A144482 -id:A144482 - cp's OEIS frontend

A335882 Numbers k for which A331410(k) = 2.

5, 9, 10, 11, 13, 18, 20, 21, 22, 23, 26, 36, 40, 42, 44, 46, 47, 49, 52, 61, 72, 80, 84, 88, 92, 93, 94, 98, 104, 122, 144, 160, 168, 176, 184, 186, 188, 191, 196, 208, 217, 223, 244, 288, 320, 336, 352, 368, 372, 376, 381, 382, 383, 392, 416, 434, 446, 488, 576, 640, 672, 704, 736, 744, 752, 762, 764, 766, 784, 832, 868, 889, 892, 961

Offset: 1

Antti Karttunen, Jun 28 2020

nonn

Numbers k such that A000265(k) is either in A144482 or in A335874.

Each term is either of the form A335874(n)*2^k, for some n >= 2, and k >= 0, or a product of two terms of A335431, whether distinct or not.

Antti Karttunen, Table of n, a(n) for n = 1..600

Row 2 of A335430.
Cf. A331410, A335431, A335874 (after its initial 2, gives the primes in this sequence), A144482 (odd semiprimes in this sequence).
Cf. also A334102.

A000265(n) = (n>>valuation(n,2));
isA000668(n) = (isprime(n)&&!bitand(n,1+n));
isA144482(n) = ((2==bigomega(n))&&isA000668(vecmin(factor(n)[,1]))&&isA000668(vecmax(factor(n)[,1])));
isA335874(n) = ((n>2)&&isprime(n)&&isA000668(A000265(1+n)));
isA335882(n) = (isA335874(A000265(n))||isA144482(A000265(n)));

A144856 Semiprimes that are a product of distinct Mersenne primes.

Original entry on oeis.org

21, 93, 217, 381, 889, 3937, 24573, 57337, 253921, 393213, 917497, 1040257, 1572861, 3670009, 4063201, 16252897, 16646017, 66584449, 1073602561, 4294434817, 6442450941, 15032385529, 66571993057, 68718821377, 272730423169

Offset: 1

G. L. Honaker, Jr., Sep 22 2008

nonn

Since each Mersenne prime is congruent to -1 (mod 4), it is easy to see that a(n) == 1 (mod 4). - Timothy L. Tiffin, Jul 07 2021

			a(1) = 3*7 = 21, a(2) = 3*31 = 93, a(3) = 7*31 = 217, ... - _Timothy L. Tiffin_, Jul 07 2021

T. D. Noe, Table of n, a(n) for n = 1..90
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 21
Googology Wiki,Largest known squarefree semiprime

Cf. A000668, A001358. Subsequence of A144482 (semiprimes that are the product of Mersenne primes).

Mp = 2^{2, 3, 5, 7, 13, 17, 19, 31, 61} - 1; Take[ Union[ Times @@@ Subsets[ Mp, {2}]], 25] (* Robert G. Wilson v, Sep 25 2008 *)

More terms from Robert G. Wilson v, Sep 25 2008

More terms from T. D. Noe, Sep 25 2008

A335912 Numbers k for which A335885(k) = 2.

Original entry on oeis.org

9, 11, 13, 15, 18, 19, 21, 22, 23, 25, 26, 29, 30, 35, 36, 38, 41, 42, 44, 46, 47, 49, 50, 51, 52, 58, 60, 61, 67, 70, 72, 76, 79, 82, 84, 85, 88, 92, 93, 94, 97, 98, 100, 102, 104, 113, 116, 119, 120, 122, 134, 137, 140, 144, 152, 155, 158, 164, 168, 170, 176, 184, 186, 188, 191, 193, 194, 196, 200, 204, 208, 217, 223

Offset: 1

Antti Karttunen, Jun 30 2020

nonn

Numbers n such that when we start from k = n, and apply in some combination the nondeterministic maps k -> k - k/p and k -> k + k/p, (where p can be any of the odd prime factors of k), then for some combination we can reach a power of 2 in exactly two steps (but with no combination allowing 0 or 1 steps).

			For n = 70 = 2*5*7, if we first take p = 7 and apply the map n -> n + (n/p), we obtain 80 = 2^4 * 5. We then take p = 5, and apply the map n -> n - (n/p), to obtain 80-16 = 64 = 2^16. Thus we reached a power of 2 in two steps (and there are no shorter paths), therefore 70 is present in this sequence.
For n = 769, which is a prime, 769 - (769/769) yields 768 = 3 * 256. For 768 we can then apply either map to obtain a power of 2, as 768 - (768/3) = 512 = 2^9 and 768 + (768/3) = 1024 = 2^10. On the other hand, 769 + (769/769) = 770 and A335885(770) = 4, so that route would not lead to any shorter paths, therefore 769 is a term of this sequence.

Row 2 of A335910.
Cf. A000265, A335911, A335885.
Subsequences of semiprimes (union gives all odd semiprimes present): A144482, A333788, A336115.
Cf. also A334092, A335874, A336116, A336117 and A334102, A335882, A336122.

A335885(n) = { my(f=factor(n)); sum(k=1,#f~,if(2==f[k,1],0,f[k,2]*(1+min(A335885(f[k,1]-1),A335885(f[k,1]+1))))); };
isA335912(n) = (2==A335885(n));

A279389 3 times Mersenne primes A000668.

Original entry on oeis.org

9, 21, 93, 381, 24573, 393213, 1572861, 6442450941, 6917529027641081853, 1856910058928070412348686333, 486777830487640090174734030864381, 510423550381407695195061911147652317181

Offset: 1

Omar E. Pol, Dec 20 2016

nonn

Also sum of n-th Mersenne prime and the radical of n-th even perfect number.

The binary representation of a(n) has only two zeros, starting with "10" and ending with "01". The sequence begins: 1001, 10101, 1011101, 101111101, 101111111111101,...

Cf. A000010, A000203, A000396, A000668, A051027, A147757.
Subsequence of A001748, and of A147758, and of A174055, and possibly of other sequences, see below:
Cf. A060482, A068156, A072087, A136252, A144482, A144856, A249780, A269633.

a(n) = 3*A000668(n) = A000668(n) + A139257(n).

a(n) = phi(M(n)) + sigma(sigma(M(n))) = A000010(A000668(n)) + A000203(A000203(A000668(n))) = A000010(A000668(n)) + A051027(A000668(n)).

A335879 a(n) = A332215(A335882(n)).

Original entry on oeis.org

15, 5, 30, 63, 255, 10, 60, 13, 126, 2047, 510, 20, 120, 26, 252, 4094, 262143, 11, 1020, 4194303, 40, 240, 52, 504, 8188, 61, 524286, 22, 2040, 8388606, 80, 480, 104, 1008, 16376, 122, 1048572, 140737488355327, 44, 4080, 59, 4503599627370495, 16777212, 160, 960, 208, 2016, 32752, 244, 2097144, 253, 281474976710654, 2417851639229258349412351

Offset: 1

Antti Karttunen, Jul 10 2020

nonn

For all n, a(n) <> A335882(n). Proof: We need to consider only the odd terms, because for n > 1, A332215(2^k * n) = 2^k * A332215(n). The odd terms of A335882 are either primes or semiprimes whose both factors are Mersenne primes, terms of A144482.
  (A) If A335882(n) is a prime, then a(n) = A332215(A335882(n)) is a term of A000225 (of the form 2^k - 1, a binary repunit), while primes in A335882 are certainly not of that form, as all Mersenne primes (A000668) are on a different row in array A335430 (on row 1, A335431).
  (B) For any semiprime k in A335882, there is only one non-leading zero in the binary representation of A332215(k). On the other hand, a product of two Mersenne primes always contains more than one non-leading zero in its base-2 representation: for three times a Mersenne prime, there are two such zeros, as explained in A279389, and products of two Mersenne primes > 3 are always of the form 8k+1, with at least two zeros immediately left of the least significant 1-bit.

Cf. A000225, A000668, A007814, A279389, A331410, A332215, A335430, A335431, A335882.

a(n) = A332215(A335882(n)).

For all n >= 1, A007814(a(n)) = A007814(A335882(n)).

A336115 Semiprimes that are product of a Fermat prime and a Mersenne prime.

Original entry on oeis.org

9, 15, 21, 35, 51, 93, 119, 155, 381, 527, 635, 771, 1799, 2159, 7967, 24573, 32639, 40955, 139247, 196611, 393213, 458759, 655355, 1572861, 2031647, 2105087, 2228207, 2621435, 8323199, 8912879, 33685247, 134741759, 536813567, 6442450941, 8590000127, 10737418235

Offset: 1

Antti Karttunen, Jul 09 2020

nonn

As 3 is both a Fermat prime and a Mersenne prime, A019434(1) * A000668(1) = 9 is also a term. It is the only square in this sequence.

Cf. A000265, A000668, A019434, A144482, A333788, A335885.

Subsequence of A335912.

isA000668(n) = (isprime(n)&&!bitand(n,1+n));
isA019434(n) = ((n>2)&&isprime(n)&&!bitand(n-2,n-1));
isA336115(n) = if(9==n,1, if((2!=omega(n))||(2!=bigomega(n)),0,my(ps=factor(n)[,1]); (isA019434(ps[1])&&isA000668(ps[2]))||(isA019434(ps[2])&&isA000668(ps[1])))); \\ Antti Karttunen, Jul 11 2020

A335885(a(n)) = 2.

Missing terms and more terms added by Jinyuan Wang, Jul 11 2020

A344780 Semiprimes that are product of two distinct Honaker primes.

Original entry on oeis.org

34453, 59867, 120191, 136109, 137419, 142921, 170431, 178291, 187723, 205801, 250603, 253223, 273257, 275887, 280471, 286933, 290951, 297763, 319771, 339421, 342163, 348853, 354617, 356189, 357499, 357943, 367193, 376879, 401777, 410947, 413173, 422999, 449723

Offset: 1

K. D. Bajpai, May 28 2021

Subsequence of A006881.

a(1) = 34453 is the only number <= 5*10^6 that is a triangular number.

			34453 = 131*263 which are distinct Honaker primes.
120191 = 263*457 which are distinct Honaker primes.

Cf. A006881, A033548, A144482, A144856, A333788.

isA006881 := proc(n)
    if numtheory[bigomega](n) =2 and A001221(n) = 2 then
        true ;
    else
        false ;
    end if;
end proc:
isA344780 := proc(n)
    if isA006881(n) then
        for p in ifactors(n)[2] do
            if not isA033548(op(1,p)) then
                return false;
            end if;
        end do:
        true ;
    else
        false;
    end if;
end proc:
for n from 1  do
    if isA344780(n) then
        printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Jul 07 2021

fHQ[n_] := Plus @@ IntegerDigits@n == Plus @@ IntegerDigits@PrimePi@n;
lst = {}; Do[If[Plus @@ Last /@ FactorInteger[n] == 2, a = Length[First /@ FactorInteger[n]]; If[a == 2, b = First /@ FactorInteger[n]; c = b[[1]]; d = b[[2]]; If[fHQ[c] && fHQ[d], AppendTo[lst, {n,c,d}]]]], {n, 2000000}]; lst

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A335882 Numbers k for which A331410(k) = 2.

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A144856 Semiprimes that are a product of distinct Mersenne primes.

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A335912 Numbers k for which A335885(k) = 2.

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A279389 3 times Mersenne primes A000668.

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A335879 a(n) = A332215(A335882(n)).

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A336115 Semiprimes that are product of a Fermat prime and a Mersenne prime.

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PARI

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A344780 Semiprimes that are product of two distinct Honaker primes.

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