A098026 -id:A098026 - cp's OEIS frontend

A073918 Smallest prime which is 1 more than a product of n distinct primes: a(n) is a prime and a(n) - 1 is a squarefree number with n prime factors.

2, 3, 7, 31, 211, 2311, 43891, 870871, 13123111, 300690391, 6915878971, 200560490131, 11406069164491, 386480064480511, 18826412648012971, 693386350578511591, 37508276737897976011, 3087649419126112110271, 183452981525059000664911, 11465419967969569966774411

Offset: 0

Amarnath Murthy, Aug 18 2002

nonn

Apparently the same as record values of A055734: least k such that phi(k) has n distinct prime factors, where phi is Euler's totient function. If the Mathematica program is used for large n, then "fact" should be reduced to, say, 1.1 in order to increase the search speed. - T. D. Noe, Dec 17 2003

			a(0) = 1 + 1 = 2 (empty product of zero primes).
a(1) = 1 + 2 = 3.
a(2) = 1 + 2*3 = 7.
a(3) = 1 + 2*3*5 = 31.
a(4) = 1 + 2*3*5*7 = 211.
a(5) = 1 + 2*3*5*7*11 = 1 + 11# = 2311.
a(6) = 1 + 2*3*5*7*11*19 = 43891, since 13# + 1 and 11#*17 + 1 = 17#/13 + 1 is not prime, and 17#/p + 1 is larger than a(6) for all p in {2, ..., 11}.
The index of the smallest prime which is not a factor of a(n)+1 is (1, 2, 3, 4, 5, 6, 6, 7, 7, 9, 10, 12, 11, 12, 13, 15, 16, 15, 16, 18, 19, 20, 21, 22, 22, 23, 25, 27, 26, 29, 29, ...) for n = 0, 1, 2, ... - _M. F. Hasler_, May 31 2018

Max Alekseyev, Table of n, a(n) for n = 0..100 (terms for n = 0..24 from M. F. Hasler)

Cf. A055734 (number of distinct prime factors of phi(n)).
Cf. A073917, A098026.
Cf. A000040 (primes), A002110 (primorial), A081545 (same with composite instead of primes).

Generate[pIndex_, i_] := Module[{p2, t}, p2=pIndex; While[p2[[i]]++; Do[p2[[j]]=p2[[i]]+j-i, {j, i+1, Length[p2]}]; t=Times@@Prime[p2]; t

A073918(n, b=0 /*best*/, p=1 /*product*/, f=[]/*factors*/)={ if( #f= f[n+1] ) || !b = A073918( n-1, b, p*f[n], f), f[n]= nextprime( f[n]+1 ) ); b } \\ then, e.g.: apply(A073918, [0..30]). - M. F. Hasler Jun 16 2007

From M. F. Hasler, Jun 16 2007 (Start):
Conjecture: For any m > 0 there is K > 0 such that for all k > K, a(k)-1 is divisible by the first m primes.
Corollary: For any m > 1 there is K > 0 such that for all k > K, a(k) = 1 (mod m).
Conjecture 2: Let K(m) be the smallest possible K satisfying the above Conjecture. Then K(m) ~ m, i.e., a(k) ~ A002110(k), only very few of the last factors will be a bit larger. (End)
Remark: the last "~" above was not intended to mean asymptotic equivalence. It appears that lim inf a(n)/A002110(n) = 1, but the lim sup might well be larger. It would be interesting to know whether it has a finite value. - M. F. Hasler, May 31 2018

More terms from Vladeta Jovovic, Aug 20 2002

Edited by M. F. Hasler, May 31 2018

A118883 Smallest prime p with bigomega(p+1)=n, where bigomega(m)=A001222(m) is the number of prime divisors of m (counted with multiplicity).

Original entry on oeis.org

2, 3, 7, 23, 31, 223, 127, 383, 1151, 3583, 5119, 6143, 8191, 129023, 73727, 245759, 131071, 917503, 524287, 5505023, 10616831, 14680063, 18874367, 109051903, 169869311, 654311423, 738197503, 2264924159, 2818572287, 3758096383, 2147483647, 24159191039

Offset: 1

Rick L. Shepherd, May 03 2006

nonn

Equivalently, smallest prime p such that p+1 is an n-almost prime. For smallest prime p such that p+1 is a squarefree n-almost prime, see A098026.

			a(4) = 23 because 23 is prime and 23+1 = 2*2*2*3 has 4 prime factors (24 is a 4-almost prime).

Robert G. Wilson v, Table of n, a(n) for n = 1..421

Cf. A073919, A098026, A073918.

(* copied directly from A073919 with only a sign change *) ptns[n_, 0] := If[n==0, {{}}, {}]; ptns[n_, k_] := Module[{r}, If[n v, Return[v]]; minp = Min@@ Select[l - 1, ProvablePrimeQ]; If[minp < v, v = minp]]] (* First do <Robert G. Wilson v *) Array[a, 32] (* Robert G. Wilson v, Jul 21 2011 *)

almost_primes(A, B, n) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, listput(list, m*q)), forprime(q=p, sqrtnint(B\m, n), list=concat(list, f(m*q, q, n-1)))); list); vecsort(Vec(f(1, 2, n)));
a(n) = my(x=2^n, y=2*x); while(1, my(v=almost_primes(x, y, n)); for(k=1, #v, if(isprime(v[k]-1), return(v[k]-1))); x=y+1; y=2*x); \\ Daniel Suteu, Jan 07 2025

a(26)-a(32) from Donovan Johnson, Feb 02 2011

A257865 Smallest k such that phi(k) = n*phi(k+1), where phi(n) = A000010(n) gives the value of Euler's totient function at n.

Original entry on oeis.org

1, 5, 119, 629, 17907119

Offset: 1

Felix Fröhlich, May 11 2015

From Manfred Scheucher, May 27 2015: (Start)
a(6)>=3*10^8 (calculation)
a(7)>=3.5*10^13, a(8)>=4.5*10^25, a(9)>=3.0*10^47, and so on... (doubly exponential lower bound, see uploaded pdf)
239719159679 and 239742643139 admit a ratio of 5.998... and 6.008..., resp.
There might be a relation to the sequence A098026. (End)

			a(3) = 119, because phi(119) == 3*phi(120) = 96 and 119 is the smallest k where this equality holds for n = 3.

Manfred Scheucher, A lower bound on A257865(n)

Cf. A001274, A171262, A256907, A256937, A257550.

Table[k = 1; While[EulerPhi[k] != n EulerPhi[k + 1], k++]; k, {n, 4}] (* Michael De Vlieger, May 12 2015 *)

a(n) = my(k=1); while(eulerphi(k)!=n*eulerphi(k+1), k++); k

a(n) >= exp(exp(c(n-3))) with c=exp(gamma) and gamma being the Euler-Mascheroni_constant (see pdf). - Manfred Scheucher, May 27 2015

A075591 Smallest squarefree number with n prime divisors which is the average of a twin prime pair.

Original entry on oeis.org

6, 30, 462, 2310, 43890, 1138830, 17160990, 300690390, 15651726090, 239378649510, 12234189897930, 568815710072610, 19835154277048110, 693386350578511590, 37508276737897976010, 3338236629672919864890

Offset: 2

Amarnath Murthy, Sep 26 2002

nonn

			a(4) = 462 because 462 = 2*3*7*11 and the twin primes are 461 and 463.

Cf. A075590.

Cf. A073918 (least prime p such that p-1 has exactly n distinct prime factors), A098026 (least prime p such that p+1 has exactly n distinct prime factors).

Generate[pIndex_, i_] := Module[{p2, t}, p2=pIndex; While[p2[[i]]++; Do[p2[[j]]=p2[[i]]+j-i, {j, i+1, Length[p2]}]; t=Times@@Prime[p2]; t

More terms from T. D. Noe, Dec 13 2004

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A073918 Smallest prime which is 1 more than a product of n distinct primes: a(n) is a prime and a(n) - 1 is a squarefree number with n prime factors.

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A118883 Smallest prime p with bigomega(p+1)=n, where bigomega(m)=A001222(m) is the number of prime divisors of m (counted with multiplicity).

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A257865 Smallest k such that phi(k) = n*phi(k+1), where phi(n) = A000010(n) gives the value of Euler's totient function at n.

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A075591 Smallest squarefree number with n prime divisors which is the average of a twin prime pair.

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Mathematica

Extensions