A001358 -id:A001358 - cp's OEIS frontend

A115654 Semiprimes (A001358) which are the sum of distinct double factorials (A006882).

4, 6, 9, 10, 14, 15, 21, 25, 26, 49, 51, 57, 58, 62, 65, 69, 74, 77, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 155, 158, 159, 161, 166, 169, 177, 178, 386, 393, 394, 395, 398, 403, 407, 411, 413, 437, 445, 446, 447, 451, 453, 458, 489, 493, 497

Offset: 1

Giovanni Resta, Jan 28 2006

nonn

Double factorials 0!! and 1!! are not considered distinct. Note that double factorial (n!!) is different from (n!)!.

			384 = 2*19 = 8!!+2!!.

Cf. A006882, A001358, A115651, A115652, A115653.

Union[Select[Total/@Subsets[Range[10]!!,10],PrimeOmega[#]==2&]] (* Harvey P. Dale, Aug 24 2012 *)

A115670 Semiprimes (A001358) whose digit reversal is prime.

Original entry on oeis.org

14, 34, 35, 38, 74, 91, 95, 106, 118, 119, 133, 134, 142, 145, 146, 166, 194, 301, 305, 334, 346, 358, 361, 362, 365, 371, 377, 382, 386, 391, 395, 703, 706, 713, 721, 731, 745, 746, 749, 755, 758, 763, 778, 779, 785, 791, 793, 799, 901, 905, 914, 917, 922

Offset: 1

Giovanni Resta, Jan 31 2006

			35=5*7 is semiprime and 53 is prime.

Cf. A001358, A083815, A085778, A097393.

A124319 Semiprime(3almostprime(n))-3almostprime(semiprime(n)). Commutator[A001358, A014612] at n.

Original entry on oeis.org

2, 6, 7, 12, 16, 17, -11, 24, 23, 20, -1, 10, 48, 40, 39, 26, 14, 4, -1, 51, 60, 48, 48, 43, 31, 39, 22, 15, 37, 32, 39, 60, 90, 82, 68, 63, 64, 58, 66, 51, 53, 48, 28, 34, 42, 24, 28, 39, 87, 96, 106, 124, 124, 135, 131, 131, 88, 91, 72, 96, 103, 83, 83, 81, 91

Offset: 1

Jonathan Vos Post, Oct 26 2006

			a(1) = semiprime(3almostprime(1)) - 3almostprime(semiprime(1)) = 22 - 20 = 2.
a(2) = semiprime(3almostprime(2)) - 3almostprime(semiprime(2)) = 34 - 28 = 6.
a(3) = semiprime(3almostprime(3)) - 3almostprime(semiprime(3)) = 51 - 44 = 7.
a(4) = semiprime(3almostprime(4)) - 3almostprime(semiprime(4)) = 57 - 45 = 12.
a(7) = semiprime(3almostprime(7)) - 3almostprime(semiprime(7)) = 87 - 98 = -11, which is the first negative value in the commutators we have seen in these related set of sequences, exposing an incorrect assumption.

Cf. A124317 Semiprimes indexed by 3-almost primes. A124318 3-almost primes indexed by semiprimes. A124319 semiprime(3almostprime(n)) - 3almostprime(semiprime(n)). A124308 Primes indexed by 5-almost primes. A124309 5-almost primes indexed by primes. A124310 prime(5almostprime(n)) - 5almostprime(prime(n)). 4-almost primes indexed by primes = A124283. prime(4almostprime(n)) - 4almostprime(prime(n)) = A124284. Primes indexed by 3-almost primes = A124268. 3-almost primes indexed by primes = A124269. prime(3almostprime(n)) - 3almostprime(prime(n)) = A124270. See also A106349 Primes indexed by semiprimes. See also A106350 Semiprimes indexed by primes. See also A122824 Prime(semiprime(n)) - semiprime(prime(n)). Commutator [A000040, A001358] at n.

Cf. A000040, A001358, A014612, A065516, A114403, A122824, A124269, A124270, A124282-A124284, A124309, A124310, A124317, A124318.

p[k_] := p[k] = Select[Range[1000], PrimeOmega[#] == k &]; p[2][[ Take[p[3], 70]]] - p[3][[Take[p[2], 70]]] (* Giovanni Resta, Jun 13 2016 *)

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A124319(n):
    def f(x): return int(x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    def g(x): return int(x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    def A001358(n):
        m, k = n, g(n)+n
        while m != k:
            m, k = k, g(k)+n
        return m
    m, k = n, f(n)+n
    while m != k:
        m, k = k, f(k)+n
    r, k = (p:=A001358(n)), f(p)+p
    while r != k:
        r, k = k, f(k)+p
    return A001358(m)-r # Chai Wah Wu, Aug 17 2024

a(18) corrected and a(22)-a(65) from Giovanni Resta, Jun 13 2016

A172348 Index k of the semiprime A001358(k) = prime(n) * prime(n+1).

Original entry on oeis.org

2, 6, 13, 26, 48, 75, 103, 135, 199, 270, 338, 443, 508, 581, 706, 878, 1001, 1124, 1305, 1413, 1565, 1764, 1978, 2299, 2571, 2724, 2886, 3052, 3213, 3710, 4259, 4581, 4859, 5259, 5668, 5954, 6409, 6797, 7184, 7696, 8029, 8515, 9062, 9325, 9608, 10246, 11444

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Feb 01 2010

nonn

The positions of products of 2 successive primes in A001358. - Juri-Stepan Gerasimov, Apr 14 2010

			n=1: 6 = 2 * 3 = prime(1) * prime(2) = semiprime(2). Therefore a(1) = 2.
n=2: 15 = 3 * 5 = prime(2) * prime(3) = semiprime(6). Therefore a(2) = 6.
n=3: 35 = 5 * 7 = prime(3) * prime(4) = semiprime(13). Therefore a(3) = 13.

Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Band I, B. G. Teubner, Leipzig u. Berlin, 1909.
Derrick H. Lehmer, Guide to Tables in the Theory of Numbers Washington, D.C. 1941.

Donovan Johnson, Table of n, a(n) for n = 1..1000
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.

Cf. A001358, A006094, A006881.

A001358 := proc(n) option remember; local a; if n = 1 then 4; else for a from procname(n-1)+1 do if numtheory[bigomega](a)= 2 then return a; end if; end do ; end if; end proc:
A006094 := proc(n) ithprime(n)*ithprime(n+1) ; end proc:
A172348 := proc(n) pp := A006094(n) ; for k from 1 do if A001358(k) = pp then return k; end if; end do ; end proc:
seq(A172348(n),n=1..70) ; # R. J. Mathar, Feb 09 2010

semiPrimePi[n_] := Sum[ PrimePi[n/Prime@ i] - i + 1, {i, PrimePi@ Sqrt@ n}];  semiPrimePi@# & /@ Table[ Prime[n] Prime[n + 1], {n, 47}] (* Robert G. Wilson v, Feb 02 2013 *)
nn=50000;Flatten[Module[{sp=Select[Range[nn+PrimePi[nn]],PrimeOmega[#] == 2&]},Table[ Position[sp,Prime[n]Prime[n+1]],{n,PrimePi[nn]}]]] (* Harvey P. Dale, Sep 07 2013 *)

a(n) = {k: A001358(k) = A006094(n)}.

Entries checked by R. J. Mathar, Feb 09 2010

A181522 Number of subsets of {1,2,...,n} whose sum is semiprime (cf. A001358, A064911).

Original entry on oeis.org

0, 0, 2, 6, 13, 25, 47, 92, 184, 367, 721, 1416, 2769, 5407, 10662, 21135, 41866, 83220, 166617, 334852, 670725, 1334868, 2650263, 5280475, 10567613, 21145411, 42103939, 83382359, 164843079, 326791838, 650995628, 1301718424, 2605360702, 5205671338, 10369588530

Offset: 1

Reinhard Zumkeller, Oct 27 2010

nonn

			a(4) = #{{1,3}, {4}, {1,2,3}, {2,4}, {2,3,4}, {1,2,3,4}} = 6.

Alois P. Heinz, Table of n, a(n) for n = 1..400

Cf. A127542, A126024.

Cf. A064911.

import Data.List (subsequences)
a181522 = length . filter ((== 1) . a064911 . sum) .
                          subsequences . enumFromTo 1
-- Reinhard Zumkeller, Feb 22 2012, Oct 27 2010

A184728 a(n) = largest k such that A001358(n+1) = A001358(n) + (A001358(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 8, 6, 13, 9, 20, 19, 24, 19, 32, 33, 32, 37, 32, 43, 47, 47, 53, 56, 54, 59, 61, 64, 71, 72, 79, 84, 85, 83, 89, 92, 93, 84, 101, 107, 112, 117, 117, 120, 121, 117, 125, 132, 127, 140, 141, 141, 144, 137, 152, 157, 157

Offset: 1

Rémi Eismann, Jan 20 2011

a(n) = A001358(n) - A065516(n) if A001358(n) - A065516(n) > A065516(n), 0 otherwise.

A001358(n): semiprimes; A065516(n): first difference of semiprimes.

			For n = 1 we have A001358(n) = 4, A001358(n+1) = 6; there is no k such that 6 - 4 = 2 = (4 mod k), hence a(1) = 0.
For n = 3 we have A001358(n) = 9, A001358(n+1) = 10; 8 is the largest k such that 10 - 9 = 1 = (9 mod k), hence a(3) = 8; a(3) = A001358(3) - A065516(3) = 8.
For n = 20 we have A001358(n) = 57, A001358(n+1) = 58; 56 is the largest k such that 58 - 57 = 1 = (57 mod k), hence a(20) = 56; a(20) = A001358(20) - A065516(20) = 56.

Rémi Eismann, Table of n, a(n) for n = 1..10000

Cf. A001358, A065516, A130533, A184729, A117078, A117563, A001223, A118534.

A259676 Heptagonal numbers (A000566) that are semiprimes (A001358).

Original entry on oeis.org

34, 55, 235, 403, 469, 697, 1177, 1651, 2059, 2839, 4141, 5221, 6943, 9211, 9517, 13213, 13579, 21949, 23377, 25351, 29539, 31753, 34633, 37027, 53071, 62173, 68641, 74563, 78943, 93799, 96727, 118483, 130759, 144841, 164737, 171217, 187279, 191407, 196981

Offset: 1

Colin Barker, Jul 03 2015

nonn

For these semiprimes k*(5*k-3)/2, the corresponding k are listed in A114517.

			The heptagonal number 34 is in the sequence because 34 = 2 * 17.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A129521, A245365, A259677.

IsSemiprime:=func; [s: n in [2..300] | IsSemiprime(s) where s is n*(5*n-3) div 2]; // Vincenzo Librandi, Jul 04 2015

a={}; Do[If[PrimeOmega[n (5 n - 3) / 2]==2, AppendTo[a, n(5 n - 3) / 2]], {n, 1, 200}]; a (* Vincenzo Librandi, Jul 04 2015 *)
Select[PolygonalNumber[7,Range[300]],PrimeOmega[#]==2&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 07 2021 *)

pg(m, n) = (n^2*(m-2)-n*(m-4))/2 \\ n-th m-gonal number
select(n->bigomega(n)==2, vector(2000, n, pg(7, n)))

Equals A000566 intersect A001358.

A085367 Semiprimes that can be expressed as the sum or difference of two cubes: intersection of A001358 and A045980.

Original entry on oeis.org

9, 26, 35, 65, 91, 133, 169, 215, 217, 218, 335, 341, 386, 407, 469, 485, 511, 559, 721, 737, 793, 817, 866, 973, 1027, 1115, 1141, 1241, 1261, 1267, 1339, 1343, 1385, 1387, 1538, 1603, 1685, 1727, 1843, 1853, 1981, 2071, 2189, 2402, 2413, 2611, 2743, 2771

Offset: 1

Hugo Pfoertner, Jun 25 2003

nonn

			a(1)=9 because 2^3+1^3=3*3, a(2)=26=3^3-1^3=2*13.
a(5)=91 is the smallest semiprime expressible in two different ways: 91=4^3+3^3=6^3-5^3=7*13.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001358, A045980, A085366.

T=thueinit('z^3+1);
is(n)=bigomega(n)==2 && #thue(T, n)
list(lim)=my(v=List()); forprime(p=2,lim\2, forprime(q=2,min(lim\p,p), if(#thue(T, p*q), listput(v,p*q)))); Set(v) \\ Charles R Greathouse IV, Nov 29 2014

A104623 Indices of semiprime (A001358) values of Heptanacci-Lucas numbers A104621.

Original entry on oeis.org

4, 8, 9, 11, 12, 14, 15, 16, 22, 23, 32, 34, 37, 41, 42, 50, 52, 57, 58, 66, 69, 76, 77, 81, 90, 120, 139

Offset: 0

Jonathan Vos Post, Mar 17 2005

nonn

The 7th-order linear recurrence A104622 (heptanacci-Lucas numbers) is a generalization of the Lucas sequence A000032. T. D. Noe and I have noted that the heptanacci-Lucas numbers have many more primes than the corresponding heptanacci (see A104414) which he found has only the first 3 primes that I identified through the first 5000 values, whereas these heptanacci-Lucas numbers have 17 primes among the first 100 values. For primes in Heptanacci-Lucas numbers, see A104622.

			A104621(4) = 15 = 3 * 5,
A104621(8) = 247 = 13 * 19,
A104621(9) = 493 = 17 * 29,
A104621(11) = 1959 = 3 * 653,
A104621(12) = 3903 = 3 * 1301,
A104621(14) = 15487 = 17 * 911,

Mario Catalani, Polymatrix and Generalized Polynacci Numbers, arXiv:math.CO/0210201 v1, 2002
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001358.

A115646 Semiprimes (A001358) that are sums of distinct factorials.

Original entry on oeis.org

6, 9, 25, 26, 33, 121, 122, 123, 129, 145, 146, 721, 723, 745, 746, 753, 841, 842, 843, 849, 865, 866, 871, 5041, 5042, 5065, 5071, 5161, 5163, 5169, 5186, 5191, 5761, 5767, 5793, 5905, 5906, 5911, 40321, 40322, 40323, 40345, 40346, 40353, 40441

Offset: 1

Giovanni Resta, Jan 27 2006

nonn

Factorials 0! and 1! are not considered distinct.

			721 = 6!+1! = 7*103.

Cf. A001358, A025494, A115645, A115644, A089359.

semipQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; fac=Range[10]!;lst={}; Do[ n = Plus@@(fac*IntegerDigits[k, 2, 10]); If[semipQ[n], AppendTo[lst, n]], {k, 2^10-1}]; Union[lst]

cp's OEIS Frontend

A115654 Semiprimes (A001358) which are the sum of distinct double factorials (A006882).

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A115670 Semiprimes (A001358) whose digit reversal is prime.

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A124319 Semiprime(3almostprime(n))-3almostprime(semiprime(n)). Commutator[A001358, A014612] at n.

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A172348 Index k of the semiprime A001358(k) = prime(n) * prime(n+1).

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A181522 Number of subsets of {1,2,...,n} whose sum is semiprime (cf. A001358, A064911).

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A184728 a(n) = largest k such that A001358(n+1) = A001358(n) + (A001358(n) mod k), or 0 if no such k exists.

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A259676 Heptagonal numbers (A000566) that are semiprimes (A001358).

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A085367 Semiprimes that can be expressed as the sum or difference of two cubes: intersection of A001358 and A045980.

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A104623 Indices of semiprime (A001358) values of Heptanacci-Lucas numbers A104621.