A003628 -id:A003628 - cp's OEIS frontend

A377179 Primes p such that -9/2 is a primitive root modulo p.

5, 13, 23, 29, 31, 47, 53, 61, 71, 79, 101, 109, 149, 151, 157, 167, 173, 191, 197, 199, 223, 229, 239, 263, 269, 277, 293, 311, 317, 359, 367, 373, 383, 389, 461, 463, 479, 487, 503, 509, 557, 599, 613, 647, 653, 661, 677, 701, 709, 719, 733, 743, 757, 773, 797, 821, 823, 829, 839, 853, 863, 887, 911, 967, 983, 991

Offset: 1

Jianing Song, Oct 18 2024

If p is a term in this sequence, then -9/2 is not a square modulo p (i.e., p is in A003628).

Conjecture: this sequence has relative density equal to Artin's constant (A005596) with respect to the set of primes.

Wikipedia, Artin's conjecture on primitive roots.
Index entries for primes by primitive root

Primes p such that +a/2 is a primitive root modulo p: A320384 (a=3), A377174 (a=5), A377176 (a=7), A377178 (a=9).
Primes p such that -a/2 is a primitive root modulo p: A377172 (a=3), A377175 (a=5), A377177 (a=7), this sequence (a=9).
Cf. A003628, A005596.

forprime(p=5, 10^3, if(znorder(Mod(-9/2, p))==p-1, print1(p, ", ")));

A124923 a(n) = n^(n-1) + 1.

Original entry on oeis.org

2, 3, 10, 65, 626, 7777, 117650, 2097153, 43046722, 1000000001, 25937424602, 743008370689, 23298085122482, 793714773254145, 29192926025390626, 1152921504606846977, 48661191875666868482, 2185911559738696531969

Offset: 1

Alexander Adamchuk, Nov 12 2006

Prime p divides a(p-1). n divides a(n-1) for all prime n and all odd composite n.
p divides a((p+1)/2) for prime p = {3, 5, 11, 13, 19, 29, 37, 43, 53, 59, 61, 67, 83, ...} = A003629 (Primes congruent to {3,5} mod 8).
p divides a((p+3)/4) for prime p = {13, 73, 97, 109, 181, 229, 241, 277, 337, 409, 421, 457, 541, 709, 733, 757, 829, ...} = A107141 (Primes of the form 4x^2+9y^2).
p divides a((p+5)/6) for prime p = {43, 61, 79, 109, 151, 163, 181, 193, 313, 337, 433, 523, 577, 631, 643, 673, 787, 829, 907, 991, ...}.
p divides a((p+7)/8) for prime p = {113, 137, 569, 641, 673, 1129, 1289, 1297, 1481, 1801, ...}.
p divides a((3p-1)/2) for prime p = {5, 7, 13, 23, 29, 31, 37, 47, 53, 61, 71, 79, 101, 103, 109, 127, 149, 151, 157, 167, 173, 181, 191, 197, 199, ...} = A003628 (Primes congruent to {5, 7} mod 8).
p^2 divides a((3p-1)/2) for prime p = {5, 13, 173, 5501, ...} = A124924.

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A000169, A003628, A003629, A107141, A124924.

List([1..20], n-> n^(n-1) + 1); # G. C. Greubel, Nov 19 2019

[n^(n-1) + 1: n in [1..20]]; // Vincenzo Librandi, Aug 14 2012

seq(n^(n-1) + 1, n=1..20); # G. C. Greubel, Nov 19 2019

Mathematica
```
Table[n^(n-1)+1, {n,20}]
```

vector(20, n, n^(n-1) + 1) \\ G. C. Greubel, Nov 19 2019

[n^(n-1) + 1 for n in (1..20)] # G. C. Greubel, Nov 19 2019

a(n) = n^(n-1) + 1.
a(n) = A000169(n) + 1.
E.g.f.: -1 + exp(x) - W(-x), where W(x) is the Lambert w-function. - G. C. Greubel, Nov 19 2019

A124924 Primes p such that p^2 divides A124923((3p-1)/2) = ((3p-1)/2)^(3(p-1)/2) + 1.

Original entry on oeis.org

5, 13, 173, 5501

Offset: 1

Alexander Adamchuk, Nov 12 2006

p divides A124923((3p-1)/2) for primes p in A003628. Hence this sequence is a subsequence of A003628.
Also, primes p such that (-2)^((p-1)/2) == -1-3p/2 (mod p^2).
No other terms below 10^11.

			5 is in this sequence because A124923((3*5-1)/2) = A124923(7) = 7^8 + 1 = 117650 is divisible by 5^2 = 25.

Cf. A124923, A003628.

Do[ p = Prime[n]; m = (3p-1)/2; f = PowerMod[ m, m-1, p^2 ] + 1; If[ IntegerQ[ f/p^2 ], Print[p] ], {n,2,10000} ]

Edited by Max Alekseyev, Jan 28 2012

A165815 Prime congruent numbers (A003273).

Original entry on oeis.org

5, 7, 13, 23, 29, 31, 37, 41, 47, 53, 61, 71, 79, 101, 103, 109, 127, 137, 149, 151, 157, 167, 173, 181, 191, 197, 199, 223, 229, 239, 257, 263, 269, 271, 277, 293, 311, 313, 317, 349, 353, 359, 367, 373, 383, 389, 397, 421, 431, 439, 457, 461, 463, 479, 487

Offset: 1

T. D. Noe, Sep 28 2009

nonn

Heegner proved that every prime p with p = 5 or 7 (mod 8) is a congruent number. See A003628 for those primes.

Kurt Heegner, Diophantische Analysis und Modulfunktionen, Math. Zeitschrift 56 (1952), 227-253.

Kent E. Morrison, Congruent Numbers

Cf. A003273, A003628.

A263458 Deal a pack of n cards into two piles and gather them up, n/2 times. All n such that this reverses the order of the deck.

Original entry on oeis.org

4, 6, 12, 22, 28, 30, 36, 46, 52, 60, 70, 78, 100, 102, 108, 126, 148, 150, 156, 166, 172, 180, 190, 196, 198, 222, 228, 238, 262, 268, 270, 276, 292, 310, 316, 348, 358, 366, 372, 382, 388, 396, 420, 430, 438, 460, 462, 478, 486, 502, 508, 540, 556, 598

Offset: 1

Christian Perfect, Oct 19 2015

nonn

This seems to be A003628(n)-1; that is, each element of this sequence is one less than a prime congruent to 5 or 7 modulo 8.

			Take a deck of 52 playing cards. Deal it into two piles, then pick up the first pile and put it on top of the other. Do this 26 times. The order of the deck is reversed, so 52 belongs to this sequence.
6 is in the sequence because the 3 shuffles are [1, 2, 3, 4, 5, 6] -> [5, 3, 1, 6, 4, 2] -> [4, 1, 5, 2, 6, 3] -> [6, 5, 4, 3, 2, 1], original reversed. 8 is not in the sequence because the 4 shuffles are [1, 2, 3, 4, 5, 6, 7, 8] -> [7, 5, 3, 1, 8, 6, 4, 2] -> [4, 8, 3, 7, 2, 6, 1, 5] -> [1, 2, 3, 4, 5, 6, 7, 8] -> [7, 5, 3, 1, 8, 6, 4, 2], not the original reversed. - _R. J. Mathar_, Aug 02 2024

Cf. A003628, A373416.

isA263458 := proc(n)
    local L,itr ;
    L := [seq(i,i=1..n)] ;
    for itr from 1 to n/2 do
        L := pileShuf(L) ; # function code in A323712
    end do:
    for i from 1 to nops(L) do
        if op(-i,L) <> i then
            return false ;
        end if;
    end do:
    true ;
end proc:
n := 1;
for k from 2 do
    if isA263458(k) then
        printf("%d %d\n",n,k) ;
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 02 2024

from itertools import cycle
def into_piles(r,deck):
    packs = [[] for i in range(r)]
    for card, pack in zip(range(1,deck+1),cycle(range(r))):
        packs[pack].insert(0,card)
    out = sum(packs,[])
    return Permutation(out)
def has_reversing_property(deck):
    p = power(into_piles(2,deck), deck/2)
    return p==into_piles(1,deck)
[i for i in range(2,400,2) if has_reversing_property(i)]

A165816 Prime congruent numbers (A165815) that are not equal to 5 or 7 (mod 8).

Original entry on oeis.org

41, 137, 257, 313, 353, 457, 761, 1201, 1217, 1249, 1321, 2113, 2273, 2777, 2833, 2953, 3001, 3433, 3593, 3761, 3881, 4441, 4481, 4649, 4793, 4889, 5273, 5449, 5569, 5657, 5849, 6073, 6529, 7001, 7321, 7417, 7561, 7793, 8521, 8609, 9049, 9257, 9281

Offset: 1

T. D. Noe, Sep 28 2009

nonn

Heegner proved that every prime p with p = 5 or 7 (mod 8) is a congruent number. See A003628 for those primes. All primes in this sequence equal 1 (mod 8).

Monsky proved that no prime of the form 8k+3 is a congruent number. - Jonathan Sondow, Nov 15 2017

T. D. Noe, Primes less than 10^7
Kurt Heegner, Diophantische Analysis und Modulfunktionen, Math. Zeitschrift 56 (1952), 227-253.
P. Monsky, Mock Heegner points and congruent numbers, Math. Zeit., 204 (1990), 45-67.
Kent E. Morrison, Congruent Numbers

Cf. A003273 (congruent numbers), A165815 (prime congruent numbers).

A369863 Inert rational primes in the field Q(sqrt(-21)).

Original entry on oeis.org

13, 29, 43, 47, 53, 59, 61, 67, 73, 79, 83, 97, 113, 127, 131, 137, 149, 151, 157, 163, 167, 181, 197, 211, 227, 229, 233, 241, 251, 281, 311, 313, 317, 331, 349, 379, 383, 389, 397, 401, 409, 419, 433, 449, 463, 467, 479, 487, 499, 503, 547, 557, 563, 569, 571, 577, 587

Offset: 1

Dimitris Cardaris, Feb 03 2024

nonn

Primes p such that Legendre(-21,p) = -1.

Cf. inert rational primes in the imaginary quadratic field Q(sqrt(-d)) for the first squarefree positive integers d: A002145 (1), A003628 (2), A003627 (3), A003626 (5), A191059 (6), A003625 (7), A296925 (10), A191060 (11), A105885 (13), A191061 (14), A191062 (15), A296930 (17), A191063 (19), this sequence (21), A191064 (22), A191065 (23).

Select[Range[3,600], PrimeQ[#] && JacobiSymbol[-21,#]==-1 &] (* Stefano Spezia, Feb 04 2024 *)

[p for p in prime_range(3, 600) if legendre_symbol(-21, p) == -1]

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A377179 Primes p such that -9/2 is a primitive root modulo p.

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A124923 a(n) = n^(n-1) + 1.

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A124924 Primes p such that p^2 divides A124923((3p-1)/2) = ((3p-1)/2)^(3(p-1)/2) + 1.

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A165815 Prime congruent numbers (A003273).

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A263458 Deal a pack of n cards into two piles and gather them up, n/2 times. All n such that this reverses the order of the deck.

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A165816 Prime congruent numbers (A165815) that are not equal to 5 or 7 (mod 8).

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A369863 Inert rational primes in the field Q(sqrt(-21)).

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