",$n++; } # _Dana Jacobsen - cp's OEIS frontend

A275879 Nonstandard Jacobi primes.

13, 97, 193, 409, 769, 2593, 4729, 6481, 12289, 15361, 55681, 331777, 417793, 737281, 786433, 2752513, 6684673, 8650753, 36175873, 69206017, 75079681, 155344897, 270532609, 435486721, 824717353, 860301577, 1380974593, 1845657601, 3221225473, 3255828481, 3281584129, 8531146753

Offset: 1

N. J. A. Sloane, Aug 17 2016

nonn

Dana Jacobsen, Table of n, a(n) for n = 1..45, (from Cosgrave-Dilcher paper)
J. B. Cosgrave and K. Dilcher, A role for generalized Fermat Numbers, Math. Comp. 86 (2017) preprint paper #10 See Tables 7.4, 7.5..

Cf. A275878.

a(26)-a(32) from Vincenzo Librandi, Aug 18 2016

A275878 Standard Jacobi primes.

Original entry on oeis.org

7, 61, 331, 547, 1951, 2437, 3571, 4219, 7351, 8269, 9241, 10267, 13669, 23497, 25117, 55897, 60919, 74419, 89269, 92401, 102121, 112327, 115837, 126691, 145861, 170647, 202021, 231019, 241117, 246247, 251431, 267307, 283669, 329677, 347821, 360187, 372769

Offset: 1

N. J. A. Sloane, Aug 17 2016

nonn

From Peter Bala, Feb 20 2022: (Start)
Primes of the form (3*k + 2)^3 - (3*k + 1)^3 = 27*k^2 + 27*k + 7.
Equivalently, primes p such that 4*p = 27*x^2 + 1, where x is odd.
Primes p of the form 6*m + 1, where 8*m + 1 is an odd square.
A prime p is in this list iff binomial(2*(p-1)/3,(p-1)/3) == -1 (mod p). See Cosgrave and Dilcher, Theorem 5, Corollary 3. (End)
Subsequence of cuban primes (A002407). - Bernard Schott, Jul 28 2022

Dana Jacobsen, Table of n, a(n) for n = 1..10000
John B. Cosgrave and Karl Dilcher, An Introduction to Gauss Factorials, Amer. Math. Monthly, Vol. 118, No. 9 (November 2011), pp. 812-829.
J. B. Cosgrave and Karl Dilcher, A role for generalized Fermat numbers, Math. Comp., to appear 2016; see also Paper #10, See Table 7.1.

Cf. A002407, A275879.

use ntheory ":all"; forprimes { if (($%3)==1) { my $z = znorder(factorial(($-1)/3),$); $z/=3 unless $z%3; say if $z==1; } } 1e6; # _Dana Jacobsen, Aug 18 2016

use ntheory ":all"; for (0..1000) { my $p = 27*$*$ + 27*$ + 7; say $p if is_prime($p); } # _Dana Jacobsen, Aug 18 2016

Terms a(21) and beyond from Dana Jacobsen, Aug 18 2016

A275450 Numbers n such that primorial(n) contains n as a string of digits.

Original entry on oeis.org

3, 9, 21, 27, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 50, 51, 52, 53, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 73, 74, 75, 76, 77, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 103, 107

Offset: 1

Ivan N. Ianakiev, Jul 28 2016

			Primorial(3) equals 30, which contains 3, therefore 3 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002110 (primorial numbers), A275451.

count:= 0:
P:= 2; p:= 2;
for n from 2 while count < 1000 do
  p:= nextprime(p);
  P:= P*p;
  if StringTools:-Search(sprintf("%d",n),sprintf("%d",P))<>0 then
     count:= count+1;
     A[count]:= n;
  fi
od:
seq(A[i],i=1..count); # Robert Israel, Jul 29 2016

primorial[n_]:=Product[Prime[i],{i,1,n}];
Select[Range@500,StringContainsQ[ToString[primorial[#]],ToString[#]]&]

use ntheory ":all"; my @a275450 = grep { index(pn_primorial($),$) >= 0 } 1 .. 1000; say "@a275450"; # Dana Jacobsen, Aug 09 2016

A275451(a(n)) > 0.

A275631 Primes p such that the product of the distinct prime factors of p^2-1 is less than p.

Original entry on oeis.org

3, 7, 17, 31, 97, 127, 251, 449, 487, 577, 1151, 1249, 1567, 1999, 2663, 4801, 4999, 7937, 8191, 12799, 13121, 13183, 15551, 31249, 31751, 32257, 33857, 35153, 39367, 65537, 79201, 81919, 85751, 115249, 117127, 124001, 126001, 131071

Offset: 1

N. J. A. Sloane, Aug 07 2016, following a suggestion from J. M. Bergot

nonn

Dana Jacobsen, Table of n, a(n) for n = 1..340

Cf. A007947, A275630.

Select[Prime[Range[13000]],Times@@FactorInteger[#^2-1][[All,1]]<#&] (* Harvey P. Dale, Apr 10 2017 *)

use ntheory ":all"; forprimes { say if vecprod(map { $->[0] } factor_exp($**2-1)) < $ } 1e6; # _Dana Jacobsen, Aug 09 2016

Terms a(10) and beyond from Dana Jacobsen, Aug 09 2016

A275629 The least odd prime leaving a prime remainder when divided by each of the first n odd primes.

Original entry on oeis.org

5, 17, 17, 47, 863, 863, 887, 887, 887, 887, 133877, 858707, 886013, 11891003, 11891003, 324840323, 1364103977, 7151489963, 7151489963, 10402068437, 10402068437, 875858298023, 7859437730303, 7859437730303, 10784807012723, 304858842631367, 1939577853386297, 4237718864973557

Offset: 1

N. J. A. Sloane, Aug 07 2016

nonn

			47 mod 3,5,7,11 is 2,2,5,3, which are all primes.

Bert Dobbelaere, Table of n, a(n) for n = 1..36
J. M. Bergot (Proposer), The least odd prime leaving prime remainder when divided by the first nn odd primes, Mathematics Stack Exchange, 2016.

use ntheory ":all"; sub a275629 { my($n)=@; my $it=prime_iterator(3); my @fnp = map { $it->() } 1..$n; while (1) { $p=$it->(); return $p if vecall {is_prime($p%$)} @fnp; } } say a275629($) for 1..20; # _Dana Jacobsen, Aug 09 2016

Corrected and extended by Dana Jacobsen, Aug 09 2016

a(22) onwards from Bert Dobbelaere, Sep 13 2019

A072456 Annihilating primes for A000522.

Original entry on oeis.org

3, 7, 11, 17, 47, 53, 61, 67, 73, 79, 89, 101, 139, 151, 157, 191, 199, 229, 233, 241, 263, 269, 277, 283, 311, 317, 337, 347, 359, 367, 379, 397, 433, 449, 467, 487, 503, 521, 541, 563, 569, 571, 577, 593, 607, 613, 619, 647, 659, 673, 683, 691, 727, 743, 769, 773, 809, 823, 827, 911, 919, 929, 953, 971, 991

Offset: 1

N. J. A. Sloane, Aug 02 2002

nonn

Primes p such that A072453(p) = 0.

Amiram Eldar, Table of n, a(n) for n = 1..3000
C. Cobeli and A. Zaharescu, Promenade around Pascal Triangle-Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 1, 2013, pp. 73-98. - From _N. J. A. Sloane_, Feb 16 2013
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999) 138-150. (ps, pdf)
Wikipedia, Annihilating primes

Cf. A000522, A072453.

use warnings;
  use strict;
  use ntheory ":all";
  use Math::GMPz;
  use Memoize;  memoize 'a000522';
  sub a000522 {
    my($n, $sum, $fn) = (shift, 0, Math::GMPz->new(1));
    do {  $sum += $fn;  $fn *= ($n-$_);  } for 0 .. $n;
    $sum;
  }
  sub a072453 {
    my $n = shift;
    vecsum( map { a000522($_) % $n == 0 } 0 .. $n-1 );
  }
  forprimes { print "$\n" unless a072453($) } 1000;
# Dana Jacobsen, Feb 16 2016

More terms from Vladeta Jovovic, Aug 02 2002

Offset corrected by Amiram Eldar, May 15 2020

A076847 Ramanujan function tau(p) as p runs through the primes.

Original entry on oeis.org

-24, 252, 4830, -16744, 534612, -577738, -6905934, 10661420, 18643272, 128406630, -52843168, -182213314, 308120442, -17125708, 2687348496, -1596055698, -5189203740, 6956478662, -15481826884, 9791485272, 1463791322, 38116845680, -29335099668, -24992917110

Offset: 1

N. J. A. Sloane, Nov 23 2002

sign

From Wolfdieter Lang, May 15 2016: (Start)
This sequence determines all values of Ramanujan's tau function A000594 due to alpha-multiplicativity with alpha(x) = x^11 (the weight of the modular cusp form eta^{24}(z) with the Dedekind eta function is k = 12). See the Apostol reference, p. 138, eq. (54) for alpha-multiplicativity and p. 114, eq. (3) for the tau function. This implies multiplicativity of tau with tau(prime(n)^k) = sqrt(prime(n)^11)^k*S(k, a(n) / sqrt(prime(n)^11)), with the Chebyshev S polynomials (A049310), for n >= 1 and k >= 2. See the Apostol Exercise 6 on p. 139.
Note that the product representation of the Dirichlet series Sum_{n >=1} tau(n)/Sum_{n >= 1} tau(n)/n^s = Prod_{n >= 1} 1/(1 - a(n)/prime(n)^s + prime(n)^(11) / prime(n)^(2*s)) (see the Mordell reference, eq. (2)) leads also to this formula for tau(p^k) for primes p after expanding the factors of the product and collecting powers of 1/p^(k*s). If one insists on convergence of the product one can use s >= 7, if one uses Ramanujan's 1916 conjecture (proved by P. Deligne 1974) |tau(p)| <= 2*p^(11/2), i.e., |a(n)| <= 2*sqrt(prime(n)^11).
(End)

			84480 = A000594(2^3) = sqrt(2^(11))^3*S(3, -24/sqrt(2^(11))) = (-24)*((-24)^2 -2*2^11) = 84480. - _Wolfdieter Lang_, May 15 2016

Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Second edition, Springer, 1990, pp. 114, 138-139.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Denis Xavier Charles, Computing the Ramanujan tau function, Ramanujan J. 11:2 (2006), pp. 221-224.
D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433.
D. H. Lehmer, The Vanishing of Ramanujan's Function tau(n), Duke Mathematical Journal, 14 (1947), pp. 429-433. [Annotated scanned copy]
Louis J. Mordell, On Mr. Ramanujan's empirical expansions of modular functions, Proceedings of the Cambridge Philosophical Society 19 (1917), pp. 117-124.
H. P. F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, pp. 1-55 of Modular Functions of One Variable III (Antwerp 1972), Lect. Notes Math., 350, 1973.
Jan Vonk, Overconvergent modular forms and their explicit arithmetic, Bulletin of the American Mathematical Society 58.3 (2021): 313-356.
Wikipedia, Ramanujan-Petersson conjecture

Cf. A000594, A049310, A278577 (prime powers).

RamanujanTau[Prime[Range[30]]] (* Jean-François Alcover, Dec 01 2015 *)

taup(p)=(65*sigma(p,11)+691*sigma(p,5)-691*252*sum(k=1,p-1,sigma(k,5)*sigma(p-k,5)))/756
a(n)=taup(prime(n)) \\ Charles R Greathouse IV, Apr 22 2013

H(n)=sumdiv(core(n,1)[2],d,my(D=-n/d^2);if(D%4<2,qfbclassno(D)/max(1,D+6)))
taup(p)=my(x='x,P=x^5-9*p*x^4+28*p^2*x^3-35*p^3*x^2+15*p^4*x-p^5);p^5*H(4*p)/2-1-sum(t=1,sqrtint(4*p),subst(P,x,t^2)*H(4*p-t^2))
a(n)=taup(prime(n)) \\ Charles R Greathouse IV, Apr 25 2013

use ntheory ":all"; forprimes { say ramanujan_tau($) } 100 # _Dana Jacobsen, Sep 05 2015

from sympy import prime, divisor_sigma
def A076847(n): return -24 if n == 1 else (q:=(p:=prime(n))**4)*(p+1)-24*(sum((i*(i*(i*(70*i - 140*p) + 90*p**2) - 20*p**3) + q)*divisor_sigma(i)*divisor_sigma(p-i) for i in range(1,p+1>>1))) # Chai Wah Wu, Nov 09 2022

[p for (n,p) in enumerate(list(delta_qexp(100))) if is_prime(n)] # Peter Luschny, May 16 2016

a(n)*a(m) = A000594(prime(n)*prime(m)) for n != m (from the tau multiplicativity). - Wolfdieter Lang, May 15 2016

a(n)^2 = A000594(prime(n)^2) + prime(n)^11 (from alpha-multiplicativity). - Wolfdieter Lang, May 15 2016

A034778 Dirichlet convolution of Ramanujan numbers (A000594) with themselves.

Original entry on oeis.org

1, -48, 504, -2368, 9660, -24192, -33488, 239616, -163782, -463680, 1069224, -1193472, -1155476, 1607424, 4868640, 86016, -13811868, 7861536, 21322840, -22874880, -16877952, -51322752, 37286544, 120766464, -27669550, 55462848, -203834232

Offset: 1

N. J. A. Sloane

Multiplicative because A000594 is. - Christian G. Bower, May 16 2005

			G.f. = x - 48*x^2 + 504*x^3 - 2368*x^4 + 9660*x^5 - 24192*x^6 - 33488*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000594.

a[n_] := DivisorSum[n, RamanujanTau[#]*RamanujanTau[n/#]&]; Array[a, 30] (* Jean-François Alcover, Nov 14 2015 *)

{a(n) = local(A); if( n<1, 0, A = Vec( eta(x + x^n*O(x))^24); sumdiv(n, d, A[d] * A[n/d]))}; /* Michael Somos, Jul 16 2004 */

use ntheory ":all"; for my $n (1..50) { say divisor_sum($n, sub { my $d=shift; ramanujan_tau($d)*ramanujan_tau($n/$d) } # Dana Jacobsen, Sep 05 2015

a(n) = Sum_{d|n} tau(d)tau(n/d) where tau(n) = A000594(n) is Ramanujan's tau function.

A033181 Absolute Euler pseudoprimes: odd composite numbers n such that a^((n-1)/2) == +-1 (mod n) for every a coprime to n.

Original entry on oeis.org

1729, 2465, 15841, 41041, 46657, 75361, 162401, 172081, 399001, 449065, 488881, 530881, 656601, 670033, 838201, 997633, 1050985, 1615681, 1773289, 1857241, 2113921, 2433601, 2455921, 2704801, 3057601, 3224065, 3581761, 3664585, 3828001, 4463641, 4903921

Offset: 1

N. J. A. Sloane, Dec 11 1999

nonn

These numbers n have the property that, for each prime divisor p, p-1 divides (n-1)/2. E.g., 2465 = 5*17*29; 1232/4 = 308; 1232/16 = 77; 1232/28 = 44. - Karsten Meyer, Nov 08 2005
All these numbers are Carmichael numbers (A002997). - Daniel Lignon, Sep 12 2015
These are odd composite numbers n such that b^((n-1)/2) == 1 (mod n) for every base b that is a quadratic residue modulo n and coprime to n. There are no odd composite numbers n such that b^((n-1)/2) == -1 (mod n) for every base b that is a quadratic non-residue modulo n and coprime to n. Note: the absolute Euler-Jacobi pseudoprimes do not exist. Theorem: if an absolute Euler pseudoprime n is a Proth number, then b^((n-1)/2) == 1 for every b coprime to n; by Proth's theorem. Such numbers are 1729, 8355841, 40280065, 53282340865, ...; for example, 1729 = 27*2^6 + 1 with 27 < 2^6. However, it seems that all absolute Euler pseudoprimes n satisfy the stronger congruence b^((n-1)/2) == 1 (mod n) for every b coprime to n, as evidenced by the modified Korselt's criterion (see the first comment). It should be noted that these are odd numbers n such that Carmichael's lambda(n) divides (n-1)/2. Also, these are odd numbers n > 1 coprime to Sum_{k=1..n-1} k^{(n-1)/2}. - Amiram Eldar and Thomas Ordowski, Apr 29 2019
Carmichael numbers k such that (p-1)|(k-1)/2 for each prime p|k. These are odd composite numbers k with half (the maximal possible fraction) of the numbers 1 <= b < k coprime to k that are bases in which k is an Euler-Jacobi pseudoprime, i.e. A329726((k-1)/2)/phi(k) = 1/2. - Amiram Eldar, Nov 20 2019
By Karsten Meyer's and Amiram Eldar's comment, this sequence is numbers k > 1 such that 2*psi(k) | (k-1), where psi = A002322. This means that if k is a term in this sequence, then we actually have a^((k-1)/2) == 1 (mod k) for every a coprime to k. - Jianing Song, Sep 03 2024

Daniel Lignon and Dana Jacobsen, Table of n, a(n) for n = 1..10000 (first 124 terms from Daniel Lignon)
Lorenzo Di Biagio, Euler Pseudoprimes for Half of the Bases, Integers, Vol. 12, No. 6 (2012), pp. 1231-1237, arXiv preprint, arXiv:1109.3596 [math.NT] (2011).
Math Help Forum, how many absolute euler pseudoprimes less than a million, Sep 2009.
Louis Monier, Evaluation and comparison of two efficient primality testing algorithms, Theoretical Computer Science, Vol. 11 (1980), pp. 97-108.
Index entries for sequences related to pseudoprimes

Cf. A002997, A006970, A047713, A080075, A329726.

filter:=  proc(n)
  local q;
  if isprime(n) then return false fi;
  if 2 &^ (n-1) mod n <> 1 then return false fi;
  if not numtheory:-issqrfree(n) then return false fi;
  for q in numtheory:-factorset(n) do
    if (n-1)/2 mod (q-1) <> 0 then return false fi
  od:
  true;
end proc:
select(filter, [seq(i,i=3..10^7,2)]); # Robert Israel, Nov 24 2015

absEulerpspQ[n_Integer?PrimeQ]:=False;
absEulerpspQ[n_Integer?EvenQ]:=False;
absEulerpspQ[n_Integer?OddQ]:=Module[{a=2},
While[aDaniel Lignon, Sep 09 2015 *)
aQ[n_] := Module[{f = FactorInteger[n], p},p=f[[;;,1]]; Length[p] > 1 && Max[f[[;;,2]]]==1 && AllTrue[p, Divisible[(n-1)/2, #-1] &]];Select[Range[3, 2*10^5], aQ] (* Amiram Eldar, Nov 20 2019 *)

use ntheory ":all"; my $n; foroddcomposites { say if is_carmichael($) && vecall { (($n-1)>>1) % ($-1) == 0 } factor($n=$); } 1e6; # _Dana Jacobsen, Dec 27 2015

a(n) == 1 (mod 4). - Thomas Ordowski, May 02 2019

"Absolute Euler pseudoprimes" added to name by Daniel Lignon, Sep 09 2015

Definition edited by Thomas Ordowski, Apr 29 2019

A007811 Numbers k for which 10k+1, 10k+3, 10k+7 and 10k+9 are primes.

Original entry on oeis.org

1, 10, 19, 82, 148, 187, 208, 325, 346, 565, 943, 1300, 1564, 1573, 1606, 1804, 1891, 1942, 2101, 2227, 2530, 3172, 3484, 4378, 5134, 5533, 6298, 6721, 6949, 7222, 7726, 7969, 8104, 8272, 8881, 9784, 9913, 10111, 10984, 11653, 11929, 12220, 13546, 14416, 15727

Offset: 1

N. J. A. Sloane and J. H. Conway, Mar 15 1996

nonn

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

Cf. A024912, A102338, A102342, A102700, A007530, A014561, A008471, A032352, A216292, A216293, A125855.

Cf. A010051, A245304, A245305.

a007811 n = a007811_list !! (n-1)
a007811_list = map (pred . head) $ filter (all (== 1) . map a010051') $
   iterate (zipWith (+) [10, 10, 10, 10]) [1, 3, 7, 9]
-- Reinhard Zumkeller, Jul 18 2014

[n: n in [0..10000] | forall{10*n+r: r in [1,3,7,9] | IsPrime(10*n+r)}]; // Bruno Berselli, Sep 04 2012

for n from 1 to 10000 do m := 10*n: if isprime(m+1) and isprime(m+3) and isprime(m+7) and isprime(m+9) then print(n); fi: od: quit

Select[ Range[ 1, 10000, 3 ], PrimeQ[ 10*#+1 ] && PrimeQ[ 10*#+3 ] && PrimeQ[ 10*#+7 ] && PrimeQ[ 10*#+9 ]& ]
Select[Range[15000], And @@ PrimeQ /@ ({1, 3, 7, 9} + 10#) &] (* Ray Chandler, Jan 12 2007 *)

p=2;q=3;r=5;forprime(s=7,1e5,if(s-p==8 && r-p==6 && q-p==2 && p%10==1, print1(p", ")); p=q;q=r;r=s) \\ Charles R Greathouse IV, Mar 21 2013

use ntheory ":all"; my @s = map { ($-1)/10 } sieve_prime_cluster(10,1e9, 2,6,8); say for @s; # _Dana Jacobsen, May 04 2017

a(n) = 3*A014561(n) + 1. - Zak Seidov, Sep 21 2009

cp's OEIS Frontend

User: ",$n++; } # _Dana Jacobsen

A275879 Nonstandard Jacobi primes.

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A275878 Standard Jacobi primes.

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A275450 Numbers n such that primorial(n) contains n as a string of digits.

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Maple

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A275631 Primes p such that the product of the distinct prime factors of p^2-1 is less than p.

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Mathematica

Perl

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A275629 The least odd prime leaving a prime remainder when divided by each of the first n odd primes.

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Perl

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A072456 Annihilating primes for A000522.

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Perl

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A076847 Ramanujan function tau(p) as p runs through the primes.

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PARI

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A034778 Dirichlet convolution of Ramanujan numbers (A000594) with themselves.

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