A092109 -id:A092109 - cp's OEIS frontend

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

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

A145488 Numbers k such that 6k+13 is prime and 12k+13 is also prime.

Original entry on oeis.org

0, 4, 5, 8, 14, 15, 19, 25, 28, 30, 33, 35, 44, 49, 50, 54, 60, 68, 70, 85, 88, 93, 99, 100, 103, 120, 123, 133, 140, 144, 145, 149, 154, 168, 170, 173, 175, 179, 184, 190, 198, 215, 228, 235, 245, 253, 259, 264, 268, 274, 275, 280, 285, 288, 294

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.

select(k -> isprime(6*k+13) and isprime(12*k+13), [$0..1000]); # Robert Israel, Jan 23 2017

aa = {}; k = 13; Do[If[PrimeQ[(k + Prime[n])/2], AppendTo[aa, (Prime[n] - k)/12]], {n, 1, 500}]; aa

a(n) = (A145474(n)-13)/12.

Definition corrected by Ivan Neretin, Jan 23 2017

A145489 Numbers k such that 6k + 11 is prime and 12k + 5 is also prime.

Original entry on oeis.org

0, 1, 2, 3, 7, 8, 12, 16, 21, 23, 26, 37, 38, 42, 43, 47, 51, 56, 58, 63, 68, 73, 78, 91, 92, 98, 101, 106, 107, 108, 133, 136, 141, 142, 156, 157, 162, 173, 192, 196, 201, 203, 212, 218, 227, 233, 236, 238, 246, 247, 257, 267, 268, 271, 287, 296, 306, 313, 316, 323, 327, 332, 346, 353, 357, 366, 367, 371, 376, 387, 401, 406, 411, 423, 441, 442, 448, 453, 471, 472, 478, 483, 488, 491, 498

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.

Select[Range[0,500], PrimeQ[6# + 11 ] && PrimeQ[12# + 5]&]

isok(n) = isprime(6*n+11) && isprime(12*n+5); \\ Michel Marcus, Jan 24 2017

a(n) = (A145475(n) - 5)/12.

Corrected by Artur Jasinski, Apr 01 2011

A353702 Composite k such that tau(k') = (tau(k))', where tau(k) is the number of divisors of k (A000005) and k' is the arithmetic derivative of k (A003415).

Original entry on oeis.org

12, 15, 21, 26, 27, 33, 38, 57, 62, 69, 74, 85, 88, 93, 106, 108, 129, 133, 134, 145, 166, 177, 178, 205, 213, 217, 218, 226, 237, 248, 249, 253, 254, 262, 265, 278, 309, 314, 328, 362, 375, 376, 393, 398, 417, 422, 424, 445, 459, 466, 469, 488, 489, 493, 502

Offset: 1

Marius A. Burtea, May 07 2022

nonn

Since for any prime number p, p' = 1 and (tau(p))' = 2' = 1 = tau(1) = tau (p'), the sequence requires only composite numbers that satisfy the given relation.
For p in A092109 the number m = 3*p is a term. Indeed, (tau(m))' = (tau(3*p))' = 4' = 4 and tau(m') = tau((3*p)') = tau(p + 3) = 4, so m is a term.
If p is in A045536 then p, p + 2 and 2*p + 1 are prime numbers and m = 3*(2*p + 1) is a term. Indeed, tau(m') = tau((3*(2*p + 1))') = tau(2*p + 4) = tau(2*(p+2)) = 4 and (tau(m))' = (tau((3*(2*p + 1)))' = 4' = 4, so m is a term.
If k is in A174100 then the numbers 2*k + 1 and 6*k + 1 are prime numbers and the numbers m = 2*(6*k + 1) is a term. Indeed, (tau(m))' = (tau(2*(6*k + 1)) )' = 4' = 4 and tau(m') = tau(2*(6*k + 1))') = tau(6*k + 3) = tau(2*(2*k + 1)) = 4, so m is a term.

			12' = 16, (tau(12)) = 6' = 5 and tau(12') = tau(16) = tau(2^4) = 5, so 12 is a term.
15' = 8, (tau(15))’ = 4' = 4 and tau(15') = tau(8) = tau(2^3) = 4, so 15 is a term.

Cf. A000005, A003415, A045536, A092109, A174100, A260961.

f:=func; [p:p in [3..550]|not IsPrime(p) and  #Divisors(Floor(f(p))) eq Floor(f(#Divisors(p)))];

isA353702 := proc(n)
    if not isprime(n) and numtheory[tau](A003415(n)) = A003415( numtheory[tau](n) ) then
        true ;
    else
        false;
    end if;
end proc:
for n from 2 to 1000 do
    if isA353702(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, May 05 2023

d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[500], CompositeQ[#] && DivisorSigma[0, d[#]] == d[DivisorSigma[0, #]] &] (* Amiram Eldar, May 07 2022 *)

ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(k) = (k>1) && !isprime(k) && numdiv(ad(k)) == ad(numdiv(k)); \\ Michel Marcus, May 08 2022

cp's OEIS Frontend

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

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

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

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A145488 Numbers k such that 6k+13 is prime and 12k+13 is also prime.

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A145489 Numbers k such that 6k + 11 is prime and 12k + 5 is also prime.

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PARI

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Extensions

A353702 Composite k such that tau(k') = (tau(k))', where tau(k) is the number of divisors of k (A000005) and k' is the arithmetic derivative of k (A003415).

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Magma

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