Richard N. Smith - cp's OEIS frontend

A309129 Numbers n such that -n is a quadratic nonresidue modulo all odd primes p <= sqrt(n) which do not divide n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 30, 33, 37, 40, 42, 43, 45, 48, 57, 58, 60, 63, 67, 70, 72, 78, 85, 88, 93, 100, 102, 105, 112, 120, 130, 133, 135, 147, 148, 163, 165, 168, 177, 190, 210, 232, 240, 247, 253, 267, 268, 273, 280, 312, 330, 333, 345, 357, 385, 408, 462, 520, 522, 652, 708, 760, 840, 928, 1320, 1365, 1467, 1848

Offset: 1

Richard N. Smith, Jul 13 2019

nonn

Contains A000926 and A003173 (except the term 11) as subsequences.

Conjecture: 1848 is the last term of this sequence.

			42 is in this sequence because sqrt(42) = 6.480740..., and -42 is quadratic nonresidue mod all odd primes < 6.480740... not dividing 42 (only mod 5).
67 is in this sequence because sqrt(67) = 8.185352..., and -67 is quadratic nonresidue mod all odd primes < 8.185352... not dividing 67 (mod 3, mod 5 and mod 7).
17 is not in this sequence because -17 is quadratic residue mod 3 and 3 < sqrt(17) and 3 does not divide 17.
90 is not in this sequence because -90 is quadratic residue mod 7 and 7 < sqrt(90) and 7 does not divide 90.
For n < 9, the range of p is empty, thus the numbers n < 9 are trivially in this sequence.

Cf. A000926, A003173.

a(n)=forprime(p=3, ,if(kronecker(-n,p)==1,return(p)))
for(k=1, 10^6,if(a(k)>sqrt(k),print1(k, ", ")))

A309130 Smallest prime factor of A077586(n).

Original entry on oeis.org

7, 127, 2147483647, 170141183460469231731687303715884105727, 47, 338193759479, 231733529, 62914441, 2351, 1399, 295257526626031, 18287, 106937, 863, 4703, 138863, 22590223644617

Offset: 1

Richard N. Smith, Jul 13 2019

A263686 is a subsequence.
Agrees with A263686 in the first four terms, but then the two sequences differ for the first time at n = 5, because prime(5) = 11 is not in A000043.
a(18) = A263686(9) is greater than 1.56*10^17*(2^61-1), see link.
a(n) = A077586(n) iff A077586(n) is prime, A077586(n) is prime for 1 <= n <= 4, but composite for 5 <= n <= 17. The status of A077586(18) = 2^(2^61-1)-1 is unknown. It is conjectured that A077586(n) is composite for all n >= 5.
a(20) = 456959, a(21) = 18384329, a(22) = 198839, a(23) = 2349023, a(24) = A263686(10) is greater than 1.25*10^16*(2^89-1).
Conjecture: All terms are in A122094 (all terms in A263686 are in A122094).
For examples related to that conjecture, see A322568. - Jeppe Stig Nielsen, Aug 29 2019
a(30) = 46559, a(32) = 23671, a(36) = 7151489, a(39) = 4698047, a(41) = 719, a(43) = 1440847, a(45) = 179689, a(47) = 11759383, a(48) = 23602441, a(50) = 9024439, a(51) = 28875361, a(52) = 6301423, a(54) = 2493983, a(56) = 33518137, a(59) = 6727783, a(66) = 95111, a(72) = 1439, a(73) = 99833, a(78) = 38119, a(81) = 26849, a(83) = 8258911, a(86) = 16173559, a(89) = 625343, a(93) = 9743. - Chai Wah Wu, Oct 16 2019

Double Mersennes Prime Search, History.
Eric Weisstein's World of Mathematics, Double Mersenne Number.

Cf. A077586, A263686, A001348, A049479.

A309130(n)=A020639(2^(2^prime(n)-1)-1) \\ For efficiency, use addprimes([large terms of this sequence]). - M. F. Hasler, Mar 01 2025

a(n) = A020639(A077586(n)).

a(n) = A049479(A001348(n)). - M. F. Hasler, Mar 01 2025

A326615 a(n) is the smallest prime p such that Sum_{primes q <= p} Kronecker(n,q) > 0, or 0 if no such prime exists.

Original entry on oeis.org

2, 11100143, 61981, 3, 2082927221, 5, 2, 11100143, 2, 3, 577, 61463, 2083, 11, 2, 3, 2, 11100121, 5, 2082927199, 1217, 3, 2, 5, 2, 17, 61981, 3, 719, 7, 2, 11100143, 2, 3, 23, 5, 11, 31, 2, 3, 2, 13, 17, 7, 2082927199, 3, 2, 61463, 2, 11100121, 7, 3, 17, 5, 2, 11, 2, 3, 31, 7, 5, 41, 2, 3

Offset: 1

Richard N. Smith, Jul 15 2019

nonn

Richard N. Smith, Table of n, a(n) for n = 1..4096

Cf. A306499, A306500.

a(n) = my(i=0); forprime(p=2, oo, i+=kronecker(n, p); if(i>0, return(p))) \\ after Jianing Song in A306499

a(A003658(n)) = A306499(n).
a(n) = 2 iff n == 1 or 7 mod 8 (see A047522).
a(n) = 3 iff n == 4 mod 6 (see A016957).

A326609 Largest minimal prime in base n (written in base 10).

Original entry on oeis.org

3, 13, 5, 3121, 5209, 2801, 76695841, 811, 66600049, 29156193474041220857161146715104735751776055777, 388177921

Offset: 2

Richard N. Smith, Jul 13 2019

a(13) is (probably) 13^32020*8+183, it has 35670 digits, a(14) = 14^85*4+65, it has 99 digits, a(15) = (15^106*66-619)/7, it has 126 digits, a(16) = 16^3544*9+145, it has 4269 digits.
a(17) is the smallest prime of the form (4105*17^k-9)/16 if it exists, otherwise (probably) (73*17^111333-9)/16 (136991 digits), a(18) = 18^31*304+1 (42 digits).
Other known terms: a(20) = (20^449*16-2809)/19 (585 digits), a(22) = 22^763*20+7041 (1026 digits), a(23) is (probably) (23^800873*106-7)/11 (1090573 digits), a(24) = (24^99*512-121)/23 (138 digits), a(30) = 30^1023*12+1 (1513 digits), a(42) = (42^487*27-1093)/41 (791 digits).
a(19) is the smallest prime of the form (15964*19^k-1)/3 if it exists, otherwise (probably) (904*19^110984-1)/3 (141924 digits), a(21) is the smallest prime of the form 16*21^k+335 if it exists, otherwise (probably) (51*21^479149-1243)/4 (633542 digits).

Richard N. Smith, Table of n, a(n) for n = 2..16
Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, preprint, 2014.
Curtis Bright, Raymond Devillers, and Jeffrey Shallit, Minimal Elements for the Prime Numbers, Experimental Mathematics, Volume 25, 2016 - Issue 3.
C. K. Caldwell, The Prime Glossary, minimal prime
Richard N. Smith, List of all known terms
Top PRPs, Search for 8*13^n+183
Top PRPs, Search for (73*17^n-9)/16
Top PRPs, Search for (106*23^n-7)/11
Wikipedia, Minimal prime (recreational mathematics)

Cf. A071062 (base 10 minimal primes), A110600 (base 12 minimal primes).

Cf. A293142 (largest non-repunit permutable prime), A317689 (largest non-repunit circular prime), A103443 (largest left-truncatable prime), A023107 (largest right-truncatable prime), A323137 (largest two-sided prime), A084738 (smallest repunit prime), A186995 (smallest weakly prime).

A326655 Numbers k such that 3*4^k+1 is prime.

Original entry on oeis.org

1, 3, 4, 6, 9, 15, 18, 33, 138, 204, 219, 267, 1104, 1408, 1584, 1956, 17175, 21147, 24075, 27396, 27591, 40095, 354984, 400989, 916248, 1145805, 2541153, 5414673

Offset: 1

Richard N. Smith, Jul 16 2019

One half of the even terms in A002253.

Numbers k such that r*(r+1)^k+1 is prime: A003306 (r=2), this sequence (r=3), A204322 (r=4), A247260 (r=5), A245241 (r=6), A269544 (r=7), A056799 (r=8), A056797 (r=9), A057462 (r=10), A251259 (r=11).

A326618 a(n) = n^18 + n^9 + 1.

Original entry on oeis.org

1, 3, 262657, 387440173, 68719738881, 3814699218751, 101559966746113, 1628413638264057, 18014398643699713, 150094635684419611, 1000000001000000001, 5559917315850179173, 26623333286045024257, 112455406962561892503, 426878854231297789441, 1477891880073843750001

Offset: 0

Richard N. Smith, Jul 15 2019

nonn

a(n) = Phi_27(n) where Phi_k(x) is the k-th cyclotomic polynomial.

Richard N. Smith, Table of n, a(n) for n = 0..2000
OEIS Wiki, Cyclotomic Polynomials at x=n, n! and sigma(n)
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
Index to values of cyclotomic polynomials of integer argument

Sequences of the type Phi_k(n), where Phi_k is the k-th cyclotomic polynomial: A000012 (k=0), A023443 (k=1), A000027 (k=2), A002061 (k=3), A002522 (k=4), A053699 (k=5), A002061 (k=6), A053716 (k=7), A002523 (k=8), A060883 (k=9), A060884 (k=10), A060885 (k=11), A060886 (k=12), A060887 (k=13), A060888 (k=14), A060889 (k=15), A060890 (k=16), A269442 (k=17), A060891 (k=18), A269446 (k=19), A060892 (k=20), A269483 (k=21), A269486 (k=22), A060893 (k=24), A269527 (k=25), A266229 (k=26), this sequence (k=27), A270204 (k=28), A060894 (k=30), A060895 (k=32), A060896 (k=36).

Cf. A153440 (indices of prime terms).

[n^18+n^9+1: n in [0..17]]; // Vincenzo Librandi, Jul 15 2019

Table[n^18 + n^9 + 1, {n, 0, 17}] (* Vincenzo Librandi, Jul 15 2019 *)
Table[Cyclotomic[27, n], {n, 0, 17}]

a(n) = polcyclo(27, n); \\ Michel Marcus, Jul 20 2019

A326614 Smallest Euler-Jacobi pseudoprime to base n.

Original entry on oeis.org

9, 561, 121, 341, 781, 217, 25, 9, 91, 9, 133, 91, 85, 15, 1687, 15, 9, 25, 9, 21, 221, 21, 169, 25, 217, 9, 121, 9, 15, 49, 15, 25, 545, 33, 9, 35, 9, 39, 133, 39, 21, 451, 21, 9, 481, 9, 65, 49, 25, 49, 25, 51, 9, 55, 9, 55, 25, 57, 15, 481, 15, 9, 529, 9, 33, 65, 33, 25, 35, 69, 9

Offset: 1

Richard N. Smith, Jul 14 2019

nonn

a(n) = 9 for n == 1 or 8 mod 9 (see A056020).

Richard N. Smith, Table of n, a(n) for n = 1..5000
Wikipedia, Euler-Jacobi pseudoprime

Cf. A047713, A048950, A090086 (least Fermat pseudoprime to base n), A298756 (least strong pseudoprime to base n).

ejpspQ[n_,b_] := CoprimeQ[n,b] && CompositeQ[n] && Mod[b^((n - 1)/2) - JacobiSymbol[b, n], n] == 0; leastEJpsp[b_] := Module[{k=9}, While[!ejpspQ[k, b], k+=2]; k]; Array[leastEJpsp, 100] (* Amiram Eldar, Jul 15 2019 *)

isok(k, n) = ((k%2==1) && (gcd(k, n)==1) && Mod(n, k)^((k-1)/2)==kronecker(n, k) && !isprime(k));
a(n) = my(k=2); while (! isok(k, n), k++); k; \\ Michel Marcus, Jul 15 2019

A326612 Indices where A001175 (Pisano period) sets a new record value.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 25, 30, 50, 98, 125, 150, 206, 243, 250, 490, 566, 590, 625, 750, 1030, 1046, 1094, 1154, 1214, 1226, 1250, 2450, 2738, 2830, 2846, 2894, 2906, 3086, 3125, 3750, 4802, 5534, 5594, 5606, 5666, 5714, 5770, 5834, 5906, 5990, 6070, 6130, 6250

Offset: 1

Richard N. Smith, Jul 14 2019

nonn

Record values: 1, 3, 8, 20, 24, 60, 100, 120, 300, 336, 500, 600, 624, 648, 1500, 1680, 1704, 1740, 2500, 3000, ...

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (n = 1..1547 from Richard N. Smith)

Cf. A001175.

fibmod(n, m)=((Mod([1, 1; 1, 0], m))^n)[1, 2]
entryp(p)=my(k=p+[0, -1, 1, 1, -1][p%5+1], f=factor(k)); for(i=1, #f[, 1], for(j=1, f[i, 2], if((Mod([1, 1; 1, 0], p)^(k/f[i, 1]))[1, 2], break); k/=f[i, 1])); k
entry(n)=if(n==1, return(1)); my(f=factor(n), v); v=vector(#f~, i, if(f[i, 1]>1e14, entryp(f[i, 1]^f[i, 2]), entryp(f[i, 1])*f[i, 1]^(f[i, 2] - 1))); if(f[1, 1]==2&&f[1, 2]>1, v[1]=3<r, r=a(n); print1(", "n))) \\ after Charles R Greathouse IV in A001175

A326610 Least k such that A000790(k) = A108574(n).

Original entry on oeis.org

0, 3, 26, 11, 14, 59, 83, 23, 443, 338, 263, 578, 38, 662, 47, 227, 3467, 1823, 842, 4898, 983, 4622, 4847, 2747, 4127, 11567, 347, 542, 17483, 2867, 22367, 43067, 18527, 5042, 5063, 12422, 66047, 2858, 87302, 11702, 11147, 3062, 24602, 158, 94763, 247838, 1202

Offset: 1

Richard N. Smith, Jul 14 2019

The largest term is a(106) = 10009487 (for the primary pretender 453).

Richard N. Smith, Table of n, a(n) for n = 1..132
N. J. A. Sloane, The primary pretenders
Richard N. Smith, Least positive and negative bases such that the primary pretender is n (format: n (in A108574), least positive base, least negative base)

Cf. A000790, A108574.

cp's OEIS Frontend

User: Richard N. Smith

A309129 Numbers n such that -n is a quadratic nonresidue modulo all odd primes p <= sqrt(n) which do not divide n.

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A309130 Smallest prime factor of A077586(n).

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A326615 a(n) is the smallest prime p such that Sum_{primes q <= p} Kronecker(n,q) > 0, or 0 if no such prime exists.

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A326609 Largest minimal prime in base n (written in base 10).

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A326655 Numbers k such that 3*4^k+1 is prime.

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A326618 a(n) = n^18 + n^9 + 1.

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A326614 Smallest Euler-Jacobi pseudoprime to base n.

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A326612 Indices where A001175 (Pisano period) sets a new record value.

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A326610 Least k such that A000790(k) = A108574(n).

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