A055003 -id:A055003 - cp's OEIS frontend

A072677 a(n) = prime(prime(n)+1) where prime(k) is the k-th prime.

5, 7, 13, 19, 37, 43, 61, 71, 89, 113, 131, 163, 181, 193, 223, 251, 281, 293, 337, 359, 373, 409, 433, 463, 521, 557, 569, 593, 601, 619, 719, 743, 787, 809, 863, 881, 929, 971, 997, 1033, 1069, 1091, 1163, 1181, 1213, 1223, 1301, 1423, 1439, 1451, 1481, 1511, 1531, 1601, 1627, 1693, 1733, 1747, 1789, 1831, 1861, 1931

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jul 05 2002

Prime numbers q for which the number of prime numbers smaller than q is also a prime number.

			a(4)=prime(prime(4)+1), prime(4)=7, hence a(4)=prime(8)=19.
163 is in the sequence because (1) it is a prime number, (2) there are 37 prime numbers smaller than 163 and 37 is also a prime number.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000720, A006450, A055003, A072677.

with(numtheory): seq(ithprime(ithprime(i)+1),i=1..51);

Prime[Prime[Range[60]]+1] (* Harvey P. Dale, Nov 07 2016 *)

a(n) = prime(prime(n)+1); \\ Michel Marcus, Nov 17 2017

Edited by N. J. A. Sloane, Nov 04 2018 at the suggestion of Georg Fischer, Nov 03 2018, merging a duplicate entry with this one.

A064554 a(n) = Min {k | A064553(k) = n}.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 13, 8, 9, 14, 29, 12, 37, 26, 21, 16, 53, 18, 61, 28, 39, 58, 79, 24, 49, 74, 27, 52, 107, 42, 113, 32, 87, 106, 91, 36, 151, 122, 111, 56, 173, 78, 181, 116, 63, 158, 199, 48, 169, 98, 159, 148, 239, 54, 203, 104, 183, 214, 271, 84, 281, 226, 117, 64

Offset: 1

Reinhard Zumkeller, Sep 21 2001

nonn

A064553(a(n)) = n and A064553(a(k)) <> k for k < a(n). For prime p, a(p)=prime(p-1), which is sequence A055003. - T. D. Noe, Dec 12 2004
a(n) is not multiplicative because a(7*13) = a(91) = 463, but a(7)*a(13) = 13*37 = 481 and 91 is the smallest possible such n. - Christian G. Bower, May 19 2005
a(n) = A080688(n,1). - Reinhard Zumkeller, Oct 01 2012
Minimal shifted Heinz number of a factorization of n, where the shifted Heinz number of a factorization (y_1, ..., y_k) is prime(y_1 - 1) * ... * prime(y_k - 1). - Gus Wiseman, Sep 05 2018

T. D. Noe, Table of n, a(n) for n = 1..8000
T. D. Noe, Plot of A064554

Cf. A064555, A001055, A064553.
Cf. A055003 (prime(prime(n)-1)).
Cf. A007716, A056239, A162247, A215366, A246868, A318871.

a064554 = head . a080688_row  -- Reinhard Zumkeller, Oct 01 2012

facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Min[Times@@Prime/@(#-1)&/@facs[n]],{n,100}] (* Gus Wiseman, Sep 05 2018 *)

A140853 a(n) = prime(prime(n) - 1) - 1, where prime(n) is the n-th prime.

Original entry on oeis.org

1, 2, 6, 12, 28, 36, 52, 60, 78, 106, 112, 150, 172, 180, 198, 238, 270, 280, 316, 348, 358, 396, 420, 456, 502, 540, 556, 576, 592, 612, 700, 732, 768, 786, 856, 862, 910, 952, 982, 1020, 1060, 1068, 1150, 1162, 1192, 1212, 1290, 1398, 1428, 1438, 1458

Offset: 1

Juri-Stepan Gerasimov, Jul 30 2008

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A006093, A055003.

Table[Prime[Prime[n - 1] - 1] - 1, {n, 2, 60}] (* Vincenzo Librandi, Apr 05 2015 *)

PARI
```
apply(p->prime(p-1)-1, primes(100))
```

a(n) = A055003(n) - 1. - N. J. A. Sloane, Jul 31 2008

a(n) ~ n (log n)^2. - Charles R Greathouse IV, Apr 07 2015

Corrected (382 replaced by 396) and extended by R. J. Mathar, Apr 27 2010

A290641 Multiplicative with a(p^e) = prime(p-1)^e.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 13, 8, 9, 14, 29, 12, 37, 26, 21, 16, 53, 18, 61, 28, 39, 58, 79, 24, 49, 74, 27, 52, 107, 42, 113, 32, 87, 106, 91, 36, 151, 122, 111, 56, 173, 78, 181, 116, 63, 158, 199, 48, 169, 98, 159, 148, 239, 54, 203, 104, 183, 214, 271, 84, 281, 226, 117, 64, 259

Offset: 1

Michel Marcus, Aug 08 2017

a(n) = A064554(n) for 1 <= n < 91, but a(91) = 481 differs from A064554(91) = 463. - Georg Fischer, Oct 23 2018

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A046523, A055003, A064554, A064988, A064989.

Array[If[# == 1, 1, Times @@ Map[Prime[#1 - 1]^#2 & @@ # &, FactorInteger[#]]] &, 65] (* Michael De Vlieger, Apr 22 2021 *)

a(n) = {my(f = factor(n)); for (k=1, #f~, f[k, 1] = prime(f[k, 1]-1);); factorback(f);}

from sympy import factorint, prime
from operator import mul
from functools import reduce
def a(n):
    return 1 if n==1 else reduce(mul, [prime(p - 1)**e for p, e in factorint(n).items()])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Aug 08 2017

(define (A290641 n) (if (= 1 n) n (* (A000040 (+ -1 (A020639 n))) (A290641 (A032742 n))))) ;; Antti Karttunen, Aug 08 2017

From Antti Karttunen, Aug 08 2017: (Start)
a(n) = A064989(A064988(n)).
A046523(a(n)) = A046523(n). [Preserves the prime signature of n].
(End)

A050298 Triangle read by rows: T(n,k) = p(r), where r is the (n-k+1)-th number such that A001222(r+1) = k, and p(r) is the r-th prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 13, 19, 31, 47, 29, 23, 59, 83, 127, 37, 41, 67, 149, 211, 307, 53, 43, 101, 167, 353, 499, 709, 61, 71, 103, 241, 401, 823, 1153, 1613, 79, 73, 109, 257, 587, 937, 1873, 2647, 3659, 107, 89, 179, 277, 607, 1319, 2113, 4201, 5843, 8147

Offset: 1

Alford Arnold, Apr 09 2003

The first column is A055003 and the main diagonal is A051438. When viewed as a sequence, this is a permutation of the prime numbers.

			a(14) = T(5,4) = p(23) = 83 because A001222(23+1) = A001222(24) = 4 since 24 has four prime factors, and this is the (5-4+1) = 2nd number with A001222 = 4.
The table begins:
2
3  5
7  11 17
13 19 31 47
29 23 59 83  127
37 41 67 149 211 307
...

Nathaniel Johnston, Rows n=1..20, flattened

Cf. A000040, A001222, A064553.

with(numtheory): A050298ind := proc(n,k) option remember: local f,m: if(n=k)then return 2^n-1: fi: for m from procname(n-1,k)+1 do if(bigomega(m+1)=k)then return m: fi: od: end: for n from 1 to 6 do seq(ithprime(A050298ind(n,k)),k=1..n);od; # Nathaniel Johnston, May 07 2011

Better name and extended by Nathaniel Johnston, May 07 2011

A143444 A054525 * the primes prefaced by 0.

Original entry on oeis.org

0, 2, 3, 3, 7, 6, 13, 12, 16, 14, 29, 17, 37, 26, 33, 30, 53, 32, 61, 41, 55, 42, 79, 40, 82, 58, 82, 59, 107, 44, 113, 80, 99, 82, 119, 70, 151, 94, 123, 88, 173, 74, 181, 115, 134, 116, 199, 98, 210, 122, 173, 133, 239, 100, 215, 142, 199, 160, 271, 107, 281, 168, 206

Offset: 1

Gary W. Adamson, Aug 15 2008

nonn

a(5) = 7 = (-1, 0, 0, 0, 1) dot (0, 2, 3, 5, 7).

a(p(n)) = p(n-1) since prime rows of the Mobius transform have only one nonzero multiplier besides the (-1) which is multiplied * the 0 in (0, 2, 3, 5, 7,...). Cf. A055003: (1, 3, 7, 13, 29, 37,...) a subset of primes in the sequences, but other terms may be primes.

			a(6) = 6 = (1, -1, -1, 0, 0, 1) dot (0, 2, 3, 5, 7, 11) where (1, -1, -1, 0, 0, 1) = row 6 of A051731, the Mobius transform.

Cf. A000040, A051731, A055003.

Mobius transform of the primes prefaced by 0.

a(12) replaced by 17 and extended by R. J. Mathar, Jan 19 2009

A275990 a(n) = prime(prime(n)-1) - prime(n).

Original entry on oeis.org

0, 0, 2, 6, 18, 24, 36, 42, 56, 78, 82, 114, 132, 138, 152, 186, 212, 220, 250, 278, 286, 318, 338, 368, 406, 440, 454, 470, 484, 500, 574, 602, 632, 648, 708, 712, 754, 790, 816, 848, 882, 888, 960, 970, 996, 1014, 1080, 1176, 1202, 1210, 1226, 1254, 1270, 1332, 1362, 1404

Offset: 1

Terry D. Grant, Aug 15 2016

nonn

			For n=3, prime(prime(3)-1) = 7, and prime(3) = 5, therefore a(3) = 7 - 5 = 2.

Cf. A000040, A006093, A055003, A275989.

Table[Prime[Prime[n]-1] - Prime[n], {n, 1, 100}]

a(n) = prime(prime(n)-1) - prime(n); \\ Michel Marcus, Aug 18 2016

a(n) = A000040(A006093(n)) - A000040(n) = A055003(n) - A000040(n).

A175248 Noncomposites (A008578) with noncomposite (A008578) subscripts.

Original entry on oeis.org

1, 2, 3, 7, 13, 29, 37, 53, 61, 79, 107, 113, 151, 173, 181, 199, 239, 271, 281, 317, 349, 359, 397, 421, 457, 503, 541, 557, 577, 593, 613, 701, 733, 769, 787, 857, 863, 911, 953, 983, 1021, 1061, 1069, 1151, 1163, 1193, 1213, 1291, 1399, 1429, 1439, 1459

Offset: 1

Jaroslav Krizek, Mar 13 2010

nonn

Noncomposite numbers of order 2.

a(n) = noncomposite(noncomposite(n)) = A008578(A008578(n)). a(1) = 1, a(n) = A055003(n) for n >=2. a(n) U A175249(n+1) = A008578 for n >= 1.

			a(5) = 13 because a(5) = q(q(5)) = q(7) = 13, q = noncomposite.

Cf. A006450, A102615.

a(n)=A008578(A008578(n)).

More terms from Juri-Stepan Gerasimov, Apr 15 2010

Edited by N. J. A. Sloane, Apr 21 2010 at the suggestion of R. J. Mathar

cp's OEIS Frontend

A072677 a(n) = prime(prime(n)+1) where prime(k) is the k-th prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A064554 a(n) = Min {k | A064553(k) = n}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

A140853 a(n) = prime(prime(n) - 1) - 1, where prime(n) is the n-th prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A290641 Multiplicative with a(p^e) = prime(p-1)^e.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Scheme

Formula

A050298 Triangle read by rows: T(n,k) = p(r), where r is the (n-k+1)-th number such that A001222(r+1) = k, and p(r) is the r-th prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Extensions

A143444 A054525 * the primes prefaced by 0.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A275990 a(n) = prime(prime(n)-1) - prime(n).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

A175248 Noncomposites (A008578) with noncomposite (A008578) subscripts.

Views