A055494 -id:A055494 - cp's OEIS frontend

A260571 Numbers n such that (n^83+1)/(n+1) is prime.

49, 75, 458, 471, 634, 734, 798, 809, 932, 1139, 1268, 1400, 1498, 1963, 1989, 2112, 2177, 2233, 2252, 2349, 2365, 2446, 2729, 2841, 2861, 2887, 3013, 3048, 3239, 3262, 3403, 3464, 3703, 3855, 3883, 4534, 5147, 5189, 5523, 5611, 5778, 6041, 6200, 6336, 6682, 7068

Offset: 1

Tim Johannes Ohrtmann, Jul 29 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Cf. A055494, A246392, A250174, A250178, A250181, A250185, A250187, A250193, A260558-A260573.

[n: n in [1..10000] |IsPrime((n^83 + 1) div (n + 1))]

Select[Range[1, 10000], PrimeQ[(#^83 + 1)/(# + 1)] &]

for(n=1,10000, if(isprime((n^83+1)/(n+1)), print1(n,", ")))

A260572 Numbers n such that (n^89+1)/(n+1) is prime.

Original entry on oeis.org

16, 20, 93, 195, 227, 325, 465, 758, 888, 911, 1075, 1301, 1590, 1640, 1783, 1807, 2168, 2204, 2231, 2376, 2528, 2591, 2627, 2648, 2909, 2959, 3063, 3109, 3650, 3688, 3709, 3784, 3910, 3943, 4132, 4162, 4385, 4417, 4443, 4613, 5183, 5465, 5574, 5750, 5854, 5975

Offset: 1

Tim Johannes Ohrtmann, Jul 29 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Cf. A055494, A246392, A250174, A250178, A250181, A250185, A250187, A250193, A260558-A260571, A260573.

[n: n in [1..10000] |IsPrime((n^89 + 1) div (n + 1))]

Select[Range[1, 10000], PrimeQ[(#^89 + 1)/(# + 1)] &]

for(n=1,10000, if(isprime((n^89+1)/(n+1)), print1(n,", ")))

A379749 a(n) is the first prime that has digit sum n in base n and n+1 in base n+1.

Original entry on oeis.org

5, 7, 13, 41, 31, 43, 113, 73, 181, 331, 397, 157, 547, 211, 241, 1361, 307, 2053, 761, 421, 463, 1013, 1657, 601, 1301, 3511, 757, 2437, 1741, 1861, 5953, 2113, 1123, 2381, 2521, 6661, 4219, 1483, 3121, 13121, 1723, 3613, 9461, 9901, 6211, 12973, 4513, 7057, 7351, 2551, 15913, 8269, 25759, 2971

Offset: 2

Robert Israel, Jan 01 2025

For n >= 3, the least number with digit sum n in base n and n+1 in base n+1 is A002061(n) = n^2 - n + 1. This is prime for n in A055494. Thus (if it exists) a(n) >= A002061(n), with equality for n in A055494.

			For n = 8, a(8) = 113 because 113 is prime, 113 = 161_8 = 135_9 has digit sums 8 in base 8 and 9 in base 9, and no smaller prime works.

Robert Israel, Table of n, a(n) for n = 2..1000

Cf. A002061, A055494, A379743.

f:= proc(n) local k,v,x;
  for k from 1 do
    v:= convert(convert(k,base,n),`+`);
    if v <= n then
      x:= k*n + n-v;
      if convert(convert(x,base,n+1),`+`) = n+1 and isprime(x) then return x fi
    fi
  od;
end proc:
map(f, [$2 .. 100]);

a[n_]:=Module[{k=1}, While[DigitSum[Prime[k],n]!=n || DigitSum[Prime[k],n+1]!=n+1, k++]; Prime[k]]; Array[a,54,2] (* Stefano Spezia, Jan 01 2025 *)

a(n) = my(p=2); while ((sumdigits(p, n) != n) || (sumdigits(p, n+1) != n+1), p=nextprime(p+1)); p; \\ Michel Marcus, Jan 02 2025

from sympy import isprime
from sympy.ntheory import digits
def nextsod(n, base):
    c, b, w = 0, base, 0
    while True:
        d = n%b
        if d+1 < b and c:
            return (n+1)*b**w + ((c-1)%(b-1)+1)*b**((c-1)//(b-1))-1
        c += d; n //= b; w += 1
def A226636gen(sod=3, base=3): # generator of terms for any sod, base
    an = (sod%(base-1)+1)*base**(sod//(base-1))-1
    while True: yield an; an = nextsod(an, base)
def a(n):
    for k in A226636gen(sod=n, base=n):
        if sum(digits(k, n+1)[1:]) == n+1 and isprime(k):
            return k
print([a(n) for n in range(2, 56)]) # Michael S. Branicky, Jan 04 2025

A108809 Numbers n such that both n+(n-1)^2 and n+(n+1)^2 are primes.

Original entry on oeis.org

2, 3, 4, 7, 9, 15, 18, 25, 34, 55, 58, 63, 67, 100, 102, 139, 144, 148, 154, 162, 163, 168, 190, 195, 219, 232, 247, 267, 280, 289, 330, 349, 379, 384, 417, 427, 448, 454, 477, 568, 580, 643, 645, 669, 672, 727, 762, 793, 802, 813, 837, 847, 900, 975, 988, 993

Offset: 1

Walter Kehowski, Jul 04 2005

			34 is in the sequence because 34 + 33^2 = 1123 and 34 + 35^2 = 1259 are both prime.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A027861.

Intersection of A055494 and A094210. - Michel Marcus, Feb 08 2017

L:=[]; for k from 1 to 1000 do if isprime(k+(k-1)^2) and isprime(k+(k+1)^2) then L:=[op(L),k] fi od;

Select[Range@1000, PrimeQ[#^2 - # + 1] && PrimeQ[#^2 + 3 # + 1] &] (* Ivan Neretin, Feb 08 2017 *)

isok(n) = isprime(n+(n-1)^2) && isprime(n+(n+1)^2); \\ Michel Marcus, Feb 08 2017

A193351 Numbers k such that A071324(k) is prime.

Original entry on oeis.org

3, 4, 8, 9, 16, 18, 49, 50, 64, 81, 169, 225, 288, 324, 392, 578, 625, 729, 882, 900, 1024, 1458, 1568, 1936, 2304, 2450, 2592, 3042, 3136, 3200, 3362, 3600, 4096, 4489, 4802, 4900, 5000, 6241, 6272, 6400, 6962, 7744, 7938, 8100, 10082, 11025, 11552, 12996

Offset: 1

Michel Lagneau, Dec 20 2012

nonn

Numbers k such that the alternating sum of all divisors of k (divisors nonincreasing, starting with k) is prime.
The corresponding primes are 2, 3, 5, 7, 11, 13, 43, 31, 43, 61, 157, ...
It is interesting to note that all numbers except the first are squares, or twice a square, hence the conjecture:
For k > 3, if the alternating sum of all divisors of k (divisors nonincreasing, starting with k) is a prime number, then k is either a square or twice a square.
This conjecture is verified up to k <= 10^8. - Shreyansh Jaiswal, Apr 25 2025
Nevertheless, the difficulty is only for the even numbers. The odd numbers k are necessarily square because the number of divisors of an integer j is odd if and only if j is a square => the alternate sum of the odd divisors is odd.
For large k, the asymptotic law of the ratio r = (number of squares) / (number of twice squares) seems to be r ~ 1. For example, for k < 10^6 we obtain 213 primes corresponding to 118 squares and 95 twice squares.
Similar considerations such as those discussed for the odd squares also exist for the sequence A023194 (numbers k such that sigma(k) is prime). All numbers k except the first are squares, and each number of this sequence is a prime power => sigma(k) is odd.
From Shreyansh Jaiswal, Apr 19 2025: (Start)
For k < 10^6, the ratio r is 118/95, approximately 1.24. For k <= 10^8, there are 815 perfect squares and 546 twice squares. This gives r = 815/546, which is approximately 1.49. This computational data does not hint that r ~ 1.
Interestingly, out of the first 400 terms, 245 terms can be represented as a sum of two distinct positive integer squares.
Another interesting aspect is terms of the form 10^n. For 1 <= n <= 500, only n = 8 (corresponding to 10^8) is a term in the sequence. It appears that terms of the form 10^n are quite uncommon. (End)

			169 is in the sequence because the divisors of 169 are 1, 13, 169 and 169 - 13 + 1 = 157 is prime.

Shreyansh Jaiswal, Table of n, a(n) for n = 1..1362 (terms 1..400 from Harvey P. Dale) [All terms upto 10^8]

Cf. A023194, A055494, A071324.

with(numtheory):for n from 1 to 20000 do:x:=divisors(n):n1:=nops(x):s:=0: s:=sum('((-1)^(i+1))*x[n1-i+1]', 'i'=1..n1): if type(s,prime)=true then printf(`%d, `,n):else fi:od:

Select[Range[13000],PrimeQ[Total[Times@@@Partition[Riffle[ Reverse[ Divisors[ #]],{1,-1},{2,-1,2}],2]]]&] (* Harvey P. Dale, Feb 04 2015 *)

from sympy import *; from functools import lru_cache
cached_divisors = lru_cache()(divisors)
def A071324(n):  return sum(d if i%2==0 else -d for i, d in enumerate(reversed(cached_divisors(n))))
for n in range(1,13001):
    if isprime(A071324(n)):
        print(n, end=", ") # Shreyansh Jaiswal, Apr 17 2025

A212738 a(n) = (7^p - 6^p - 1)/(1806p) where p is the n-th prime.

Original entry on oeis.org

1, 43, 81271, 3570505, 7025726485, 314435374639, 639872336584027, 60775577624897675065, 2794429652350970000851, 276858360603194024261113585, 600808083611945729624598396925, 28083738921571587634894783049047, 61728002094732427074308383210511683

Offset: 3

Michel Lagneau, May 27 2012

nonn

7^p - 6^p - 1 is divisible by 1806p = 6*7*43*p where p prime > 3 (see the proof with the general case).
The sequence is generalizable with the form a(n) = ((k^p - (k-1)^p - 1)) /(k*(k-1)*p*q) where p = prime(n), k integer such that q = k*(k-1) + 1 prime (q = A002383(n) with k = A055494(n)).
k*(k-1)*p*q divides k^p - (k-1)^p - 1, proof :
(1)   p divides k^p - (k-1)^p - 1 (Fermat’s theorem)
(2)   k*(k-1) divides k^p - (k-1)^p - 1
(3)   q = k*(k-1) + 1 divides k^p - (k-1)^p - 1. Suppose  k^p - (k-1)^p - 1 ==r (mod q). Then ((k-1)^p)*k^p - ((k-1)^p)*(k-1)^p - (k-1)^p ==r*(k-1)^p (mod q). But the first term is congruent to -1 (mod q), the second term is congruent to k^p (mod q) and the last term is congruent to (k-1)^p (mod q). We obtain r (mod q) = r*(k-1)^p (mod q) => r = 0.

Andrew Howroyd, Table of n, a(n) for n = 3..100
Peter Vandendriessche and Hojoo Lee, Problems in elementary number theory, Problem A43.

Cf. A002383, A055494.

with(numtheory): for n from 3 to 25 do:p:=ithprime(n):x:=(7^p - 6^p - 1)/(1806*p): printf(`%d, `, x):od:

a(n)={my(p=prime(n)); (7^p - 6^p - 1)/(1806*p)} \\ Andrew Howroyd, Feb 25 2018

A258185 Primes p such that p^2 - q + 1 is prime, where p, q are consecutive primes and p

Original entry on oeis.org

2, 3, 5, 11, 17, 29, 71, 101, 149, 197, 269, 419, 523, 599, 617, 641, 683, 761, 857, 997, 1061, 1063, 1091, 1151, 1201, 1277, 1289, 1409, 1531, 1571, 1607, 1753, 1789, 1987, 2027, 2039, 2111, 2129, 2161, 2267, 2309, 2339, 2503, 2687, 2753, 2999, 3049, 3067, 3257

Offset: 1

K. D. Bajpai, May 23 2015

			a(4) = 11 is prime: 13 is next prime. 11^2 - 13 + 1 = 109 which is also prime.
a(5) = 17 is prime: 19 is next prime. 17^2 - 19 + 1 = 271 which is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..10000

Cf. A000040, A055494, A236477, A236478.

[n: n in [0..10^4] | IsPrime(n) and IsPrime(n^2 - NextPrime(n) +1)]; // Vincenzo Librandi, May 23 2015

Select[Prime[Range[1000]], PrimeQ[#^2 - NextPrime[#] + 1] &]
Select[Partition[Prime[Range[500]],2,1],PrimeQ[#[[1]]^2-#[[2]]+1]&][[All,1]] (* Harvey P. Dale, Sep 06 2016 *)

c=0; forprime(p = 1,1e6, if(isprime(p^2 - nextprime(p+1) + 1), c++; print(c,"  ",p)))

Definition clarified by Harvey P. Dale, Sep 06 2016

A260561 Numbers n such that (n^41+1)/(n+1) is prime.

Original entry on oeis.org

61, 63, 99, 144, 230, 312, 360, 401, 413, 424, 451, 515, 542, 567, 610, 618, 622, 651, 690, 732, 817, 871, 1007, 1100, 1156, 1278, 1403, 1427, 1460, 1535, 1572, 1604, 1681, 1742, 1802, 1820, 1847, 1903, 1910, 1913, 1978, 2019, 2104, 2134, 2152, 2169, 2309, 2383

Offset: 1

Tim Johannes Ohrtmann, Jul 29 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Cf. A055494, A246392, A250174, A250178, A250181, A250185, A250187, A250193, A260558-A260573.

[n: n in [1..10000] |IsPrime((n^41 + 1) div (n + 1))]

Select[Range[1, 10000], PrimeQ[(#^41 + 1)/(# + 1)] &]

for(n=1,10000, if(isprime((n^41+1)/(n+1)), print1(n,", ")))

A260562 Numbers n such that (n^43+1)/(n+1) is prime.

Original entry on oeis.org

2, 3, 6, 22, 59, 83, 91, 95, 120, 148, 195, 196, 201, 247, 252, 264, 315, 360, 378, 458, 555, 680, 792, 893, 1025, 1088, 1158, 1171, 1240, 1280, 1416, 1437, 1632, 1661, 1677, 1681, 1849, 1946, 1960, 2007, 2090, 2092, 2225, 2242, 2244, 2377, 2483, 2547, 2596, 2641

Offset: 1

Tim Johannes Ohrtmann, Jul 29 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Cf. A055494, A246392, A250174, A250178, A250181, A250185, A250187, A250193, A260558-A260573.

[n: n in [1..10000] |IsPrime((n^43 + 1) div (n + 1))]

Select[Range[1, 10000], PrimeQ[(#^43 + 1)/(# + 1)] &]

for(n=1,10000, if(isprime((n^43+1)/(n+1)), print1(n,", ")))

A260563 Numbers n such that (n^47+1)/(n+1) is prime.

Original entry on oeis.org

6, 7, 17, 90, 126, 139, 143, 257, 293, 295, 319, 387, 482, 519, 603, 720, 819, 884, 896, 903, 905, 921, 952, 954, 956, 1042, 1058, 1147, 1170, 1237, 1253, 1279, 1295, 1343, 1366, 1370, 1406, 1465, 1514, 1593, 1595, 1607, 1609, 1622, 1701, 1705, 1709, 1736, 1772

Offset: 1

Tim Johannes Ohrtmann, Jul 29 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Cf. A055494, A246392, A250174, A250178, A250181, A250185, A250187, A250193, A260558-A260573.

[n: n in [1..10000] |IsPrime((n^47 + 1) div (n + 1))]

Select[Range[1, 10000], PrimeQ[(#^47 + 1)/(# + 1)] &]

for(n=1,10000, if(isprime((n^47+1)/(n+1)), print1(n,", ")))

cp's OEIS Frontend

A260571 Numbers n such that (n^83+1)/(n+1) is prime.

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A260572 Numbers n such that (n^89+1)/(n+1) is prime.

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A379749 a(n) is the first prime that has digit sum n in base n and n+1 in base n+1.

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A108809 Numbers n such that both n+(n-1)^2 and n+(n+1)^2 are primes.

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A193351 Numbers k such that A071324(k) is prime.

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A212738 a(n) = (7^p - 6^p - 1)/(1806p) where p is the n-th prime.

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A258185 Primes p such that p^2 - q + 1 is prime, where p, q are consecutive primes and p

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Extensions

A260561 Numbers n such that (n^41+1)/(n+1) is prime.