A079704 -id:A079704 - cp's OEIS frontend

A350638 Numbers of the form p^2*q, with odd primes p > q, such that q divides p-1.

147, 507, 605, 1083, 2883, 4107, 4805, 5547, 5819, 5887, 8405, 11163, 12943, 13467, 15987, 18605, 18723, 25205, 28227, 31827, 35287, 35643, 36517, 48387, 49379, 50807, 51005, 57963, 68403, 73947, 79707, 81133, 85805, 87131, 89383, 98283, 100949, 111747, 112903

Offset: 1

Bernard Schott, Jan 10 2022

nonn

For these terms m there are precisely (q+9)/2 groups of order m.

Only two of these groups are abelian: C_{p^2*q} and (C_p X C_p) X C_q. The (q+5)/2 groups that are nonabelian are C_{p^2} : C_q and the (q+3)/2 semidirect products of the form (C_p X C_p) : C_q that are not isomorphic, where C means cyclic groups of the stated order, the symbols X and : mean direct and semidirect products respectively.

			147 = 7^2 * 3, 3 and 7 are odd primes, 3 divides 7-1 = 6, hence 147 is a term.

Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, problème 1.35, pp. 70-74, 2004.

N. S. Wedd, Groups of orders 147.

Other subsequences of A054753 linked with order of groups: A079704, A143928, A349495, A350115, A350245, A350332, A350421, A350422.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; e == {1, 2} && Divisible[p[[2]] - 1, p[[1]]]]; Select[Range[1, 120000, 2], q] (* Amiram Eldar, Jan 11 2022 *)

isok(m) = if (m%2, my(f=factor(m)); if (f[, 2] == [1, 2]~, my(p=f[1, 1], q=f[2, 1]); ((q-1) % p) == 0)); \\ Michel Marcus, Jan 11 2022

from sympy import integer_nthroot, primerange
def aupto(limit):
    aset, maxp = set(), integer_nthroot(limit**2, 3)[0]
    for p in primerange(5, maxp+1):
        pp = p*p
        for q in primerange(3, min(p, limit//pp+1)):
            if (p-1)%q == 0:
                aset.add(pp*q)
    return sorted(aset)
print(aupto(113000)) # Michael S. Branicky, Jan 10 2022

More terms from Michael S. Branicky, Jan 10 2022

A179262 a(n) = 2*prime(n)^2 - 1.

Original entry on oeis.org

7, 17, 49, 97, 241, 337, 577, 721, 1057, 1681, 1921, 2737, 3361, 3697, 4417, 5617, 6961, 7441, 8977, 10081, 10657, 12481, 13777, 15841, 18817, 20401, 21217, 22897, 23761, 25537, 32257, 34321, 37537, 38641, 44401, 45601, 49297, 53137, 55777

Offset: 1

Frank Wals (frank.wals(AT)gmail.com), Jul 06 2010, Jul 09 2010

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000040.

Array[2 Prime[#]^2 - 1 &, 10^4] (* Michael De Vlieger, Oct 06 2019 *)

a(n) = A079704(n) - 1 = 2*prime(n)^2 - 1. - R. J. Mathar, Jul 08 2010

More terms from R. J. Mathar, Jul 08 2010

A255586 Composite k such that Sum_{i=1..t-1} d(i+1)/d(i) is an integer, where d(1), ..., d(t) are the divisors of k in ascending order.

Original entry on oeis.org

4, 8, 9, 16, 18, 25, 27, 32, 48, 49, 50, 64, 81, 98, 108, 121, 125, 128, 162, 169, 242, 243, 256, 289, 338, 343, 361, 375, 512, 529, 578, 625, 722, 729, 841, 961, 1024, 1029, 1058, 1250, 1331, 1369, 1458, 1681, 1682, 1849, 1920, 1922, 2048, 2187, 2197, 2209

Offset: 1

Michel Lagneau, Feb 27 2015

nonn

The sequence is infinite because the powers of 2 (A000079) are in the sequence.
The prime powers with even exponents (A056798) are in the sequence.
The cubes of primes (A030078) are in the sequence.
The numbers of the form 2p^2 (A079704) with p prime are in the sequence.
The corresponding integers are 4, 6, 6, 8, 9, 10, 9, 10, 14, 14, 11, 12, 12, 13, 17, 22, 15, 14, 16, 26, 17, 15, 16, 34, 19, ...

			18 is in the sequence because the divisors of 18 are {1, 2, 3, 6, 9, 18} and 2/1 + 3/2 + 6/3 + 9/6 + 18/9 = 9 is an integer.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000079, A030078, A079704, A227993, A255585.

lst={};Do[s=0;Do[s=s+Divisors[n][[i+1]]/Divisors[n][[i]],{i,1,Length[Divisors[n]]-1}];If[IntegerQ[s]&&!PrimeQ[n],AppendTo[lst,n]],{n,2300}];lst
Select[Range[2210],CompositeQ[#]&&IntegerQ[Total[#[[2]]/#[[1]]&/@Partition[ Divisors[ #],2,1]]]&] (* Harvey P. Dale, Jul 09 2019 *)

A153480 a(n) = 2*prime(n)^2 - 4.

Original entry on oeis.org

4, 14, 46, 94, 238, 334, 574, 718, 1054, 1678, 1918, 2734, 3358, 3694, 4414, 5614, 6958, 7438, 8974, 10078, 10654, 12478, 13774, 15838, 18814, 20398, 21214, 22894, 23758, 25534, 32254, 34318, 37534, 38638, 44398, 45598, 49294, 53134, 55774

Offset: 0

Roger L. Bagula, Dec 27 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A079704, A049001.

[2*NthPrime(n)^2-4: n in [1..40]]; // Vincenzo Librandi, Aug 19 2016

Clear[a, k]; a[k_] := 2*Prime[k]^2 - 4; Table[a[k], {k, 1, 30}]
2*Prime[Range[25]]^2 - 4 (* G. C. Greubel, Aug 18 2016 *)
Table[2 Prime[n]^2 - 4, {n, 60}] (* Vincenzo Librandi, Aug 19 2016 *)

a(n) = A079704(n)-4 = 2*A049001(n). - R. J. Mathar, Jan 03 2009

Extended by R. J. Mathar, Jan 03 2009

A349546 Composite numbers k such that k+1 is divisible by (k+1 mod A001414(k)) and k-1 is divisible by (k-1 mod A001414(k)).

Original entry on oeis.org

4, 8, 20, 32, 50, 55, 64, 77, 80, 98, 110, 115, 125, 128, 152, 170, 216, 242, 243, 256, 275, 290, 329, 338, 341, 343, 364, 371, 416, 506, 511, 512, 544, 551, 578, 583, 611, 638, 663, 722, 729, 731, 741, 851, 870, 920, 987, 1024, 1025, 1054, 1058, 1079, 1144, 1219, 1243, 1298, 1325, 1331, 1421

Offset: 1

J. M. Bergot and Robert Israel, Nov 21 2021

nonn

			a(3) = 20 is a term because A001414(20) = 2+2+5 = 9, 20+1 = 21 is divisible by 21 mod 9 = 3, and 20-1 = 19 is divisible by 19 mod 9 = 1.

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

Cf. A001414.

Includes all members of A079704 except 18.

filter:= proc(n) local s, t,r,q;
  if isprime(n) then return false fi;
  s:= add(t[1]*t[2],t = ifactors(n)[2]);
  r:= (n+1) mod s;
  q:= (n-1) mod s;
  r<> 0 and q <> 0 and (n+1) mod r = 0 and (n-1) mod q = 0
end proc:
select(filter, [$4..2000]);

filter[n_] := Module[{s, t, r, q},
   If[ PrimeQ[n], Return[False]];
   s = Sum[t[[1]]*t[[2]], {t, FactorInteger[n]}];
   r = Mod[n+1, s];
   q = Mod[n-1, s];
   r != 0 && q != 0 && Mod[n+1, r] == 0 && Mod[n-1, q ] == 0];
Select[Range[4, 2000], filter] (* Jean-François Alcover, Sep 29 2024, after Maple program *)

A352081 Numbers of the form k*p^k, where k>1 and p is a prime.

Original entry on oeis.org

8, 18, 24, 50, 64, 81, 98, 160, 242, 324, 338, 375, 384, 578, 722, 896, 1029, 1058, 1215, 1682, 1922, 2048, 2500, 2738, 3362, 3698, 3993, 4374, 4418, 4608, 5618, 6591, 6962, 7442, 8978, 9604, 10082, 10240, 10658, 12482, 13778, 14739, 15309, 15625, 15842, 18818

Offset: 1

Amiram Eldar, Apr 16 2022

nonn

Each term in this sequence has a single presentation in the form k*p^k.

			8 is a term since 8 = 2*2^2.
18 is a term since 18 = 2*3^2.
24 is a term since 24 = 3*2^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Lindqvist and Jaak Peetre, On the remainder in a series of Mertens, Expositiones Mathematicae, Vol. 15 (1997), pp. 467-478. See eq. (3).
Eric Weisstein's World of Mathematics, Mertens Constant. See eq. (5).

Subsequences: A036289 \ {0, 2}, A036290 \ {0, 3}, A036291 \ {0, 5}, A036293 \ {0, 7}, A073113 \ {2}, A079704, A100042, A104126.

Cf. A001620, A077761, A143524.

addP[p_, n_] := Module[{k = 2, s = {}, m}, While[(m = k*p^k) <= n, k++; AppendTo[s, m]]; s]; seq[max_] := Module[{m = Floor[Sqrt[max/2]], s = {}, ps}, ps = Select[Range[m], PrimeQ]; Do[s = Join[s, addP[p, max]], {p, ps}]; Sort[s]]; seq[2*10^4]

Sum_{n>=1} 1/a(n) = -A143524 = gamma - B_1, where gamma is Euler's constant (A001620), and B_1 is Mertens's constant (A077761).

A164986 Numbers of the form 2p^2 = q^2 + 1, where p and q are primes.

Original entry on oeis.org

50, 1682, 3971273138702695316402, 367680737852094722224630791187352516632102802

Offset: 1

Rick L. Shepherd, Sep 03 2009

nonn

A079704 INTERSECT A002522. Subsequence of A088920 (Solutions k to the Diophantine equation k = 2n^2 = m^2+1): those terms for which associated m in A002315 and n in A001653 are both prime.
Corresponding p are prime Pell numbers (prime denominators of continued fraction convergents to sqrt(2)).
Corresponding q are prime numerators of the continued fraction convergents to sqrt(2).
Corresponding p, q, p^2, q^2, (p,q), (q,p), etc., form subsequences of many other OEIS sequences; see cross-references.
Any further terms are too large to include here.

			a(1) = 50 as 50 = 2*5^2 = 7^2 + 1, where 5 and 7 are prime.

Cf. A088920, A118612, A086397, A086395, A002315 (NSW numbers), A088165 (prime NSW numbers = prime RMS numbers (A140480)), A001653, A000129 (Pell numbers), A086383, A101411, A079704, A002522, A008843, A104683, A163742, etc.

a(n) = 2*(A118612(n+1))^2 = (A086397(n+1))^2 + 1.

A240838 Primes p such that prime(p) + 2*p^2 is prime.

Original entry on oeis.org

2, 3, 5, 13, 41, 43, 139, 173, 227, 239, 359, 463, 541, 691, 743, 761, 821, 823, 827, 887, 1021, 1117, 1289, 1427, 1489, 1637, 1723, 1933, 1999, 2081, 2287, 2309, 2719, 2791, 2833, 2843, 2953, 3329, 3541, 3803, 3823, 3929, 4003, 4007, 4079, 4139, 4297, 4451, 4561, 4597, 4691, 4703, 4817, 4931, 4943

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 13 2014

nonn

The associated primes are: 11, 23, 41, 379, 3541, ...

			2 is in this sequence because 2 and prime(2) + 2*2^2 = 3 + 8 = 11 are both prime.

Cf. A006450, A079704.

[n: n in {1..5000} | IsPrime(n) and IsPrime(s) where s is (2*n^2 + NthPrime(n))];

Select[Prime[Range[700]],PrimeQ[Prime[#]+2#^2]&] (* Harvey P. Dale, Mar 19 2018 *)

isok(p) = isprime(p) && isprime(prime(p) + 2*p^2); \\ Michel Marcus, Apr 13 2014

A257404 Numbers of the form p * q^p where p and q are primes, in increasing order.

Original entry on oeis.org

8, 18, 24, 50, 81, 98, 160, 242, 338, 375, 578, 722, 896, 1029, 1058, 1215, 1682, 1922, 2738, 3362, 3698, 3993, 4418, 5618, 6591, 6962, 7442, 8978, 10082, 10658, 12482, 13778, 14739, 15309, 15625, 15842, 18818

Offset: 1

William Brian Repko, Apr 22 2015

nonn

			(2,2):8, (2,3):18, (3,2):24, (2,5):50, (3,3):81, (2,7):98.

Cf. some subsequences: A079704, A104126.

primes = [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97];
results = [];
max = 2 * Math.pow(primes[primes.length-1],2);
for (i = 0; i < primes.length; i++) {
    for (j = 0; j  < primes.length; j++) {
        p = primes[i];
        q = primes[j];
        n = p * Math.pow(q,p);
        if (n <= max) {
            // add it
            results.push(n);
        } else {
            // break out of this loop
            break;
        }
    }
}
// sort results and print them
results.sort(function(a, b){return a-b}).valueOf();

max=10^5; p=q=2; Sort[Reap[While[2*q^2 <= max, While[(n=p*q^p) <= max, Sow@n; p=NextPrime@p]; p=2; q=NextPrime@q ]][[2,1]]] (* Giovanni Resta, May 19 2015 *)

is(n)={bittest(6,#n=factor(n)~)||return;#n==1&&return(n[1,1]+1==n[2,1]);(n[2,1]==1&&n[2,2]==n[1,1])||(n[2,2]==1&&n[1,2]==n[2,1])} \\ M. F. Hasler, May 04 2015

A308707 a(n) = gcd(n, phi(n) + sigma(n)), where phi is A000010 and sigma is A000203.

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 7, 1, 1, 2, 11, 4, 13, 2, 1, 1, 17, 9, 19, 10, 1, 2, 23, 4, 1, 2, 1, 4, 29, 10, 31, 1, 1, 2, 1, 1, 37, 2, 1, 2, 41, 6, 43, 4, 3, 2, 47, 4, 1, 1, 1, 2, 53, 6, 1, 8, 1, 2, 59, 4, 61, 2, 7, 1, 1, 2, 67, 2, 1, 14, 71, 3, 73, 2, 1, 4, 1, 6, 79, 2, 1, 2, 83, 4, 1, 2, 1, 44, 89, 6

Offset: 1

Juri-Stepan Gerasimov, Jun 18 2019

nonn

If 2p = phi(p) + sigma(p), where p is A000040, then:
(i) primes m such that a(m-1) is equal to 1: 2, 5, 17, 37, 101, 257, 401, 577, 677, 1297, 1601, 2917, 3137, 4357, 5477, 8101, ...
Conjecture: ALL m are primes of the form i^2 + 1 (see A002496);
(ii) the smallest prime k such that a(k-1) is equal to n: 2, 3, 73, 13, 1464101, 43, 197, 113, 19, 31, 156817, 397, 9096257, 71, 405001, 387, ...
(iii) primes r such that a(r-1) is equal to r-1: 2, 3, 313, 23761, 3343777, 12558913, 45326161, 1178491681, ...
From Bernard Schott, Jun 23 2019: (Start)
There are distinct families of integers that satisfy a(k) = 1:
(i) k = p^q with p prime and q >= 2: A001597,
(ii) k = p*q with p, q primes and 2 < p < q: A046388,
(iii) k = 2*p^2 with p prime <> 3: A079704 \ {18},
(iv) conjecture: k = m^2 with m >= 1: A000290 \ {0}; if m is prime, it's not a conjecture, see (i). This conjecture is stronger than the conjecture of the 1st comment. (End)

Cf, A000010, A000040, A000203, A000290, A001597, A002496, A008578, A050873, A046388, A065387, A308470.

[Gcd(n, EulerPhi(n)+SumOfDivisors(n)): n in [1..100]];

a(n) = gcd(n, eulerphi(n) + sigma(n)); \\ Michel Marcus, Jun 19 2019

a(n) = gcd(n, A065387(n)). - Michel Marcus, Jun 19 2019
a(n) = n if n = 1 or n is prime: A008578.
a(2*p) = 2 if p prime >= 3: A100484 \ {4}. - Bernard Schott, Jun 26 2019

cp's OEIS Frontend

A350638 Numbers of the form p^2*q, with odd primes p > q, such that q divides p-1.

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A179262 a(n) = 2*prime(n)^2 - 1.

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A255586 Composite k such that Sum_{i=1..t-1} d(i+1)/d(i) is an integer, where d(1), ..., d(t) are the divisors of k in ascending order.

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A153480 a(n) = 2*prime(n)^2 - 4.

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Magma

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A349546 Composite numbers k such that k+1 is divisible by (k+1 mod A001414(k)) and k-1 is divisible by (k-1 mod A001414(k)).

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A352081 Numbers of the form k*p^k, where k>1 and p is a prime.

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A164986 Numbers of the form 2p^2 = q^2 + 1, where p and q are primes.

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A240838 Primes p such that prime(p) + 2*p^2 is prime.

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