A063908 -id:A063908 - cp's OEIS frontend

A241817 Semiprimes sp such that sp-3 is prime.

6, 10, 14, 22, 26, 34, 46, 62, 74, 82, 86, 106, 134, 142, 166, 194, 202, 214, 226, 254, 274, 314, 334, 362, 382, 386, 422, 446, 466, 482, 502, 526, 566, 622, 634, 662, 694, 746, 842, 862, 866, 886, 914, 922, 974, 1042, 1094, 1126, 1154, 1174, 1226, 1234, 1262

Offset: 1

K. D. Bajpai, Apr 29 2014

nonn

Even numbers of the form 2p, p prime, that can be expressed as the sum of two primes in at least two ways as 2p = p + p = 3 + (2p-3). For example, 34 is in the sequence because 34 = 2*17 = 17 + 17 = 3 + 31. These are the only numbers that have Goldbach partitions with both a minimum and a maximum possible difference between their prime parts, i.e., |p-p| = 0 and |(2p-3)-3| = 2p-6 respectively. - Wesley Ivan Hurt, Apr 08 2018

			a(2) = 10 = 2*5, which is semiprime and 10-3 = 7 is a prime.
a(6) = 34 = 2*17, which is semiprime and 34-3 = 31 is a prime.

K. D. Bajpai, Table of n, a(n) for n = 1..2500

Cf. A001358, A063908, A092207, A123017, A198327.

with(numtheory): A241817:= proc(); if bigomega(x)=2 and isprime(x-3) then  RETURN (x); fi; end: seq(A241817 (), x=1..3000);

2 Select [Prime[Range[5!]], PrimeQ[2 # - 3] &] (* Vincenzo Librandi, Apr 10 2018 *)
Select[Range[1500],PrimeOmega[#]==2&&PrimeQ[#-3]&] (* Harvey P. Dale, Oct 14 2018 *)

a(n) = 2 * A063908(n). - Wesley Ivan Hurt, Apr 08 2018

A243630 Primes p such that 2*p^3 - 3 is also prime.

Original entry on oeis.org

2, 7, 11, 13, 47, 101, 107, 151, 163, 167, 251, 257, 401, 443, 467, 521, 571, 641, 653, 673, 797, 907, 911, 971, 983, 997, 1013, 1151, 1153, 1181, 1187, 1223, 1231, 1277, 1291, 1303, 1361, 1433, 1481, 1511, 1597, 1637, 1723, 1741, 1811, 1951, 2027, 2081, 2083, 2141, 2287, 2311

Offset: 1

Vincenzo Librandi, Jun 08 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A063908, A023204, A153507, A174363, A243595.

[p: p in PrimesUpTo(2500) | IsPrime(2*p^3 - 3)];

Select[Prime[Range[2500]], PrimeQ[2 #^3 - 3] &]

A290838 a(n) = smallest prime p such that 2p - 2n + 1 is prime.

Original entry on oeis.org

2, 2, 3, 5, 5, 7, 7, 13, 11, 11, 11, 13, 13, 19, 17, 17, 17, 19, 19, 37, 23, 23, 23, 29, 29, 31, 29, 29, 29, 31, 31, 37, 37, 41, 37, 37, 37, 43, 41, 41, 41, 43, 43, 61, 47, 47, 47, 53, 53, 67, 53, 53, 53, 59, 59, 61, 59, 59, 59, 61, 61, 67, 67, 71, 67, 67, 67, 73

Offset: 0

XU Pingya, Aug 11 2017

a(n) > n. - Iain Fox, Nov 13 2017

Iain Fox, Table of n, a(n) for n = 0..20000

Cf. A005382, A063908, A063909, A063910, A063911, A063912, A063913, A020483, A290839.

Table[j=0; found=False; While[!found,j++; found=PrimeQ[2Prime[j]-2n+1] && 2Prime[j]-2n+1>0]; Prime[j],{n,67}]
(* Second program: *)
Table[SelectFirst[Prime@ Range[n^2], And[# > 0, PrimeQ@ #] &[2 # - 2 n + 1] &], {n, 67}] (* Michael De Vlieger, Aug 14 2017 *)

a(n) = {my(p=2); while(!isprime(2*p-2*n+1), p = nextprime(p+1)); p; } \\ Michel Marcus, Aug 12 2017

a(n) = forprime(p=n+1, , if(isprime(2*p - 2*n + 1), return(p))) \\ Iain Fox, Nov 13 2017

a(-n) = A290839(n+1) - Iain Fox, Dec 14 2017

a(0) prepended by Iain Fox, Dec 14 2017

A308643 Odd squarefree composite numbers k, divisible by the sum of their prime factors, sopfr (A001414).

Original entry on oeis.org

105, 231, 627, 805, 897, 1581, 2967, 3055, 4543, 5487, 6461, 6745, 7881, 9717, 10707, 14231, 15015, 16377, 21091, 26331, 29607, 33495, 33901, 33915, 35905, 37411, 38843, 40587, 42211, 45885, 49335, 50505, 51051, 53295, 55581, 60297

Offset: 1

David James Sycamore, Jun 13 2019

nonn

Every term has an odd number of prime divisors (A001221(k) is odd), since if not, sopfr(k) would be even, and so not divide k, which is odd.
Some Carmichael numbers appear in this sequence, the first of which is 3240392401.
From Robert Israel, Jul 05 2019: (Start)
Includes p*q*r if p and q are distinct odd primes and r=(p-1)*(q-1)-1 is prime. Dickson's conjecture implies that there are infinitely many such terms for each odd prime p.  Thus for p=3, q is in A063908 (except 3), for p=5, q is in A156300 (except 2), and for p=7, q is in A153135 (except 2). (End)

			105=3*5*7; sum of prime factors = 15 and 105 = 7*15, so 105 is a term.

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

Intersection of A005117 and A046347.

Cf. A001414, A046346, A002997, A001221, A063908, A156300, A153135.

[k:k in [2*d+1: d in [1..35000]]|IsSquarefree(k) and not IsPrime(k) and k mod &+PrimeDivisors(k) eq 0]; // Marius A. Burtea, Jun 19 2019

with(NumberTheory);
N := 500;
for n from 2 to N do
S := PrimeFactors(n);
X := add(S);
if IsSquareFree(n) and not mod(n, 2) = 0 and not isprime(n) and mod(n, X) = 0 then print(n);
end if:
end do:

aQ[n_] := Module[{f = FactorInteger[n]}, p=f[[;;,1]]; e=f[[;;,2]]; Length[e] > 1 && Max[e]==1 && Divisible[n, Plus@@(p^e)]]; Select[Range[1, 61000, 2], aQ] (* Amiram Eldar, Jul 04 2019 *)

A145481 Primes p such that 2*p - 17 is prime.

Original entry on oeis.org

11, 17, 23, 29, 53, 59, 83, 107, 137, 149, 167, 233, 239, 263, 269, 293, 317, 347, 359, 389, 419, 449, 479, 557, 563, 599, 617, 647, 653, 659, 809, 827, 857, 863, 947, 953, 983, 1049, 1163, 1187, 1217, 1229, 1283, 1319, 1373, 1409, 1427, 1439, 1487, 1493

Offset: 1

Artur Jasinski, Oct 11 2008

nonn

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

Cf. A063908-A063913, A092109, A145471-A145480.

aa = {}; k = 17; Do[If[PrimeQ[(k + Prime[n])/2], AppendTo[aa, (k + Prime[n])/2]], {n, 1, 500}];aa
(* Second program: *)
Select[Prime@ Range@ 250, And[PrimeQ@ #, # > 0] &[2 # - 17] &] (* Michael De Vlieger, Jan 23 2017 *)

a(n) = 2*A145475(n) - 17.

A145482 Primes p such that 2*p - 19 is prime.

Original entry on oeis.org

11, 13, 19, 31, 43, 61, 73, 79, 109, 151, 163, 193, 199, 229, 241, 271, 283, 313, 331, 373, 379, 421, 439, 463, 541, 571, 661, 673, 709, 733, 739, 751, 823, 859, 883, 1009, 1051, 1129, 1153, 1201, 1279, 1453, 1543, 1549, 1663, 1669, 1741, 1759, 1783, 1789

Offset: 1

Artur Jasinski, Oct 11 2008

nonn

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

Cf. A063908-A063913, A092109, A145471-A145480.

aa = {}; k = 19; Do[If[PrimeQ[(k + Prime[n])/2], AppendTo[aa, (k + Prime[n])/2]], {n, 1, 500}];aa
(* Second program: *)
Select[Prime@ Range@ 300, And[PrimeQ@ #, # > 0] &[2 # - 19] &] (* Michael De Vlieger, Jan 23 2017 *)

a(n) = 2*A145476(n) - 19.

A145483 Primes p such that 2*p - 23 is prime.

Original entry on oeis.org

13, 17, 23, 41, 47, 53, 101, 107, 131, 137, 167, 191, 227, 233, 251, 257, 263, 293, 311, 353, 383, 431, 443, 467, 503, 521, 557, 563, 587, 593, 641, 653, 761, 773, 797, 821, 947, 977, 1013, 1031, 1061, 1181, 1187, 1217, 1223, 1277, 1283, 1301, 1307, 1361

Offset: 1

Artur Jasinski, Oct 11 2008

nonn

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

Cf. A063908-A063913, A092109, A145471-A145490.

aa = {}; k = 23; Do[If[PrimeQ[(k + Prime[n])/2], AppendTo[aa, (k + Prime[n])/2]], {n, 1, 500}];aa
(* Second program: *)
Select[Prime@ Range@ 240, And[PrimeQ@ #, # > 0] &[2 # - 23] &] (* Michael De Vlieger, Jan 23 2017 *)

a(n) = 2*A145477(n) - 23.

A145484 Primes p such that 2*p - 29 is a positive prime.

Original entry on oeis.org

17, 23, 29, 41, 59, 71, 83, 89, 101, 113, 131, 149, 173, 191, 239, 269, 293, 311, 353, 401, 419, 443, 479, 491, 503, 521, 563, 569, 653, 659, 701, 719, 761, 821, 863, 881, 953, 971, 1013, 1049, 1091, 1151, 1163, 1181, 1193, 1223, 1289, 1319, 1361, 1409

Offset: 1

Artur Jasinski, Oct 11 2008

nonn

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

Cf. A063908-A063913, A092109, A145471-A145490.

aa = {}; k = 29; Do[If[PrimeQ[(k + Prime[n])/2], AppendTo[aa, (k + Prime[n])/2]], {n, 1, 500}];aa (* Artur Jasinski *)
Select[Prime[Range[7,300]],PrimeQ[2#-29]&] (* Harvey P. Dale, Dec 14 2010 *)

a(n) = 2*A145478(n) - 29.

A145485 Primes p such that 2*p - 31 is prime.

Original entry on oeis.org

17, 19, 31, 37, 67, 79, 97, 127, 151, 157, 181, 199, 277, 331, 337, 379, 409, 421, 457, 499, 541, 547, 577, 601, 631, 661, 727, 739, 751, 757, 787, 829, 877, 907, 991, 1009, 1021, 1087, 1117, 1171, 1201, 1249, 1291, 1381, 1399, 1459, 1549, 1597, 1609, 1669

Offset: 1

Artur Jasinski, Oct 11 2008

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

Cf. A063908-A063913, A092109, A145471-A145490.

aa = {}; k = 31; Do[If[PrimeQ[(k + Prime[n])/2], AppendTo[aa, (k + Prime[n])/2]], {n, 1, 500}];aa

list(lim)=my(v=List()); forprime(p=2,lim, if(isprime(2*p-31), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Jan 23 2017

a(n) = 2*A145479(n) - 31.

A145486 Primes p such that 2*p - 37 is prime.

Original entry on oeis.org

37, 67, 73, 97, 109, 139, 157, 193, 223, 229, 307, 349, 373, 397, 433, 457, 487, 523, 577, 619, 643, 709, 733, 823, 829, 853, 907, 919, 1033, 1063, 1087, 1129, 1153, 1213, 1237, 1279, 1297, 1327, 1447, 1543, 1549, 1579, 1609, 1627, 1669, 1699, 1747, 1753

Offset: 1

Artur Jasinski, Oct 11 2008

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

Cf. A063908-A063913, A092109, A145471-A145490.

aa = {}; k = 37; Do[If[PrimeQ[(k + Prime[n])/2], AppendTo[aa, (k + Prime[n])/2]], {n, 1, 500}];aa

list(lim)=my(v=List()); forprime(p=2,lim, if(isprime(2*p-37), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Jan 23 2017

a(n)=2*A145480(n)-37

cp's OEIS Frontend

A241817 Semiprimes sp such that sp-3 is prime.

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A243630 Primes p such that 2*p^3 - 3 is also prime.

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A290838 a(n) = smallest prime p such that 2p - 2n + 1 is prime.

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A308643 Odd squarefree composite numbers k, divisible by the sum of their prime factors, sopfr (A001414).

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Magma

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A145481 Primes p such that 2*p - 17 is prime.

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A145482 Primes p such that 2*p - 19 is prime.

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A145483 Primes p such that 2*p - 23 is prime.

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A145484 Primes p such that 2*p - 29 is a positive prime.

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