A103855 -id:A103855 - cp's OEIS frontend

A103890 a(n) = prime(n)! / prime(n)# + 1.

2, 2, 5, 25, 17281, 207361, 696729601, 12541132801, 115880067072001, 1366643159020339200001, 40999294770610176000001, 1854768736099424576471040000001, 109950690675973888893203251200000001, 4617929008390903333514536550400000001

Offset: 1

Reinhard Zumkeller, Feb 20 2005

nonn

Reinhard Zumkeller, p(n)! / p(n)# + 1.

Cf. A103855, A103891, A103892, A103893, A103894, A103895, A103896.

#[[1]]/#[[2]]&/@With[{nn=15},Thread[{Prime[Range[nn]]!,FoldList[ Times,Prime[ Range[nn]]]}]]+1 (* Harvey P. Dale, May 21 2019 *)

a(n) = prime(n)!/vecprod(primes(n)) + 1; \\ Michel Marcus, Nov 12 2023

a(n) = A039716(n)/A002110(n) + 1 = A092435(n) + 1.

A103856 Smallest prime factor of prime(n)! - prime(n)# + 1.

Original entry on oeis.org

1, 1, 7, 4831, 3673, 16349, 5240507, 4159, 83, 3911, 401, 61, 66935021479, 199, 73, 152249, 379, 5014363, 181, 191, 277, 155893, 35851, 431, 1499

Offset: 1

Reinhard Zumkeller, Feb 20 2005

nonn

a(26) has 160 digits and is too large to be included here.

Amiram Eldar, Table of n, a(n) for n = 1..26
Reinhard Zumkeller, p(n)! - p(n)# + 1.

Cf. A020639, A103855, A103857, A103891.

a(n) = if(n>2, factor(prime(n)!-factorback(primes(n))+1)[1, 1], 1); \\ Jinyuan Wang, Dec 30 2024

a(n) = A020639(A103855(n)).

A103857 Greatest prime factor of prime(n)! - prime(n)# + 1.

Original entry on oeis.org

1, 1, 13, 4831, 10867, 380879, 67872713, 29248641596329, 5375659901, 2260742008115495258127626261, 39302553278734052383226897, 2202226772995090243, 363934772650919, 125129894757902719183, 1330939456978870860802004244587

Offset: 1

Reinhard Zumkeller, Feb 20 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..26
Reinhard Zumkeller, p(n)! - p(n)# + 1.

Cf. A006530, A103855, A103856, A103892.

a(n) = A006530(A103855(n)).

A103858 Number of distinct prime factors of prime(n)! - prime(n)# + 1.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Feb 20 2005

a(27) >= 4. - Amiram Eldar, Jan 21 2024

Factordb.com, Status of 107! - 107# + 1.
Reinhard Zumkeller, prime(n)! - prime(n)# + 1.

Cf. A001221, A103855, A103859, A103893.

primorial[n_] := Product[Prime[i], {i, n}]; A103855[n_] := Prime[n]! - primorial[n] + 1;  PrimeNu[Array[A103855, 20]] (* G. C. Greubel, May 09 2017 *)

a(n) = omega(prime(n)! - prod(k=1, n, prime(k)) + 1); \\ Michel Marcus, Nov 06 2022

a(n) = A001221(A103855(n)).

a(27) from Jinyuan Wang, Jun 19 2025

A103859 Number of divisors of prime(n)! - prime(n)# + 1.

Original entry on oeis.org

1, 1, 4, 2, 4, 4, 4, 4, 16, 4, 8, 16, 16, 32, 32, 8, 32, 8, 16, 64, 32, 16, 32, 64, 512, 2, 16

Offset: 1

Reinhard Zumkeller, Feb 20 2005

Reinhard Zumkeller, p(n)! - p(n)# + 1.

Cf. A000005, A103855, A103858, A103860, A103894.

a(n) = numdiv(prime(n)! - prod(k=1, n, prime(k)) + 1); \\ Michel Marcus, Nov 06 2022

a(n) = A000005(A103855(n)).

a(27) from Jinyuan Wang, Jun 19 2025

A103860 Sum of divisors of prime(n)! - prime(n)# + 1.

Original entry on oeis.org

1, 1, 112, 4832, 39929032, 6227388000, 355687500698712, 121674349040732800, 26204303786955547121568, 8844022735747817449795273936944, 8243360285859647409507002537385024, 13989390090648918627329493090109449081402880, 33452526614068666040537953433982904696696861440000

Offset: 1

Reinhard Zumkeller, Feb 20 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..26
Reinhard Zumkeller, p(n)! - p(n)# + 1.

Cf. A000203, A103855, A103858, A103859, A103895.

Module[{nn=15,pr,pm},pr=Prime[Range[nn]]!;pm=FoldList[Times,Prime[Range[nn]]];DivisorSigma[1,#[[1]]-#[[2]]+1&/@Thread[{pr,pm}]]] (* Harvey P. Dale, Dec 18 2022 *)

a(n) = sigma(prime(n)! - factorback(primes(n)) + 1); \\ Jinyuan Wang, Dec 28 2024

a(n) = A000203(A103855(n)).

A103861 Euler's totient of prime(n)! - prime(n)# + 1.

Original entry on oeis.org

1, 1, 72, 4830, 39899952, 6226593544, 355687354472272, 121615851757531824, 25500701524255140352000, 8839501251731586459279018676600, 8202317101101304783411826869132800, 13538116146734258320615004890482768163319520, 33452526612258948175814325380737291579993833222336

Offset: 1

Reinhard Zumkeller, Feb 20 2005

nonn

Amiram Eldar, Table of n, a(n) for n = 1..26
Reinhard Zumkeller, p(n)! - p(n)# + 1.

Cf. A000010, A103855, A103896.

EulerPhi[Table[Prime[n]! - Product[Prime[i], {i, 1, n}] + 1, {n, 1, 10}]] (* Amiram Eldar, Feb 21 2020 *)

a(n) = A000010(A103855(n)).

cp's OEIS Frontend

A103890 a(n) = prime(n)! / prime(n)# + 1.

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A103856 Smallest prime factor of prime(n)! - prime(n)# + 1.

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A103857 Greatest prime factor of prime(n)! - prime(n)# + 1.

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A103858 Number of distinct prime factors of prime(n)! - prime(n)# + 1.

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A103859 Number of divisors of prime(n)! - prime(n)# + 1.

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A103860 Sum of divisors of prime(n)! - prime(n)# + 1.

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PARI

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A103861 Euler's totient of prime(n)! - prime(n)# + 1.

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