A001358 -id:A001358 - cp's OEIS frontend

A157414 Decimal expansion of Sum_{q = semiprimes = A001358} 1/(q*2^q).

0, 1, 8, 5, 5, 0, 2, 6, 6, 2, 7, 9, 9, 4, 9, 7, 0, 6, 5, 8, 9, 2, 6, 5, 4, 8, 5, 2, 8, 8, 2, 0, 4, 7, 7, 7, 4, 3, 0, 1, 6, 8, 9, 3, 1, 8, 6, 9, 2, 7, 5, 1, 2, 7, 0, 3, 2, 8, 2, 8, 9, 3, 0, 0, 3, 5, 0, 1, 5, 8, 8, 4, 7, 7, 6, 3, 7, 1, 6, 5, 7, 3, 8, 8, 0, 1, 5, 8, 5, 4, 6, 3, 6, 6, 7, 7, 0, 3, 8, 1, 7, 4, 1, 8, 3

Offset: 0

R. J. Mathar, Feb 28 2009

			0.0185502662799497065892654852882047774... = 1/(4*2^4)+1/(6*2^6)+1/(9*2^9)+1/(10*2^10)+... = Sum_{i>=1} 1/(A001358(i)*2^A001358(i)).

R. J. Mathar, Series of reciprocal powers of k-almost primes, arxiv:0803.0900, eq. (66).

Cf. A001358.

A002162 = Sum_{n>=1} 1/(n*2^n) = 1/2 + A157413 + this_constant_here + equivalent terms of higher order k-almost primes.

A173477 Semiprimes having no representation of the form semiprime(n)-+n, where semiprime(n) = A001358(n).

Original entry on oeis.org

10, 15, 25, 26, 35, 38, 39, 58, 65, 82, 85, 87, 91, 94, 118, 119, 121, 123, 133, 134, 142, 143, 155, 166, 183, 185, 201, 202, 209, 213, 217, 226, 237, 253, 267, 274, 278, 287, 295, 298, 299, 301, 303, 305, 314, 319, 321, 339, 355, 362, 371, 377, 381, 395, 407, 413, 415, 417, 422, 427

Offset: 1

Juri-Stepan Gerasimov, Nov 22 2010

			Listing the first eight terms of A001358 gives us:
n: 1, 2, 3,  4,  5,  6,  7,  8, ...
   4, 6, 9, 10, 14, 15, 21, 22, ...
We see that 4 can be represented as 6-2, 6 can be represented as 4+2 or 9-3 or 10-4, 9 can be represented as 14-5 or 15-6, but 10 cannot be represented by any such sum or difference as 4+1, 6+2, 9+3, 14-5, 15-6, 21-7, and also any difference A001358(n)-n after that will miss it. Thus 10 is the first semiprime included in this sequence.

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

Cf. A001358, A100493, A172096.

N:= 2000: # to use semiprimes <= N
Primes:= select(isprime, [2,seq(i,i=3..N,2)]):
Semiprimes:= select(`<=`,{seq(seq(Primes[i]*Primes[j],i=1..j),j=1..nops(Primes))},N):
sort(convert(Semiprimes minus {seq}(i+Semiprimes[i],i=1..nops(Semiprimes)) minus {seq}(Semiprimes[i]-i,i=1..nops(Semiprimes))),list)); # Robert Israel, Dec 20 2015

Corrected by D. S. McNeil, Nov 23 2010

Name clarified and Example section added by Antti Karttunen, Dec 20 2015

A174838 Numbers n such that semiprime(n)+1 is prime, where semiprime(n) is A001358.

Original entry on oeis.org

1, 2, 4, 8, 16, 21, 27, 35, 55, 58, 76, 84, 111, 113, 120, 143, 147, 155, 176, 183, 218, 252, 258, 265, 294, 304, 348, 377, 383, 387, 403, 424, 435, 444, 464, 525, 548, 582, 585, 593, 600, 633, 690, 694, 732, 787, 803, 810, 827, 841, 846, 877, 892, 900, 971

Offset: 1

Juri-Stepan Gerasimov, Mar 30 2010

nonn

N is in the sequence iff semiprime(n) is of the form 2p where p is a Sophie Germain prime (A005384).

			a(1)=1 because semiprime(1)+1=5=prime, a(2)=2 because semiprime(2)+1=7=prime, a(3)=4 because semiprime(4)+1=11=prime.

Cf. A000040, A001358, A005384.

Edited, corrected and extended by Ray Chandler, Apr 05 2010

A179463 Semiprimes A001358 containing at least one semiprime digit in base 10.

Original entry on oeis.org

4, 6, 9, 14, 26, 34, 39, 46, 49, 62, 65, 69, 74, 86, 91, 93, 94, 95, 106, 119, 129, 134, 141, 142, 143, 145, 146, 159, 161, 166, 169, 194, 206, 209, 214, 219, 226, 247, 249, 254, 259, 262, 265, 267, 274, 289, 291, 295, 298, 299, 309, 314, 319, 326, 329, 334, 339, 341

Offset: 1

Jonathan Vos Post, Jul 15 2010

Semiprimes containing at least one 4, 6, or 9 digit base 10.
This is to semiprimes A001358 as A179336 is to primes A000040.
This properly includes the subset A107342 Semiprimes with semiprime digits.

Cf. A107342 Semiprimes with semiprime digits (digits 4, 6, 9 only), A107665 Numbers with semiprime digits (digits 4, 6, 9 only), A107666 Primes with semiprime digits (digits 4, 6, 9 only), A111494, A111496, A111697, A108614 Semiprimes with non-semiprimes digits (no digits 4, 6, 9 in semiprimes), A179336.

spdQ[n_]:=Module[{idn=IntegerDigits[n]},MemberQ[idn,4] || MemberQ[ idn,6] || MemberQ[ idn,9]]; Select[Select[Range[350],PrimeOmega[#]==2&],spdQ] (* Harvey P. Dale, Jun 24 2013 *)

Corrected (a(37) added) by Harvey P. Dale, Jun 24 2013

A226834 Smallest semiprime (A001358) which is at the beginning of an arithmetic progression of n semiprimes whose largest term is as small as possible.

Original entry on oeis.org

4, 4, 6, 10, 10, 201, 133, 133, 635, 697, 2215, 2215, 4979, 2995, 13561, 22903, 1691, 5951, 72697, 72697, 72697, 172151, 172151, 1782371, 1782371, 3660743, 3660743, 3660743, 3660743, 13298267, 2235913, 41963249

Offset: 1

T. D. Noe, Jun 28 2013

Smallest number in row A226833(n).

Cf. A096003 (largest semiprime in row), A097824 (gaps).

A243428 Semiprimes A001358(n) such that A001358(n) + 2^n is also a semiprime.

Original entry on oeis.org

4, 6, 10, 14, 22, 25, 35, 39, 95, 123, 129, 177, 289, 309, 327, 355, 415, 445, 471, 497, 543, 689

Offset: 1

Juri-Stepan Gerasimov, Jun 05 2014

Generated by n = 1, 2, 4, 5, 8, 9, 13, 15, 34, 42, 43, 57, 90, 99, 105, 112, 131, 136, 145, 153, 170, 184, ...

			4 is in this sequence because A001358(1) + 2^1 = 6 is also semiprime.

Cf. A001358.

NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[ sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; SemiPrimePi[n_] := Sum[ PrimePi[ n/Prime[i]] - i + 1, {i, PrimePi[ Sqrt[ n]] }]; sp = 4; lst = {}; While[ sp < 1001, If[ PrimeOmega[sp + 2^SemiPrimePi@ sp] == 2, AppendTo[lst, sp]; Print@ sp]; sp = NextSemiPrime@ sp; c++]; lst (* Robert G. Wilson v, Jun 20 2014 *)

list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ From A001358
sp=list(700); s=[]; for(n=1, #sp, if(bigomega(sp[n]+2^n)==2, s=concat(s, sp[n]))); s \\ Colin Barker, Jun 05 2014

One term inserted, and more terms from Colin Barker, Jun 05 2014

A257933 Prime p such that sqrt(p+2) is semiprime (A001358).

Original entry on oeis.org

79, 223, 439, 1087, 1223, 2399, 3023, 4759, 5927, 8647, 14159, 14639, 21023, 24023, 25919, 28559, 31327, 33487, 42023, 47087, 56167, 61007, 64007, 67079, 70223, 71287, 89399, 90599, 91807, 95479, 104327, 112223, 116279, 126023, 137639, 152879, 172223, 199807

Offset: 1

Vladimir Shevelev, May 13 2015

nonn

The terms are not congruent to 1 (mod 10).
The sequence contains no Mersenne prime p=2^t-1. Since p > 79, t is an odd prime and p+2 = 2^t+1 is divisible by 3. So, since 2^t+1 should be square, 2^t+1 is divisible by 9, i.e., (2^t+1)/3 == 0 (mod 3).       (1)
Note that either t=6k+1 or t=6m+5. In each case, (1) is impossible.
Indeed, if t=6k+1, then (2^t+1)/3 = (2*(4^k)^3+1)/3 = (2*(3+1)^(3*k)+1)/3 == (2*binomial(3*k,1)*3+2+1)/3 == 1(mod 3), and analogously in case t=6*m+5, (2^t+1)/3 == 2 (mod 3): a contradiction.

			Prime 79 is in the sequence because sqrt(79+2) = 9 = 3*3 which is semiprime.
Prime 1223 is in the sequence because sqrt(1223+2) = 35 = 5*7 which is semiprime.

Peter J. C. Moses and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from Moses)

Cf. A000040, A001358, A046315.

Select[Prime@Range@18000,PrimeOmega[Sqrt[#+2]]==2&]//Quiet (* Ivan N. Ianakiev, May 13 2015 *)

issemi(n)=bigomega(n)==2
is(n)=isprime(n) && issquare(n+2,&n) && issemi(n) \\ Charles R Greathouse IV, May 13 2015

list(lim)=my(v=List(), k=sqrt(lim+2), t); forprime(p=2, sqrt(k), forprime(q=p, k\p, if(isprime(t=(p*q)^2-2), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, May 13 2015

use ntheory ":all"; forprimes { say if is_power($+2,2) && scalar(factor(sqrtint($+2)))==2 } 1e7; # Dana Jacobsen, May 13 2015

use ntheory ":all"; sub list { my($lim,$k,$t,$p,%v)=shift; $k=sqrt($lim+2); forprimes { $p=$; forprimes { $t=($p*$)**2-2; $v{$t}++ if is_prime($t); } $p,int($k/$p); } int(sqrt($k)); my @v=sort{$a<=>$b} keys %v; @v; } say for list(1e10); # Translation of PARI, Dana Jacobsen, May 13 2015

Trivially a(n) >> n^2 log^2 n/(log log n)^2. - Charles R Greathouse IV, May 13 2015

More terms from Peter J. C. Moses, May 13 2015

A263349 a(n) = smallest number > a(n-1) such that a(n-1) + a(n) is a semiprime (A001358), a(1)=1.

Original entry on oeis.org

1, 3, 6, 8, 13, 20, 26, 29, 33, 36, 38, 39, 43, 44, 47, 48, 58, 60, 61, 62, 67, 74, 81, 85, 92, 93, 94, 100, 101, 102, 103, 106, 107, 108, 109, 110, 111, 115, 120, 127, 132, 133, 134, 140, 147, 148, 150, 151, 152, 153, 156, 158, 161, 162, 164, 165

Offset: 1

Zak Seidov, Oct 15 2015

nonn

Similar sequences with any other initial a(1) will eventually merge with case a(1)=1.

			1 + 3 = 4 = 2*2, 3 + 6 = 9 = 3*3, 6 + 8 = 14 = 2*7, etc.

Cf. A001358, A243625.

s={a=1};Do[k=a+1;While[PrimeOmega[a+k]!=2,k++];AppendTo[s,a=k],{100}];s

lista(nn) = {print1(a = 1, ", "); for (k=1, nn, na = a+1; while (bigomega(a+na) != 2, na++); a = na; print1(a, ", "););} \\ Michel Marcus, Oct 17 2015

A277343 a(1) is 4. a(n) is the least semiprime q (A001358) greater than p = a(n-1), such that p/q is a new minimum.

Original entry on oeis.org

4, 6, 10, 21, 46, 106, 247, 579, 1363, 3214, 7586, 17915, 42311, 99931, 236023, 557455, 1316638, 3109733, 7344803, 17347513, 40972678, 96772393, 228564417, 539840885, 1275037411, 3011480697, 7112745019, 16799424206, 39678162637, 93714913738, 221343037931, 522784885426, 1234753254431

Offset: 1

Robert G. Wilson v, Oct 09 2016

nonn

Inspired by and analogous to A265418.

p/q -> 0.423392190744304142156851442297311481582158896664...

			4/6 is 0.666... is a new low or minimum;
6/9 is 0.666... is not a new minimum, but;
6/10 is 0.600... is a new minimum;
10/21 is 0.476... is a new minimum;
21/46 is 0.456... is a new minimum;
... 522784885426/1234753254431 is 0.423... is a new minimum; etc.

Robert G. Wilson v, Table of n, a(n) for n = 1..200

Cf. A001358, A265418.

NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]];
p = 4; q = 6; mx = 1; lst = {}; While[q < 10^15, r = p/q; If[r < mx, mx = r; AppendTo[lst, p]; p = q]; q = NextSemiPrime[Floor[q/r]]]; lst (* or *)
f[lst_List] := Block[{p = lst[[-2]], q = lst[[-1]]}, Append[lst, NextSemiPrime[ Floor[q^2/p]]]]; lst = {4, 6}; lst = Nest[f, lst, 30]

A288517 Least integer k such that A001358(k) + A001358(k+1) is the product of exactly n prime factors (counting multiplicity).

Original entry on oeis.org

3, 1, 28, 4, 19, 39, 48, 89, 120, 551, 447, 589, 3707, 10137, 21644, 28456, 22998, 44494, 86132, 166930, 703448, 628371, 1220814, 1608668, 11153853, 6091437, 56676014, 268389220, 146153797, 193010987, 916382785, 738246947, 4702317172, 2830095027, 12627951809

Offset: 1

Zak Seidov, Jun 10 2017

nonn

			n=1: k=3, A001358(3) + A001358(4) = 9 + 10 = 19 = A000040(8) (8th prime),
n=2: k=1, A001358(1)+A001358(2) = 4+6 = 10 = 2*5 = A001358(4) (4th semiprime),
n=11: k=447, A001358(447)+A001358(448) = 1535+1537 = 3072 = 2^10*3 = A069272(2) (2nd 11-almost prime).

Charles R Greathouse IV, Table of n, a(n) for n = 1..50

Cf. A000040, A001358, A014612, A014614, A046306, A046308, A046310, A046312, A046314, A069272.

a(21)-a(35) from Charles R Greathouse IV, Jun 10 2017

cp's OEIS Frontend

A157414 Decimal expansion of Sum_{q = semiprimes = A001358} 1/(q*2^q).

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A173477 Semiprimes having no representation of the form semiprime(n)-+n, where semiprime(n) = A001358(n).

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A174838 Numbers n such that semiprime(n)+1 is prime, where semiprime(n) is A001358.

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A179463 Semiprimes A001358 containing at least one semiprime digit in base 10.

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A226834 Smallest semiprime (A001358) which is at the beginning of an arithmetic progression of n semiprimes whose largest term is as small as possible.

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A243428 Semiprimes A001358(n) such that A001358(n) + 2^n is also a semiprime.

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A257933 Prime p such that sqrt(p+2) is semiprime (A001358).

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A263349 a(n) = smallest number > a(n-1) such that a(n-1) + a(n) is a semiprime (A001358), a(1)=1.

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A277343 a(1) is 4. a(n) is the least semiprime q (A001358) greater than p = a(n-1), such that p/q is a new minimum.