A105329 -id:A105329 - cp's OEIS frontend

A105286 Numbers k such that prime(k+1) == 1 (mod k).

1, 2, 3, 10, 24, 25, 66, 168, 182, 186, 187, 188, 438, 6462, 40071, 40084, 40085, 40091, 40108, 40118, 251745, 637224, 637306, 637336, 637338, 10553441, 10553445, 10553452, 10553479, 10553515, 10553550, 10553829, 27067032, 27067054, 27067134, 69709710, 69709713, 179992838, 179993008, 3140421868, 8179002150, 55762149074, 1003652347080, 1003652347109, 1003652347112, 1003652347352, 1003652347375

Offset: 1

Zak Seidov, Apr 25 2005

nonn

If k is a term, then prime(k+1)^prime(k+1) is a reverse Meertens number in base prime(k+1)^((prime(k+1)-1)/k). - Chai Wah Wu, Dec 14 2022

Integers k such that A004649(k+1) = 1. - Michel Marcus, Dec 30 2022

Chai Wah Wu, Meertens Number and Its Variations, arXiv:1603.08493 [math.NT], 2016.
Chai Wah Wu, Meertens number and its variations, Communications on Number Theory and Combinatorial Theory, 3 (2022), article 5.

Cf. A004649, A105287, A105288, A105290, A105329, A105451.

bb={};Do[If[1==Mod[Prime[n+1], n], bb=Append[bb, n]], {n, 1, 200000}];bb

from itertools import count, islice
from sympy import prime
def A105286_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda k: not (prime(k+1)-1)%k,count(max(startvalue,1)))
A105286_list = list(islice(A105286_gen(),10)) # Chai Wah Wu, Dec 14 2022

def A105286(max) :
    terms = []
    p = 3
    for n in range(1, max+1) :
        if (p - 1) % n == 0 : terms.append(n)
        p = next_prime(p)
    return terms
# Eric M. Schmidt, Feb 05 2013

More terms from Farideh Firoozbakht, May 12 2005
First term inserted by Eric M. Schmidt, Feb 05 2013
More terms from Michel Marcus, Dec 29 2022
a(40)-a(47) from Max Alekseyev, Aug 31 2024

A105290 Numbers k such that prime(k+1) == 4 (mod k).

Original entry on oeis.org

1, 3, 11, 13, 69, 71, 637225, 637253, 637313, 637327, 4124459, 4124685, 27067033, 179993017, 179993023, 1208198853, 8179002097, 8179002109, 55762149091

Offset: 1

Zak Seidov, Apr 25 2005

Integers k such that A004649(k+1) = 4. - Michel Marcus, Dec 30 2022

Cf. A023144, A092044.

Cf. A004649, A105286, A105287, A105288, A105329, A105451.

my(n=0, p=2); while(n++, (-4+p=nextprime(p+1))%n || print1(n, ", ")) \\ M. F. Hasler, Feb 05 2009

Missing first two terms inserted by M. F. Hasler, Feb 04 2009
a(11)-a(13) from M. F. Hasler, Feb 05 2009
a(14)-a(15) from Sean A. Irvine, Nov 25 2010
a(16) from D. S. McNeil, Nov 25 2010
a(17)-a(19) from Charles R Greathouse IV, May 05 2011

A105287 Numbers k such that prime(k+1) == 2 (mod k).

Original entry on oeis.org

1, 9, 67, 437, 441, 2615, 100349, 100353, 100359, 637197, 637305, 27066969, 27067049, 27067101, 27067113, 27067115, 179992839, 179993001, 55762149071, 382465573491

Offset: 1

Zak Seidov, Apr 25 2005

Integers k such that A004649(k+1) = 2. - Michel Marcus, Dec 30 2022

Cf. A004649, A105286, A105288, A105290, A105329, A105451.

bb={};Do[If[2==Mod[Prime[n+1], n], bb=Append[bb, n]], {n, 1, 200000}];bb
With[{nn=640000},Flatten[Position[Thread[{Range[nn],Prime[Range[2,nn+1]]}], ?(Mod[Last[#]-2,First[#]]==0&),{1},Heads->False]]] (* _Harvey P. Dale, Sep 23 2021 *)

def A105287(max) :
    terms = []
    p = 3
    for n in range(1, max+1) :
        if (p - 2) % n == 0 : terms.append(n)
        p = next_prime(p)
    return terms
# Eric M. Schmidt, Feb 05 2013

First term inserted by Eric M. Schmidt, Feb 05 2013
More terms from Harvey P. Dale, May 04 2013
a(12)-a(18) from Michel Marcus, Dec 29 2022
a(19)-a(20) from Max Alekseyev, Aug 31 2024

A105288 Numbers k such that prime(k+1) == 3 (mod k).

Original entry on oeis.org

1, 2, 4, 5, 70, 440, 1055, 1058, 6461, 6466, 6469, 251752, 4124468, 27067036, 27067112, 69709709, 69709957, 465769835, 8179002104, 145935689357, 382465573490

Offset: 1

Zak Seidov, Apr 25 2005

Integers k such that A004649(k+1) = 3. - Michel Marcus, Dec 30 2022

Cf. A004649, A105286, A105287, A105290, A105329, A105451.

[1,2] cat [n: n in [1..2*10^4] | NthPrime(n+1) mod n eq 3]; // Vincenzo Librandi, May 02 2018

n:= 0: p:= 2: count:= 0:
for n from 1 while count < 13 do
p:= nextprime(p);
if p-3 mod n = 0 then
    count:= count+1;
  A[count]:= n;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, May 02 2018

bb={};Do[If[3==Mod[Prime[n+1], n], bb=Append[bb, n]], {n, 1, 200000}];bb
Join[{1, 2}, Select[Range[2 10^7], Mod[Prime[# + 1], #]==3 &]] (* Vincenzo Librandi, May 02 2018 *)

def A105288(max) :
    terms = []
    p = 3
    for n in range(1, max+1) :
        if (p - 3) % n == 0 : terms.append(n)
        p = next_prime(p)
    return terms
# Eric M. Schmidt, Feb 05 2013

First two terms inserted by Eric M. Schmidt, Feb 05 2013
a(12)-a(13) from Robert Israel, May 02 2018
a(14)-a(21) from Giovanni Resta, May 02 2018

A105451 Numbers k such that prime(k+1) == 8 (mod k).

Original entry on oeis.org

1, 5, 15, 73, 75, 100347, 637329, 27067271, 179993015, 1208198523, 55762149023, 55762149103, 382465573515

Offset: 1

Zak Seidov, May 02 2005

No additional terms up to 7 million. - Harvey P. Dale, Jan 23 2012

Integers k such that A004649(k+1) = 8. - Michel Marcus, Dec 30 2022

Cf. A004649, A105286, A105287, A105288, A105290, A105329.

Cf. A023150 (Numbers k such that prime(n) == 8 (mod k)).

bb={};Do[If[8==Mod[Prime[n+1], n], bb=Append[bb, n]], {n, 1, 1000000}];bb
Select[Range[700000],Mod[Prime[#+1],#]==8&] (* Harvey P. Dale, Jan 23 2012 *)

from sympy import nextprime
def A105451(max):
    terms = []
    p = 3
    for n in range(1, max+1):
        if (p - 8) % n == 0: terms.append(n)
        p = nextprime(p)
    return terms
# Eric M. Schmidt, Feb 05 2013

First two terms inserted by Eric M. Schmidt, Feb 05 2013
a(8)-a(10) from Michel Marcus, Dec 29 2022
a(11)-a(13) from Max Alekseyev, Aug 31 2024

A105421 Numbers k such that prime(k + 1) == 7 (mod k).

Original entry on oeis.org

1, 2, 3, 4, 8, 30, 31, 33, 68, 72, 180, 1052, 6471, 40083, 40087, 40090, 40113, 40120, 100348, 100360, 100362, 637334, 4124588, 10553439, 10553442, 10553455, 10553478, 10553505, 10553512, 10553827, 10553849, 69709712, 69709719, 69709728, 69709958, 21338685404, 1003652347332, 1003652347349, 1003652347360, 1003652347365

Offset: 1

Zak Seidov, May 02 2005

Manfred Scheucher, Sage Script.

Cf. A105286, A105287, A105288, A105290, A105329, A105451.

Select[Range[650000],Mod[Prime[#+1]-7,#]==0&] (* Harvey P. Dale, Sep 23 2021 *)

isok(n) = Mod(prime(n + 1), n) == Mod(7, n); \\ Michel Marcus, Apr 05 2015

n=0;forprime(p=3,,if(Mod(p,n++)==7,print1(n", "))) \\ Charles R Greathouse IV, Jul 23 2015

from sympy import nextprime
def A105421(max):
    terms = []
    p = 3
    for n in range(1, max+1):
        if (p - 7) % n == 0: terms.append(n)
        p = nextprime(p)
    return terms
# Eric M. Schmidt, Feb 05 2013

a(1)-a(4) inserted by Eric M. Schmidt, Feb 05 2013
a(19)-a(22) from Harvey P. Dale, Apr 05 2015
a(23)-a(35) from Manfred Scheucher, Jul 23 2015
a(36) from Charles R Greathouse IV, Jul 23 2015
a(37)-a(40) from Max Alekseyev, Aug 31 2024

A066458 Numbers n such that Sum_{d runs through digits of n} d^d = pi(n) (cf. A000720).

Original entry on oeis.org

12, 22, 132, 34543, 612415, 27236725, 27236752, 311162281, 311163138, 327361548, 9237866583, 17499331217, 17499551725, 36475999489, 36475999498

Offset: 1

Jason Earls, Jan 02 2002

Note that only two terms, namely 34543 & 17499331217 are primes. So we have: 34543=prime(3^3+4^4+5^5+4^4+3^3), 17499331217=prime(1^1+7^7+4^4+9^9+9^9+3^3+3^3+1^1+2^2+1^1+7^7) and there is no other such prime. - Farideh Firoozbakht, Sep 23 2005

			a(3)=132 because there are exactly 1^1+3^3+2^2 (or 32) prime numbers less than or equal to 132.

C. Caldwell and G. L. Honaker, Jr., Is pi(6521)=6!+5!+2!+1! unique?

Cf. A105328, A105329.

Do[ If[ Apply[Plus, IntegerDigits[n]^IntegerDigits[n]] == PrimePi[n], Print[n]], {n, 1, 10^7} ]

More terms from Robert G. Wilson v, Jan 15 2002
Terms 27236725 onwards from Farideh Firoozbakht, Apr 21 2005 and Sep 17 2005
Sequence completed by Farideh Firoozbakht, Sep 23 2005

cp's OEIS Frontend

A105286 Numbers k such that prime(k+1) == 1 (mod k).

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A105290 Numbers k such that prime(k+1) == 4 (mod k).

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A105287 Numbers k such that prime(k+1) == 2 (mod k).

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A105288 Numbers k such that prime(k+1) == 3 (mod k).

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A105451 Numbers k such that prime(k+1) == 8 (mod k).

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A105421 Numbers k such that prime(k + 1) == 7 (mod k).

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A066458 Numbers n such that Sum_{d runs through digits of n} d^d = pi(n) (cf. A000720).

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