A062439 -id:A062439 - cp's OEIS frontend

A325709 Replace k with k! in the prime indices of n.

1, 2, 3, 4, 13, 6, 89, 8, 9, 26, 659, 12, 5443, 178, 39, 16, 49033, 18, 484037, 52, 267, 1318, 5222429, 24, 169, 10886, 27, 356, 61194647, 78, 774825383, 32, 1977, 98066, 1157, 36, 10552185239, 968074, 16329, 104, 153903050137, 534, 2394322471421, 2636, 117

Offset: 1

Gus Wiseman, May 19 2019

The union is A308299.

			The sequence of terms together with their prime indices begins:
       1: {}
       2: {1}
       3: {2}
       4: {1,1}
      13: {6}
       6: {1,2}
      89: {24}
       8: {1,1,1}
       9: {2,2}
      26: {1,6}
     659: {120}
      12: {1,1,2}
    5443: {720}
     178: {1,24}
      39: {2,6}
      16: {1,1,1,1}
   49033: {5040}
      18: {1,2,2}
  484037: {40320}
      52: {1,1,6}.

Amiram Eldar, Table of n, a(n) for n = 1..78 (calculated using the b-file at A062439)
Index entries for sequences computed from indices in prime factorization.

Cf. A000142, A056239, A062439, A064986, A076934, A112798, A115944, A284605, A308299, A322583, A325509, A325616, A325618, A325704.

Table[Times@@Prime/@(If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]!),{n,20}]

A325709(n) = { my(f=factor(n)); prod(i=1,#f~,prime(primepi(f[i, 1])!)^f[i, 2]); }; \\ Antti Karttunen, Nov 17 2019

from math import prod, factorial
from sympy import prime, primepi, factorint
def A325709(n): return prod(prime(factorial(primepi(p)))**e for p, e in factorint(n).items()) # Chai Wah Wu, Dec 26 2022

Completely multiplicative with a(prime(n)) = prime(n!).

Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(k!)) = 3.292606708493... . - Amiram Eldar, Dec 09 2022

Keyword:mult added by Antti Karttunen, Nov 17 2019

A261997 a(n) = prime(n)! - prime(n!).

Original entry on oeis.org

0, 3, 107, 4951, 39916141, 6227015357, 355687428046967, 121645100408347963, 25852016738884971417571, 8841761993739701954543554805353, 8222838654177922817725562105174617

Offset: 1

Altug Alkan, Sep 08 2015

nonn

a(n) is prime for n = 2, 3, 4, 5, 7.

			For n=2, a(n) = prime(n)! - prime(n!) = prime(2)! - prime(2!) = 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..21

Cf. A039716, A062439.

Array[Prime[#]! - Prime[#!] &, {11}] (* Michael De Vlieger, Sep 08 2015 *)

PARI
```
vector(11, n, prime(n)! - prime(n!))
```

a(n) = A039716(n) - A062439(n).

log a(n) ~ n log^2 n. - Charles R Greathouse IV, Sep 08 2015

A308299 Numbers whose prime indices are factorial numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 24, 26, 27, 32, 36, 39, 48, 52, 54, 64, 72, 78, 81, 89, 96, 104, 108, 117, 128, 144, 156, 162, 169, 178, 192, 208, 216, 234, 243, 256, 267, 288, 312, 324, 338, 351, 356, 384, 416, 432, 468, 486, 507, 512, 534, 576, 624, 648

Offset: 1

Gus Wiseman, May 19 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions using factorial numbers. The enumeration of these partitions by sum is given by A064986.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
    9: {2,2}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   18: {1,2,2}
   24: {1,1,1,2}
   26: {1,6}
   27: {2,2,2}
   32: {1,1,1,1,1}
   36: {1,1,2,2}
   39: {2,6}
   48: {1,1,1,1,2}
   52: {1,1,6}
   54: {1,2,2,2}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000142, A056239, A062439, A064986, A112798, A115944, A284605, A322583, A325616, A325709, A325618.

nn=5;
facts=Array[Factorial,nn];
Select[Range[Prime[Max@@facts]],SubsetQ[facts,PrimePi/@First/@FactorInteger[#]]&]

Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/prime(k!)) = 3.292606708493... . - Amiram Eldar, Dec 03 2022

A075489 a(n) = prime((n+1)!) - prime(n!).

Original entry on oeis.org

1, 10, 76, 570, 4784, 43590, 435004, 4738392, 55972218, 713630736, 9777359856, 143350864898, 2240419421284, 37194276947898, 653800845663788, 12132997533521320, 237076055569553246, 4865738414759433466, 104661156692004606078, 2354571975178917773640

Offset: 1

Labos Elemer, Sep 26 2002

			n=5: a(5) = prime(720) - prime(120) = 5443 - 659 = 4784.

Cf. A000040, A000142, A062439.

Table[Prime[(n + 1)! ] - Prime[n! ], {n, 1, 10}] (* Stefan Steinerberger, Apr 10 2006 *)
#[[2]]-#[[1]]&/@Partition[Prime/@(Range[15]!),2,1](* Harvey P. Dale, Jul 07 2019 *)

a(n) = A062439(n+1) - A062439(n). - Jinyuan Wang, Jun 27 2020

Definition corrected by Stefan Steinerberger, Apr 10 2006
a(15)-a(18) from Giovanni Resta, Jul 08 2019
a(19)-a(20) from Jinyuan Wang, Jun 27 2020

A076232 a(n) = prime(1+n!) - prime(n!).

Original entry on oeis.org

1, 2, 4, 8, 2, 6, 4, 24, 18, 86, 54, 18, 22, 48, 12, 10, 56, 10, 8, 12, 24

Offset: 1

Labos Elemer, Oct 03 2002

Cf. A000040, A000142, A001223, A062439.

Table[Prime[n! + 1] - Prime[n!], {n, 10}] (* Wesley Ivan Hurt, Aug 24 2019 *)

a(n) = prime(1+n!) - prime(n!); \\ Michel Marcus, Aug 24 2019

a(n) = my(p=prime(n!)); nextprime(p+1) - p; \\ Michel Marcus, Aug 24 2019

a(n) = A001223(n!). - Michel Marcus, Aug 24 2019

a(16)-a(21) from Michel Marcus, Aug 24 2019

A262185 a(n) = n^prime(n!) - prime(n!)^n.

Original entry on oeis.org

-1, -1, -1, 1592126, 383123885216472214589586756787577295904684780483158303

Offset: 0

Altug Alkan, Sep 14 2015

Inspired by A007965.

			For n = 1, a(n) = n^prime(n!) - prime(n!)^n = 1^2 - 2^1 = -1.

Cf. A000040, A062439, A007965.

[n^NthPrime(Factorial(n)) - NthPrime(Factorial(n))^n: n in [0..6]]; // Vincenzo Librandi, Sep 15 2015

Table[n^Prime[n!] - Prime[n!]^n, {n, 0, 4}] (* Michael De Vlieger, Sep 14 2015 *)

a(n) = n^prime(n!) - prime(n!)^n;
vector (6, n, a(n-1))

a(n) = n^A062439(n) - A062439(n)^n.

A262398 a(n) = prime(n)! mod prime(n!).

Original entry on oeis.org

0, 0, 3, 56, 511, 194, 46976, 104633, 546681, 41130177, 643108140, 7034542959, 65748733699, 1518781632657, 35097481516962, 396029533782911, 4146710666095789, 159899356955923308, 3662069108121609141, 109629928744379590001, 828180977946159463007

Offset: 1

Altug Alkan, Sep 21 2015

nonn

Inspired by A261997.
a(n) = n and a(n) = prime(n-1) for n=3.
a(n) = 0 only for n=1 and n=2. What is the minimum value of a(n) for n > 2? Is there a possibility of observing that a(n) = 1 or a(n) = 2?

			a(1) = prime(1)! mod prime(1!) = 2 mod 2 = 0.
a(2) = prime(2)! mod prime(2!) = 6 mod 3 = 0.
a(3) = prime(3)! mod prime(3!) = 120 mod 13 = 3.

Cf. A000040, A039716, A062439, A261997.

[Factorial(NthPrime(n)) mod NthPrime(Factorial(n)): n in [1..11]]; // Vincenzo Librandi, Sep 23 2015

Table[Mod[Prime[n]!, Prime[n!]], {n, 15}] (* Michael De Vlieger, Sep 21 2015 *)

a(n) = prime(n)! % prime(n!);
vector(11, n, a(n))

a(n) = A039716(n) mod A062439(n).

a(11)-a(15) from Michael De Vlieger, Sep 21 2015

A362056 Prime numbers of the form prime(k)! - prime(k!).

Original entry on oeis.org

3, 107, 4951, 39916141, 355687428046967

Offset: 1

Michael S. Branicky, Apr 06 2023

No other terms < prime(22)! - prime(22!) (using A062439).

Problem suggested by Carlos Rivera as a follow-on to Puzzle 1127 (see links).

			prime(2)! - prime(2!) = 3 is prime.
prime(3)! - prime(3!) = 107 is prime.
prime(4)! - prime(4!) = 4951 is prime.
prime(5)! - prime(5!) = 39916141 is prime.
prime(7)! - prime(7!) = 355687428046967 is prime.

Carlos Rivera, Puzzle 1127. Prime(11!)+Prime(11)!, The Prime Puzzles and Problems Connection.
Carlos Rivera, Puzzle 1128. Prime(n)!-Prime(n!), The Prime Puzzles and Problems Connection.

Cf. A000040, A000142, A062439.

Prime terms in A261997.

A261997(k) = A000142(A000040(k)) - A000040(A000142(k)) is in A000040.

cp's OEIS Frontend

A325709 Replace k with k! in the prime indices of n.

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A261997 a(n) = prime(n)! - prime(n!).

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A308299 Numbers whose prime indices are factorial numbers.

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A075489 a(n) = prime((n+1)!) - prime(n!).

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A076232 a(n) = prime(1+n!) - prime(n!).

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A262185 a(n) = n^prime(n!) - prime(n!)^n.

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A262398 a(n) = prime(n)! mod prime(n!).

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