A056604 -id:A056604 - cp's OEIS frontend

A254924 a(n) = (A060371(n) - A094998(n))/A056604(n) for n > 1, with a(1)=1.

1, 0, 0, 1, 130, 1329, 1707670, 27502484, 209927657739, 130904517147542068, 3673771932850374193, 69623451054783204822486486, 3724616892817543661693877073170, 149157913707716515940392007441860, 12429106799179771738076359013310638297

Offset: 1

Bruno Berselli, Feb 12 2015 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)

nonn

Let theta(p) be the smallest nonnegative solution z to the system of congruences z == 0 (mod p), z == 1 (mod v(p-1)), where p is a prime and v(p-1) = lcm(1,...,p-1). Theta(p) is unique mod lcm(p, v(p-1)), therefore it is unique mod v(p). Since both (p-1)!+1 and theta(p) are solutions to these congruences, ((p-1)!+1 - theta(p))/v(p) is always an integer. The sequence lists the values of this ratio (assuming theta(2)=0 and p=prime(n)).

			For n=5, a(5) = (A060371(5) - A094998(5))/A056604(5) = (3628801 - 25201)/27720 = 130.

Bruno Berselli, Table of n, a(n) for n = 1..50
Umberto Cerruti, Il Teorema Cinese dei Resti (in Italian), 2015. The sequence is on page 21.

Cf. A000040, A056604, A060371, A094998, A254939, A255010.

[(Factorial(p-1)+1-Modinv(p,Lcm([1..p-1]))*p)/Lcm([1..p]): p in PrimesUpTo(50)];

with(numtheory): P:=proc(q)  local a,j,k,ok,n;  print(1); a:=[1];
for n from 3 to q do k:=0; a:=[op(a),n]; if isprime(n) then ok:=0;  while ok=0 do ok:=1;
k:=k+1; for j from 2 to n-1 do if not (k*n mod j)=1 then ok:=0; break; fi; od; od;
print((((n-1)!+1)-k*n)/lcm(op(a))); fi; od; end: P(100); # Paolo P. Lava, Feb 16 2015

r[k_] := LCM @@ Range[k]; s[k_] := PowerMod[k, -1, r[k - 1]] k; w[k_] := ((k - 1)! + 1 - s[k])/r[k]; Table[w[Prime[n]], {n, 1, 20}]

A024620 Positions of primes among the powers of primes (A000961).

Original entry on oeis.org

2, 3, 5, 6, 9, 10, 12, 13, 14, 17, 18, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 93, 94

Offset: 1

Clark Kimberling

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement of A024621.
Cf. A001222 (bigomega), A025474, A056604, A027883.
Cf. A025528, A065515, A000040.

a024620 n = a024620_list !! (n-1)
a024620_list = filter ((== 1) . a025474) [1..]
-- Reinhard Zumkeller, May 01 2015

a[n_] := PrimeOmega[LCM @@ Range@Prime@n] + 1; Array[a, 100] (* Amiram Eldar, Dec 02 2018 *)

lista(nn) = my(powpr = select((i->((omega(i)==1) || (i==1))), [1..nn])); for (i = 1, #powpr, if (isprime(powpr[i]), print1(i, ", ")); ); \\ Michel Marcus, Jun 03 2021

from sympy import prime, primepi, integer_nthroot
def A024620(n):
    x = prime(n)
    return n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())) # Chai Wah Wu, Nov 05 2024

A025474(a(n)) = 1. - Reinhard Zumkeller, May 01 2015
a(n) = A001222(A056604(n)) + 1. - Eric Desbiaux, Dec 02 2018
From Ridouane Oudra, Oct 18 2020: (Start)
a(n) = A027883(n) + 1;
a(n) = A025528(A000040(n)) + 1;
a(n) = A065515(A000040(n)). (End)

A329571 a(n) = Product_{prime p} p^floor(log_p P) with P = A329570(n) the least prime with log_p P >= valuation(n,p) for all primes p.

Original entry on oeis.org

2, 2, 6, 60, 60, 6, 420, 27720, 27720, 60, 27720, 60, 360360, 420, 60, 12252240, 12252240, 27720, 232792560, 60, 420, 27720, 5354228880, 27720, 2329089562800, 360360, 2329089562800, 420, 2329089562800, 60, 72201776446800, 5342931457063200, 27720, 12252240, 420, 27720, 5342931457063200, 232792560, 360360, 27720, 219060189739591200, 420, 9419588158802421600, 27720

Offset: 1

M. F. Hasler, Jan 03 2020

nonn

Related to the inequality (54) in Ramanujan's paper about highly composite numbers (HCN) A002182, also used in A199337: This is the square root of the (not minimal) bound a(n)^2 above which all HCN are divisible by n, according to the right part of that inequality.

Like the highly composite numbers A002182, all terms in this sequence are a product of primorials.

Amiram Eldar, Table of n, a(n) for n = 1..2308
Srinivasa Ramanujan, Highly composite numbers, Proceedings of the London Mathematical Society, Ser. 2, Vol. XIV, No. 1 (1915), pp 347-409. A variant of better quality with an additional footnote is available at Ramanujan Papers.

Cf. A329570, A002182 (highly composite numbers), A199337 (number of HCN not divisible by n), A003418 (lcm(1..n)), A056604 (lcm(1..prime(n))), A025487.

a[n_] := Module[{P = NextPrime[Max[Power @@@ FactorInteger[n]] - 1], p}, p = Select[Range[P], PrimeQ]; Times @@ (p^Floor[Log[p, P]])]; a[1] = 2; Array[a, 50] (* Amiram Eldar, Jan 17 2025 *)

apply( {A329571(n)=vecprod([p^logint(n,p)|p<-primes([2,n=A329570(n)])])}, [1..44])

a(n) = lcm([1..P]) = A003418(P) = A056604(i) with P = A329570(n), i = A000720(P).

A154524 Primes p such that lcm(1,2,3,...,p-2,p-1,p) - 1 is prime.

Original entry on oeis.org

3, 5, 7, 19, 23, 29, 47, 61, 97, 181, 233, 307, 401, 887, 1021, 1087, 1361, 1481, 2053, 2293, 5407, 5857, 11059, 14281, 27277, 27803, 36497, 44987, 53017

Offset: 1

Lekraj Beedassy, Jan 11 2009

nonn

a(28) > 42000. - Daniel Suteu, Oct 06 2018

a(30) > 100000. - Michael S. Branicky, Jul 04 2025

			7 is in the sequence because it is prime and also lcm(1,2,3,4,5,6,7)-1 = 420-1 = 419 is prime. - _Emeric Deutsch_, Jan 16 2009

Cf. A056604, A154525, A154526.

P := proc(n) options operator, arrow: ilcm(seq(j, j = 1 .. n)) end proc: a := proc(n) if isprime(n) and isprime(P(n)-1) then n else end if end proc: seq(a(n), n = 1 .. 3000); # Emeric Deutsch, Jan 16 2009

A057825 INTERSECT A000040. - R. J. Mathar, Jan 14 2009

a(8)-a(17) from Ray Chandler, Jan 16 2009
a(18)-a(22) from Emeric Deutsch, Jan 16 2009
a(23)-a(27) from Daniel Suteu, Oct 06 2018
a(28)-a(29) from Michael S. Branicky, Jul 03 2025

A254939 a(n) = (A099795(n)^-1 mod p)*A099795(n), where p = prime(n).

Original entry on oeis.org

1, 4, 36, 120, 2520, 277200, 5045040, 183783600, 4655851200, 80313433200, 32607253879200, 2743667504978400, 58772246027695200, 5038384364010597600, 56517528952814529600, 34089489546705963770400, 7391221142626702144764000

Offset: 1

Bruno Berselli, Feb 12 2015 - proposed by Umberto Cerruti (Department of Mathematics "Giuseppe Peano", University of Turin, Italy)

nonn

The sequence lists the smallest nonnegative solutions z to the system of congruences z == 1 (mod p), z == 0 (mod v(p-1)), where p is a prime and v(p-1) = lcm(1,...,p-1).

			5045040 is the seventh term of the sequence because the modular inverse of A099795(7) mod A000040(7) is 7 and 7*A099795(7) = 7*720720 = 5045040.

Umberto Cerruti, Il Teorema Cinese dei Resti (in Italian), 2015. The sequence is on page 21.
Eric Weisstein's World of Mathematics, Modular Inverse.

Cf. A000040, A056604, A099795, A254924, A255010.

[Modinv(Lcm([1..p-1]),p)*Lcm([1..p-1]): p in PrimesUpTo(60)];

with(numtheory): P:=proc(q)  local a, n;  a:=[];
for n from 1 to q do a:=[op(a),n]; if isprime(n+1) then
print(lcm(op(a))*(lcm(op(a))^(-1) mod (n+1))); fi;
od; end: P(10^3); # Paolo P. Lava, Feb 16 2015

r[k_] := LCM @@ Range[k]; u[k_] := PowerMod[r[k - 1], -1, k] r[k - 1]; Table[u[Prime[n]], {n, 1, 20}]

a099795(n) = lcm(vector(prime(n)-1, k, k));
a(n) = {my(m = a099795(n)); m*lift(1/Mod(m, prime(n)));} \\ Michel Marcus, Feb 13 2015

a(n) = A255010(n)*A099795(n).

A154525 Primes p such that lcm(1,2,3,...,p-2,p-1,p) + 1 is prime.

Original entry on oeis.org

2, 3, 5, 7, 19, 31, 47, 89, 127, 139, 2251, 3271, 4253, 4373, 4549, 5449, 13331, 37123, 55291

Offset: 1

Lekraj Beedassy, Jan 11 2009

nonn

Cf. A056604, A154524, A154526.

Select[Range[1000], PrimeQ[#] && PrimeQ[LCM@@Range[#]+1] &] (* Amiram Eldar, Nov 21 2018 *)

isok(p) = isprime(p) && (isprime(lcm(vector(p, i, i)) + 1)); \\ Michel Marcus, Oct 26 2013, Feb 25 2014

A049537 INTERSECT A000040. - Ray Chandler, Jan 16 2009

a(1)=2 inserted and a(8)-a(18) from Ray Chandler, Jan 16 2009

a(19) from Daniel Suteu, Nov 21 2018

A154526 Primes p such that lcm(1,2,3,...,p-2,p-1,p) -+ 1 are both primes.

Original entry on oeis.org

3, 5, 7, 19, 47

Offset: 1

Lekraj Beedassy, Jan 11 2009

Intersection of A154524 and A154525.

Cf. A056604, A154524, A154525.

isok(p) = {if (! isprime(p), return (0)); lcmv = lcm(vector(p, i, i)); isprime(lcmv + 1) && isprime(lcmv - 1);} \\ Michel Marcus, Oct 26 2013

A109923 a(n) = lcm(1,2,3,...,prime(n))/(1 + 2 + ... + prime(n)).

Original entry on oeis.org

1, 4, 15, 420, 3960, 80080, 1225224, 19399380, 5354228880, 145568097675, 7600186994400, 254425307479200, 9957281351799600, 392482839950100900, 114779426083185063200, 5474978624167927514640, 312603618218620377448800

Offset: 2

Amarnath Murthy, Jul 16 2005

			a(4)=15 because the 4th prime is 7 and lcm(1,2,3,4,5,6,7)/(1+2+3+4+5+6+7) = 420/28 = 15.

Cf. A006254, A034953, A056604, A099795, A109922.

a:=n->lcm(seq(i,i=1..ithprime(n)))/sum(j,j=1..ithprime(n)): seq(a(n),n=2..20); # Emeric Deutsch, Jul 16 2005

a(n) = lcm(vector(prime(n), k, k))/sum(k=1, prime(n), k); \\ Michel Marcus, Mar 07 2018

a(n) = A099795(n)/A006254(n-1). - Andrey Zabolotskiy, Mar 07 2018

a(n) = A056604(n)/A034953(n). - Michel Marcus, Mar 07 2018

More terms from Emeric Deutsch, Jul 16 2005

cp's OEIS Frontend

A254924 a(n) = (A060371(n) - A094998(n))/A056604(n) for n > 1, with a(1)=1.

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A024620 Positions of primes among the powers of primes (A000961).

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A329571 a(n) = Product_{prime p} p^floor(log_p P) with P = A329570(n) the least prime with log_p P >= valuation(n,p) for all primes p.

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A154524 Primes p such that lcm(1,2,3,...,p-2,p-1,p) - 1 is prime.

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A254939 a(n) = (A099795(n)^-1 mod p)*A099795(n), where p = prime(n).

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A154525 Primes p such that lcm(1,2,3,...,p-2,p-1,p) + 1 is prime.

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A154526 Primes p such that lcm(1,2,3,...,p-2,p-1,p) -+ 1 are both primes.

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A109923 a(n) = lcm(1,2,3,...,prime(n))/(1 + 2 + ... + prime(n)).