 01 2019

a(n)=k such that A088257(n)=A002110(k).

a(22)-a(27) from Michael De Vlieger, Aug 03 2016
a(28)-a(40) from Jeppe Stig Nielsen, Aug 01 2019
a(41) from Jeppe Stig Nielsen, Oct 19 2021

A103513 Primes of the form primorial(P(k))/2-2^n with minimal n, n>=0, k>=2.

Original entry on oeis.org

2, 13, 103, 1153, 15013, 255253, 4849843, 111546433, 3234846607, 100280245063, 3710369067401, 152125131763603, 6541380632280583, 307444891294245701, 16294579238595022363, 961380175077106319471, 58644190679703485491571

Offset: 1

Lei Zhou, Feb 15 2005

nonn

The Mathematica Program does not produce a(2). Conjecture: sequence is defined for all k>=2.

			P(2)/2=3, 3-2^0=2 is prime, so a(2)=2;
P(5)/2=1155, 1155-2^1=1153 is prime, so a(5)=1153;

Cf. A002110, A005234, A014545, A018239, A006794, A057704, A057705, A103154.

nmax = 2^8192; npd = 1; n = 2; npd = npd*Prime[n]; While[npd < nmax, tt = 2; cp = npd - tt; While[(cp > 1) && (! (PrimeQ[cp])), tt = tt*2; cp = npd - tt]; If[cp < 2, Print["*"], Print[cp]]; n = n + 1; npd = npd*Prime[n]]

A104877 Semiprimes of the form primorial(k) + 1.

Original entry on oeis.org

30031, 9699691, 223092871, 13082761331670031, 117288381359406970983271, 7858321551080267055879091, 40729680599249024150621323471, 267064515689275851355624017992791

Offset: 1

Jonathan Vos Post, Mar 28 2005

			6# + 1 = 2*3*5*7*11*13 + 1 = 30031 = 59 x 509.
8# + 1 = 2*3*5*7*11*13*17*19 + 1 = 9699691 = 347 x 27953.
9# + 1 = 2*3*5*7*11*13*17*19*23 + 1 = 223092871 = 317 x 703763.
14# + 1 = 2*3*5*7*11*13*17*19*23*29*31*37*41*43 + 1 = 13082761331670031 = 167 x 78339888213593.

Sebastian Martin Ruiz, A Result on Prime Numbers, Math. Gaz. 81, 269, 1997.
Eric Weisstein's World of Mathematics, Primorial.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358, A002110, A034386, A005234, A014545, A018239, A006794, A057704, A057705, A104876.

Bigomega[n_]:=Plus@@Last/@FactorInteger[n]; SemiprimeQ[n_]:=Bigomega[n]==2; Primorial[n_]:=Product[Prime[i], {i, n}]; Select[Table[Primorial[n]+1, {n, 30}], SemiprimeQ] (* Ray Chandler, Mar 28 2005 *)
Select[FoldList[Times,Prime[Range[30]]]+1,PrimeOmega[#]==2&] (* Harvey P. Dale, Oct 13 2022 *)

n# + 1 iff semiprime. Equals {A002110(i) + 1} intersection {A001358(j)}.

A114432 Primes of the form 1 + product of the first k 4-almost primes A014613.

Original entry on oeis.org

17, 8126654054401

Offset: 1

Jonathan Vos Post, Feb 13 2006

nonn

The next term is too large to display here. - N. J. A. Sloane, Jul 30 2009
4-almost prime analog of primorial primes A005234 as indexed by A014545. In that sense, this sequence is indexed by (1, 8, ...). No more through product of first 16 of the 4-almost primes.
Terms are one more than the products of 4-almost primes up to 16, 81, 294, 513, 825, 1356, 1612, 2004, 2756, 7714, ... - Charles R Greathouse IV, Jul 28 2009

			a(1) = 17 because 1 + 16 = 1 + A014613(1) = 1 more than the first 4-almost prime is itself prime.
a(2) = 8126654054401 = 1 + (16 * 24 * 36 * 40 * 54 * 56 * 60 * 81) = 1 more than the product of the first 8 of the 4-almost primes and is prime.

Amiram Eldar, Table of n, a(n) for n = 1..8

Cf. A000040, A014613, A005234, A014545.

Select[FoldList[Times, Select[Range[200], PrimeOmega[#] == 4 &]] + 1, PrimeQ] (* Amiram Eldar, Jul 20 2025 *)

{1 + Product_{i=1..k} A014613(i)} INTERSECTION A000040.

A191623 Primes of the form 1 + Product_{k=1..n} prime(k)^(2^(k-1)).

Original entry on oeis.org

3, 19, 11251, 2980024297506569894680811251

Offset: 1

Jonathan Vos Post, Jun 09 2011

nonn

Primes of the form 1 + A191554(k), associated with positions k = 1, 2, 3, and 5 there. The next one (if it exists) occurs at k >= 15 and has > 53500 digits. [Edited by R. J. Mathar, Jun 17 2011 and Joerg Arndt, Jun 21 2011]

This connects A191554 and A191555, which are deeply about primes and monic polynomial irreducible by Eisenstein's Criterion, to primes by another way, connecting additive and multiplicative number theory analogously to the relationship in Primorial primes: A014545, n such that n-th Euclid number (A006862(n)) = 1 + (Product of first n primes) is prime.

			a(1) = 1 + 2^1 = 1 + 2 = 3 is prime.
a(2) = 1 + (2^1 * 3^2) = 1 + 18 = 19 is prime.
a(3) = 1 + (2^1 * 3^2 * 5^4) = 1 + 11250 = 11251 is prime.

Cf. A000040, A191554.

Select[Array[1 + Product[Prime[k]^(2^(k - 1)), {k, #}] &, 5], PrimeQ] (* Michael De Vlieger, Feb 15 2020 *)

A224081 Numbers k such that A112141(k) + 1 is prime.

Original entry on oeis.org

1, 4, 5, 6, 11, 12, 18, 39, 51, 98, 124, 170, 179, 208, 248, 838, 919, 939, 1233, 1352, 2177, 3070, 10714

Offset: 1

Jonathan Vos Post and Robert G. Wilson v, Apr 02 2013

This is the semiprime analog to A014545.

			4*6*9*10+1 = 2161 which is prime.

Cf. A112141, A014545, A224082.

NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; f[n_] := Times @@ NestList[NextSemiPrime, 2^2, n - 1]; k = 1; lst = {}; While[k < 4651, If[ PrimeQ[f[k] + 1], Print[k]; AppendTo[lst, k]]; k++]; lst

a(23) from Tyler Busby, Feb 12 2023

A289125 Numbers n such that phi(n)/phi(phi(n)) > phi(m)/phi(phi(m)) for all m < n.

Original entry on oeis.org

1, 3, 7, 31, 211, 2311, 43891, 60653, 870871, 1023053, 13123111, 19417793, 300690391, 446235509, 6915878971, 12939711677, 200560490131

Offset: 1

Amiram Eldar, Jun 25 2017

Erdős et al. proved that phi(n)/phi(phi(n)) is unbounded, thus this sequence is infinite.

A018239(k) = A002110(A014545(k)) + 1 is a term for k > 1. Are there terms m with omega(m) > 2? Is omega(phi(a(n + 1))) >= omega(phi(a(n)))? - David A. Corneth, Jun 28 2017

Paul Erdős, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdos, Andrew Granville, Carl Pomerance and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]

Cf. A000010, A001221, A002110, A010554, A014545, A018239.

a = {}; k=1; rmax = 0; While[Length[a]<10,s = EulerPhi[ k]; s2 = EulerPhi[ s]; r = s/s2;  If[r > rmax, AppendTo[a, k]; rmax = r]; k++]; a
DeleteDuplicates[Table[{n,EulerPhi[n]/EulerPhi[EulerPhi[n]]},{n,11*10^5}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* The program generates the first ten terms of the sequence. *) (* Harvey P. Dale, Aug 17 2024 *)

r=0; forfactored(n=1,10^10, t=eulerphi(n); t/=eulerphi(t); if(t>r, r=t; print1(n[1]", "))) \\ Charles R Greathouse IV, Jun 25 2017

a(15)-a(17) from Giovanni Resta, Jul 01 2017

A295741 a(n) is smallest divisor d of the n-th primorial such that d + prime(n)#/d is prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 5, 6, 10, 2, 1, 2, 11, 7, 2, 41, 2, 5, 5, 5, 3, 51, 10, 3, 14, 37, 10, 7, 2, 17, 17, 62, 21, 3, 38, 17, 47, 13, 17, 13, 7, 21, 83, 10, 138, 10, 26, 2, 7, 19, 31, 15, 13, 5, 23, 103, 19, 5, 95, 61, 6, 15, 2, 35, 13, 19, 65, 39, 7, 43, 73, 13, 26, 97, 1, 91, 2, 78, 19, 39, 34, 167, 82, 10, 38, 66, 58, 2, 46, 71

Offset: 0

Thomas Ordowski and Robert G. Wilson v, Nov 26 2017

nonn

Conjecture: a(n) < prime(n)^2 for n > 0.
a(n) = 1 iff n belongs to A014545.
Inspired by A293756.

			a(3) = 1 because 1 + prime(3)#/1 = 1 + 2*3*5 = 31 and 31 is a prime;
a(6) = 2 because 2 + prime(6)#/2 = 2 + 2*3*5*7*11*13/2 = 15017 and that is a prime;
a(7) = 5 because 5 + prime(7)#/5 = 5 + 2*3*5*7*11*13*17/5 = 102107 which is a prime; etc.

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

Cf. A002110, A005235, A014545.

f[n_] := Block[{d = 1, p = Fold[Times, 1, Prime@ Range@ n]}, While[ !PrimeQ[d + p/d], d++]; d]; Array[f, 90]

a(n)=my(P=vecprod(primes(n))); for(d=1,P, if(P%d==0 && ispseudoprime(d+P/d), return(d))) \\ Charles R Greathouse IV, Nov 27 2017

A114428 Primes of the form 1 + product of the first n semiprimes.

Original entry on oeis.org

5, 2161, 30241, 453601, 4495130640001, 152834441760001, 911300420785759804800001, 19660095637340203930960075575675174251117567173124497920000000001

Offset: 1

Jonathan Vos Post, Feb 13 2006

nonn

Semiprime analog of primorial primes A005234 (primes p such that 1 + product of primes up to p is prime) as indexed by A014545 (n such that n-th Euclid number (A006862(n)) = 1 + (Product of first n primes) is prime). In that sense, this sequence is indexed by (1, 4, 5, 6, 11, 12, 39, ...).

The next term has 90 digits. - Harvey P. Dale, Sep 21 2011

			a(1) = 5 = 4 + 1 = 1 + A001358(1) = 1 + A112141(1) because 4 is the first semiprime and 5 is prime.
a(2) = 2161 because 2160 + 1 = 1 + A001358(1)*A001358(2)*A001358(3)*A001358(4) = 1 + A112141(4) = 1 + (4*6*9*10) is prime.
a(3) = 1 + A112141(5).
a(4) = 1 + A112141(6).
a(5) = 1 + A112141(11).
a(6) = 1 + A112141(12).
a(7) = (4 * 6 * 9 * 10 * 14 * 15 * 21 * 22 * 25 * 26 * 33 * 34 * 35 * 38 * 39 * 46 * 49 * 51 * 55 * 57 * 58 * 62 * 65 * 69 * 74 * 77 * 82 * 85 * 86 * 87 * 91 * 93 * 94* 95 * 106 * 111 * 115 * 118 * 119) + 1 is prime.

Cf. A000040, A001358, A002110, A005234, A014545, A112141.

Select[#+1&/@FoldList[Times,1,Select[Range[200],PrimeOmega[#] == 2&]], PrimeQ] (* Harvey P. Dale, Sep 21 2011 *)

{a(n)} = {1 + A112141} INTERSECTION {A000040}.

a(7) added by Jonathan Vos Post, Dec 12 2010

cp's OEIS Frontend

A376416 a(n) = A276085(A006862(n)), where A276085 is the primorial base log-function, and A006862 is the Euclid numbers, one more than primorials.

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A088411 A088257 indexed by A002110.

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A103513 Primes of the form primorial(P(k))/2-2^n with minimal n, n>=0, k>=2.

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Mathematica

A104877 Semiprimes of the form primorial(k) + 1.

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A114432 Primes of the form 1 + product of the first k 4-almost primes A014613.

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A191623 Primes of the form 1 + Product_{k=1..n} prime(k)^(2^(k-1)).

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Mathematica

A224081 Numbers k such that A112141(k) + 1 is prime.

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Mathematica

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A289125 Numbers n such that phi(n)/phi(phi(n)) > phi(m)/phi(phi(m)) for all m < n.

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