A100484 -id:A100484 - cp's OEIS frontend

A124794 Coefficients of incomplete Bell polynomials in the prime factorization order.

1, 1, 1, 1, 1, 3, 1, 1, 3, 4, 1, 6, 1, 5, 10, 1, 1, 15, 1, 10, 15, 6, 1, 10, 10, 7, 15, 15, 1, 60, 1, 1, 21, 8, 35, 45, 1, 9, 28, 20, 1, 105, 1, 21, 105, 10, 1, 15, 35, 70, 36, 28, 1, 105, 56, 35, 45, 11, 1, 210, 1, 12, 210, 1, 84, 168, 1, 36, 55, 280, 1, 105, 1, 13, 280, 45, 126, 252, 1

Offset: 1

Max Alekseyev, Nov 07 2006

nonn

Coefficients of (D^k f)(g(t))*(D g(t))^k1*(D^2 g(t))^k2*... in the Faa di Bruno formula for D^m(f(g(t))) where k = k1 + k2 + ..., m = 1*k1 + 2*k2 + ....

Number of set partitions whose block sizes are the prime indices of n (i.e., the integer partition with Heinz number n). - Gus Wiseman, Sep 12 2018

			The a(6) = 3 set partitions of type (2,1) are {{1},{2,3}}, {{1,3},{2}}, {{1,2},{3}}. - _Gus Wiseman_, Sep 12 2018

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Bell Polynomial
Eric Weisstein's World of Mathematics, Faà di Bruno's Formula

Cf. A000110, A000258, A000670, A005651, A008277, A008480, A056239, A094416, A124794, A215366, A318762, A319182, A319225.

Cf. A065475, A100484.

with(numtheory):
a:= n-> (l-> add(i*l[i], i=1..nops(l))!/mul(l[i]!*i!^l[i],
         i=1..nops(l)))([seq(padic[ordp](n, ithprime(i)),
         i=1..pi(max(1, factorset(n))))]):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 14 2020

numSetPtnsOfType[ptn_]:=Total[ptn]!/Times@@Factorial/@ptn/Times@@Factorial/@Length/@Split[ptn];
Table[numSetPtnsOfType[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,100}] (* Gus Wiseman, Sep 12 2018 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, primepi(f[k,1])*f[k,2])!/(prod(k=1, #f~, f[k,2]!)*prod(k=1, #f~, primepi(f[k,1])!^f[k,2])); \\ Michel Marcus, Oct 11 2023

For n = p1^k1*p2^k2*... where 2 = p1 < p2 < ... are the sequence of all primes, a(n) = a([k1,k2,...]) = (k1+2*k2+...)!/((k1!*k2!*...)*(1!^k1*2!^k2*...)).

a(2*prime(n)) = n + 1, for n > 1. See A065475. - Bill McEachen, Oct 11 2023

A161344 Numbers k with A033676(k)=2, where A033676 is the largest divisor <= sqrt(k).

Original entry on oeis.org

4, 6, 8, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502, 514

Offset: 1

Omar E. Pol, Jun 20 2009

Define a sieve operation with parameter s that eliminates integers of the form s^2 + s*i (i >= 0) from the set A000027 of natural numbers. The sequence lists those natural numbers that are eliminated by the sieve s=2 and cannot be eliminated by any sieve s >= 3. - R. J. Mathar, Jun 24 2009
After a(3)=8 all terms are 2*prime; for n > 3, a(n) = 2*prime(n-1) = 2*A000040(n-1). - Zak Seidov, Jul 18 2009
From Omar E. Pol, Jul 18 2009: (Start)
A classification of the natural numbers A000027.
=============================================================
Numbers k whose largest divisor <= sqrt(k) equals j
=============================================================
j       Sequence     Comment
=============================================================
1 ..... A008578      1 together with the prime numbers
2 ..... A161344      This sequence
3 ..... A161345
4 ..... A161424
5 ..... A161835
6 ..... A162526
7 ..... A162527
8 ..... A162528
9 ..... A162529
10 .... A162530
11 .... A162531
12 .... A162532
... And so on. (End)
The numbers k whose largest divisor <= sqrt(k) is j are exactly those numbers j*m where m is either a prime >= k or one of the numbers in row j of A163925. - Franklin T. Adams-Watters, Aug 06 2009
See also A163280, the main entry for this sequence. - Omar E. Pol, Oct 24 2009
Also A100484 UNION 8. - Omar E. Pol, Nov 29 2012 (after Seidov and Hasler)
Is this the union of {4} and A073582? - R. J. Mathar, May 30 2025

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos
Omar E. Pol, Illustration: Divisors and pi(x)
Omar E. Pol, Illustration of initial terms
Omar E. Pol. Illustration of initial terms of A008578, A161344, A161345, A161424

Cf. A000005, A018253, A160811, A160812, A161205, A161346, A033676, A008578, A161345, A161424, A161835, A162526, A162527, A162528, A162529, A162530, A162531, A162532, A163925.

Second column of array in A163280. Also, second row of array in A163990.

isA := proc(n,s) if n mod s <> 0 then RETURN(false); fi; if n/s-s >= 0 then RETURN(true); else RETURN(false); fi; end: isA161344 := proc(n) for s from 3 to n do if isA(n,s) then RETURN(false); fi; od: isA(n,2) ; end: for n from 1 to 3000 do if isA161344(n) then printf("%d,",n) ; fi; od; # R. J. Mathar, Jun 24 2009

a[n_] := If[n <= 3, 2n+2, 2*Prime[n-1]]; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Nov 26 2012, after Zak Seidov *)

a(n)=if(n>3,prime(n-1),n+1)*2 \\ M. F. Hasler, Nov 27 2012

Equals 2*A000040 union {8}. - M. F. Hasler, Nov 27 2012

a(n) = 2*A046022(n+1) = 2*A175787(n). - Omar E. Pol, Nov 27 2012

More terms from R. J. Mathar, Jun 24 2009

Definition added by R. J. Mathar, Jun 28 2009

A338899 Concatenated sequence of prime indices of squarefree semiprimes (A006881).

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 2, 3, 2, 4, 1, 5, 1, 6, 2, 5, 1, 7, 3, 4, 1, 8, 2, 6, 1, 9, 2, 7, 3, 5, 2, 8, 1, 10, 1, 11, 3, 6, 2, 9, 1, 12, 4, 5, 1, 13, 3, 7, 1, 14, 2, 10, 4, 6, 2, 11, 1, 15, 3, 8, 1, 16, 2, 12, 3, 9, 1, 17, 4, 7, 1, 18, 2, 13, 2, 14, 4, 8, 1, 19, 2, 15

Offset: 1

Gus Wiseman, Nov 16 2020

This is a triangle with two columns and strictly increasing rows, namely {A270650(n), A270652(n)}.

A squarefree semiprime is a product of any two distinct prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
      6: {1,2}     57: {2,8}     106: {1,16}    155: {3,11}
     10: {1,3}     58: {1,10}    111: {2,12}    158: {1,22}
     14: {1,4}     62: {1,11}    115: {3,9}     159: {2,16}
     15: {2,3}     65: {3,6}     118: {1,17}    161: {4,9}
     21: {2,4}     69: {2,9}     119: {4,7}     166: {1,23}
     22: {1,5}     74: {1,12}    122: {1,18}    177: {2,17}
     26: {1,6}     77: {4,5}     123: {2,13}    178: {1,24}
     33: {2,5}     82: {1,13}    129: {2,14}    183: {2,18}
     34: {1,7}     85: {3,7}     133: {4,8}     185: {3,12}
     35: {3,4}     86: {1,14}    134: {1,19}    187: {5,7}
     38: {1,8}     87: {2,10}    141: {2,15}    194: {1,25}
     39: {2,6}     91: {4,6}     142: {1,20}    201: {2,19}
     46: {1,9}     93: {2,11}    143: {5,6}     202: {1,26}
     51: {2,7}     94: {1,15}    145: {3,10}    203: {4,10}
     55: {3,5}     95: {3,8}     146: {1,21}    205: {3,13}

A270650 is the first column.
A270652 is the second column.
A320656 counts multiset partitions using these rows, or factorizations into squarefree semiprimes.
A338898 is the version including squares, with columns A338912 and A338913.
A338900 gives row differences.
A338901 gives the row numbers for first appearances.
A001221 and A001222 count distinct/all prime indices.
A001358 lists semiprimes.
A004526 counts 2-part partitions, with strict case shifted right once.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes.
A046315 and A100484 list odd and even semiprimes.
A046388 lists odd squarefree semiprimes.
A166237 gives first differences of squarefree semiprimes.
Cf. A030229, A056239, A065516, A112798, A115392, A167171, A176506, A320893, A338904, A338906, A338907, A338910, A338911.

Join@@Cases[Select[Range[100],SquareFreeQ[#]&&PrimeOmega[#]==2&],k_:>PrimePi/@First/@FactorInteger[k]]

A338913 Greater prime index of the n-th semiprime.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 5, 3, 6, 5, 7, 4, 8, 6, 9, 4, 7, 5, 8, 10, 11, 6, 9, 12, 5, 13, 7, 14, 10, 6, 11, 15, 8, 16, 12, 9, 17, 7, 5, 18, 13, 14, 8, 19, 15, 20, 6, 10, 21, 11, 22, 16, 9, 23, 6, 17, 24, 18, 12, 7, 25, 19, 26, 10, 13, 27, 8, 20, 28, 14, 11, 29, 21

Offset: 1

Gus Wiseman, Nov 20 2020

nonn

A semiprime is a product of any two prime numbers. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

After the first three terms, there appear to be no adjacent equal terms.

			The semiprimes are:
  2*2, 2*3, 3*3, 2*5, 2*7, 3*5, 3*7, 2*11, 5*5, 2*13, ...
so the greater prime factors are:
  2, 3, 3, 5, 7, 5, 7, 11, 5, 13, ...
with indices:
  1, 2, 2, 3, 4, 3, 4, 5, 3, 6, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

A115392 lists positions of first appearances of each positive integer.
A270652 is the squarefree case, with lesser part A270650.
A338898 has this as second column.
A338912 is the corresponding lesser prime index.
A001221 counts distinct prime indices.
A001222 counts prime indices.
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A087794/A176504/A176506 are product/sum/difference of semiprime indices.
A338910/A338911 list products of pairs of odd/even-indexed primes.
Cf. A037143, A056239, A065516, A084126, A084127, A112798, A128301, A289182, A320732, A320892, A338900, A338904, A338909.

Table[Max[PrimePi/@First/@FactorInteger[n]],{n,Select[Range[100],PrimeOmega[#]==2&]}]

a(n) = A000720(A084127(n)).

A338898 Concatenated sequence of prime indices of semiprimes (A001358).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 1, 3, 1, 4, 2, 3, 2, 4, 1, 5, 3, 3, 1, 6, 2, 5, 1, 7, 3, 4, 1, 8, 2, 6, 1, 9, 4, 4, 2, 7, 3, 5, 2, 8, 1, 10, 1, 11, 3, 6, 2, 9, 1, 12, 4, 5, 1, 13, 3, 7, 1, 14, 2, 10, 4, 6, 2, 11, 1, 15, 3, 8, 1, 16, 2, 12, 3, 9, 1, 17, 4, 7, 5, 5, 1, 18, 2

Offset: 1

Gus Wiseman, Nov 15 2020

This is a triangle with two columns and weakly increasing rows, namely {A338912(n), A338913(n)}.

A semiprime is a product of any two prime numbers. A prime index of n is a number m such that the m-th prime number divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of semiprimes together with their prime indices begins:
      4: {1,1}     46: {1,9}      91: {4,6}     141: {2,15}
      6: {1,2}     49: {4,4}      93: {2,11}    142: {1,20}
      9: {2,2}     51: {2,7}      94: {1,15}    143: {5,6}
     10: {1,3}     55: {3,5}      95: {3,8}     145: {3,10}
     14: {1,4}     57: {2,8}     106: {1,16}    146: {1,21}
     15: {2,3}     58: {1,10}    111: {2,12}    155: {3,11}
     21: {2,4}     62: {1,11}    115: {3,9}     158: {1,22}
     22: {1,5}     65: {3,6}     118: {1,17}    159: {2,16}
     25: {3,3}     69: {2,9}     119: {4,7}     161: {4,9}
     26: {1,6}     74: {1,12}    121: {5,5}     166: {1,23}
     33: {2,5}     77: {4,5}     122: {1,18}    169: {6,6}
     34: {1,7}     82: {1,13}    123: {2,13}    177: {2,17}
     35: {3,4}     85: {3,7}     129: {2,14}    178: {1,24}
     38: {1,8}     86: {1,14}    133: {4,8}     183: {2,18}
     39: {2,6}     87: {2,10}    134: {1,19}    185: {3,12}

A112798 restricted to rows of length 2 gives this triangle.
A115392 is the row number for the first appearance of each positive integer.
A176506 gives row differences.
A338899 is the squarefree version.
A338912 is column 1.
A338913 is column 2.
A001221 counts a number's distinct prime indices.
A001222 counts a number's prime indices.
A001358 lists semiprimes.
A004526 counts 2-part partitions.
A006881 lists squarefree semiprimes.
A037143 lists primes and semiprimes.
A046315 and A100484 list odd and even semiprimes.
A046388 and A100484 list odd and even squarefree semiprimes.
A065516 gives first differences of semiprimes.
A084126 and A084127 give the prime factors of semiprimes.
A270650 and A270652 give the prime indices of squarefree semiprimes.
A320655 counts factorizations into semiprimes.
Cf. A056239, A101048, A320892, A320912, A338900, A338901, A338904, A338906, A338907, A338910, A338911.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Join@@primeMS/@Select[Range[100],PrimeOmega[#]==2&]

A338912 Lesser prime index of the n-th semiprime.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 4, 2, 3, 2, 1, 1, 3, 2, 1, 4, 1, 3, 1, 2, 4, 2, 1, 3, 1, 2, 3, 1, 4, 5, 1, 2, 2, 4, 1, 2, 1, 5, 3, 1, 3, 1, 2, 4, 1, 6, 2, 1, 2, 3, 5, 1, 2, 1, 4, 3, 1, 5, 2, 1, 3, 4, 1, 2, 6, 1, 3, 2, 6, 2, 5, 1, 4, 1, 3, 2, 1

Offset: 1

Gus Wiseman, Nov 20 2020

nonn

A semiprime is a product of any two prime numbers. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The semiprimes are:
  2*2, 2*3, 3*3, 2*5, 2*7, 3*5, 3*7, 2*11, 5*5, 2*13, ...
so the lesser prime factors are:
  2, 2, 3, 2, 2, 3, 3, 2, 5, 2, ...
with indices:
  1, 1, 2, 1, 1, 2, 2, 1, 3, 1, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

A084126 is the lesser prime factor (not index).
A084127 is the greater factor, with index A338913.
A115392 lists positions of ones.
A128301 lists positions of first appearances of each positive integer.
A270650 is the squarefree case, with greater part A270652.
A338898 has this as first column.
A001221 counts distinct prime indices.
A001222 counts prime indices.
A001358 lists semiprimes, with odds A046315 and evens A100484.
A006881 lists squarefree semiprimes, with odds A046388 and evens A100484.
A087794/A176504/A176506 are product/sum/difference of semiprime indices.
A338910/A338911 list products of pairs of odd/even-indexed primes.
Cf. A037143, A056239, A065516, A112798, A289182, A320732, A320892, A338900, A338904, A338909.

Table[Min[PrimePi/@First/@FactorInteger[n]],{n,Select[Range[100],PrimeOmega[#]==2&]}]

a(n) = A000720(A084126(n)).

A001748 a(n) = 3 * prime(n).

Original entry on oeis.org

6, 9, 15, 21, 33, 39, 51, 57, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 201, 213, 219, 237, 249, 267, 291, 303, 309, 321, 327, 339, 381, 393, 411, 417, 447, 453, 471, 489, 501, 519, 537, 543, 573, 579, 591, 597, 633, 669, 681, 687, 699, 717, 723, 753

Offset: 1

N. J. A. Sloane

Semiprimes divisible by 3. - Jianing Song, Oct 02 2022

Paolo P. Lava, Table of n, a(n) for n = 1..20000
Eric Weisstein's World of Mathematics, Skeleton

Subsequence of A001358.

Cf. A000040, A100484, A253046, A164023, A164024, A179545, A253046.

a001748 = (* 3) . a000040  -- Reinhard Zumkeller, Dec 26 2014

[3*p: p in PrimesUpTo(300)]; // Vincenzo Librandi, Mar 26 2014

Prime[Range[22]]*3 (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)

3*primes(22) \\ Charles R Greathouse IV, May 19 2011

A164023(a(n)) = A164024(a(n)) = A000040(n). - Reinhard Zumkeller, Aug 09 2009
a(n) = 3*A000040(n). - Omar E. Pol, Jan 31 2012
A253046(a(n)) < a(n). - Reinhard Zumkeller, Dec 26 2014

A001747 2 together with primes multiplied by 2.

Original entry on oeis.org

2, 4, 6, 10, 14, 22, 26, 34, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218, 226, 254, 262, 274, 278, 298, 302, 314, 326, 334, 346, 358, 362, 382, 386, 394, 398, 422, 446, 454, 458, 466, 478, 482, 502

Offset: 1

N. J. A. Sloane

When supplemented with 8, may be considered the "even primes", since these are the even numbers n = 2k which are divisible just by 1, 2, k and 2k. - Louis Zuckerman (louis(AT)trapezoid.com), Sep 12 2000
Sequence gives solutions of sigma(n) - phi(n) = n + tau(n) where tau(n) is the number of divisors of n.
Numbers n such that sigma(n) = 3*(n - phi(n)).
Except for 2, orders of non-cyclic groups k (in A060679(n)) such that x^k==1 (mod k) has only 1 solution 2<=x<=k. - Benoit Cloitre, May 10 2002
Numbers n such that A092673(n) = 2. - Jon Perry, Mar 02 2004
Except for initial terms, this sequence = A073582 = A074845 = A077017. Starting with the term 10, they are identical. - Robert G. Wilson v, Jun 15 2004
Together with 8 and 16, even numbers n such that n^2 does not divide (n/2)!. - Arkadiusz Wesolowski, Jul 16 2011
Twice noncomposite numbers. - Omar E. Pol, Jan 30 2012

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A060679, A009530, A098764.

Equals {2} UNION {A100484}.

Concatenation([2], List([1..60], n-> 2*Primes[n])); # G. C. Greubel, May 18 2019

[2] cat [2*NthPrime(n): n in [1..60]]; // G. C. Greubel, May 18 2019

Join[{2},2*Prime[Range[60]]] (* Harvey P. Dale, Jul 23 2013 *)

print1(2);forprime(p=2,97,print1(", "2*p)) \\ Charles R Greathouse IV, Jan 31 2012

[2]+[2*nth_prime(n) for n in (1..60)] # G. C. Greubel, May 18 2019

a(n) = A001043(n) - A001223(n+1), except for initial term.
a(n) = A116366(n-2,n-2) for n>2. - Reinhard Zumkeller, Feb 06 2006
A006093(n) = A143201(a(n+1)) for n>1. - Reinhard Zumkeller, Aug 12 2008
a(n) = 2*A008578(n). - Omar E. Pol, Jan 30 2012, and Reinhard Zumkeller, Feb 16 2012

A062234 From Bertrand's postulate: a(n) = 2*prime(n) - prime(n+1).

Original entry on oeis.org

1, 1, 3, 3, 9, 9, 15, 15, 17, 27, 25, 33, 39, 39, 41, 47, 57, 55, 63, 69, 67, 75, 77, 81, 93, 99, 99, 105, 105, 99, 123, 125, 135, 129, 147, 145, 151, 159, 161, 167, 177, 171, 189, 189, 195, 187, 199, 219, 225, 225, 227, 237, 231, 245, 251, 257, 267, 265, 273, 279

Offset: 1

Reinhard Zumkeller, Jun 29 2001

The theorem that a(n) > 0 for all n is known as "Bertrand's Postulate", and was proved by Tchebycheff in 1852.

The analog for Ramanujan primes is Paksoy's theorem that 2*R(n) - R(n+1) > 0 for n > 1. See A233822. - Jonathan Sondow, Dec 16 2013

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939.

Antti Karttunen, Table of n, a(n) for n = 1..10001 (first 1000 terms from Harry J. Smith)

Cf. A000040, A001223, A215808 (prime terms), A233822.
Cf. A098764, A100484, A258383 (run lengths), A258432, A258469, A257762, A258449, A257892, A257951.
When negated, forms the left edge of irregular triangle A252750, and also the leftmost column of square array A372562.

a062234 n = a062234_list !! (n-1)
a062234_list = zipWith (-) (map (* 2) a000040_list) (tail a000040_list)
-- Reinhard Zumkeller, May 31 2015

a:= n-> (p-> 2*p(n)-p(n+1))(ithprime):
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 09 2022

Table[2*Prime[n]-Prime[n+1],{n,60}] (* James C. McMahon, Apr 27 2024 *)
2#[[1]]-#[[2]]&/@Partition[Prime[Range[70]],2,1] (* Harvey P. Dale, Jul 29 2024 *)
ListConvolve[{-1, 2}, Prime[Range[100]]] (* Paolo Xausa, Nov 02 2024 *)

a(n) = 2*prime(n) - prime(n + 1); \\ Harry J. Smith, Aug 03 2009

a(n) = A000040(n) - A001223(n). - Zak Seidov, Sep 07 2012
a(n) = 2*A000040(n) - A000040(n+1). - Zak Seidov, May 12 2020
a(n) = A098764(n) - A000040(n). - Anthony S. Wright, Feb 19 2024

Edited by N. J. A. Sloane, Feb 24 2023

A070552 Semiprimes k such that k+1 is also a semiprime.

Original entry on oeis.org

9, 14, 21, 25, 33, 34, 38, 57, 85, 86, 93, 94, 118, 121, 122, 133, 141, 142, 145, 158, 177, 201, 202, 205, 213, 214, 217, 218, 253, 298, 301, 302, 326, 334, 361, 381, 393, 394, 445, 446, 453, 481, 501, 514, 526, 537, 542, 553, 565, 622, 633, 634, 694, 697

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), May 03 2002

nonn

A064911(a(n))*A064911(a(n)+1) = 1. - Reinhard Zumkeller, Dec 03 2009

D. W. Wilson, Table of n, a(n) for n = 1..10000

Cf. A001358, A007674, A039832, A100484.

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [4..700] | IsSemiprime(n) and IsSemiprime(n+1) ]; // Vincenzo Librandi, Jan 22 2016

f[n_]:=Last/@FactorInteger[n]=={1,1}||Last/@FactorInteger[n]=={2};lst={};Do[If[f[n]&&f[n+1],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Feb 25 2010 *)
Flatten[Position[Partition[Table[If[PrimeOmega[n]==2,1,0],{n,700}],2,1],{1,1}]] (* Harvey P. Dale, Feb 04 2015 *)
Select[Range[700], PrimeOmega[#] == PrimeOmega[# + 1] == 2 &] (* Vincenzo Librandi, Jan 22 2016 *)

forprime(p=3,1e3,if(bigomega(2*p-1)==2,print1(2*p-1", "));if(bigomega(2*p+1)==2,print1(2*p", "))) \\ Charles R Greathouse IV, Nov 09 2011

is(n)=if(n%2, isprime((n+1)/2) && bigomega(n)==2, isprime(n/2) && bigomega(n+1)==2) \\ Charles R Greathouse IV, Sep 08 2015

from sympy import factorint
def is_semiprime(n): return sum(e for e in factorint(n).values()) == 2
def ok(n): return is_semiprime(n) and is_semiprime(n+1)
print(list(filter(ok, range(698)))) # Michael S. Branicky, Sep 14 2021

a(n) >> n log n since either n or n+1 is in A100484. - Charles R Greathouse IV, Jul 21 2015

a(n) = A109373(n) - 1. - Zak Seidov Dec 19 2018

More terms from Vladeta Jovovic, May 03 2002

cp's OEIS Frontend

A124794 Coefficients of incomplete Bell polynomials in the prime factorization order.

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A161344 Numbers k with A033676(k)=2, where A033676 is the largest divisor <= sqrt(k).

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A338899 Concatenated sequence of prime indices of squarefree semiprimes (A006881).

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A338913 Greater prime index of the n-th semiprime.

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A338898 Concatenated sequence of prime indices of semiprimes (A001358).

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A338912 Lesser prime index of the n-th semiprime.

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A001748 a(n) = 3 * prime(n).

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