A001358 -id:A001358 - cp's OEIS frontend

A105998 Semiprime function n -> A001358(n) applied four times to n.

77, 119, 219, 235, 377, 381, 566, 634, 721, 779, 998, 1006, 1126, 1282, 1294, 1563, 1642, 1745, 1853, 1959, 1961, 2209, 2402, 2483, 2554, 2785, 3005, 3149, 3173, 3242, 3481, 3574, 3587, 3622, 4101, 4282, 4471, 4681, 4714, 4798, 4859, 4882, 5095, 5201

Offset: 1

Jonathan Vos Post, Apr 29 2005

			a(1) = semiprime(semiprime(semiprime(semiprime(1)))) = semiprime(semiprime(semiprime(4))) = semiprime(semiprime(10)) = semiprime(26) = 77.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A001358, A007097, A091022, A105997, A105999.

issp:= n-> not isprime(n) and numtheory[bigomega](n)=2:
sp:= proc(n) option remember; local k; if n=1 then 4 else
       for k from 1+sp(n-1) while not issp(k) do od; k fi end:
a:= n-> (sp@@4)(n):
seq(a(n), n=1..44);  # Alois P. Heinz, Aug 16 2024

f[n_] := Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ FactorInteger[ n]]; t = Select[ Range[ 5210], f[ # ] == 2 &]; Table[ Nest[ t[[ # ]] &, n, 4], {n, 45}] (* Robert G. Wilson v, Apr 30 2005 *)

from math import isqrt
from sympy import primepi, primerange
def A105998(n):
    def f(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, f(n)+n
        while m != k:
            m, k = k, f(k)+n
        return m
    return A001358(A001358(A001358(A001358(n)))) # Chai Wah Wu, Aug 16 2024

a(n) = A001358(A001358(A001358(A001358(n)))).

More terms from Robert G. Wilson v, Apr 30 2005

A105999 Semiprimeth recurrence: a(0) = 1, a(n+1) = semiprime(a(n)) = A001358(a(n)).

Original entry on oeis.org

1, 4, 10, 26, 77, 235, 779, 2785, 10643, 43697, 192893, 915218, 4657929, 25380749, 147721169, 916036271, 6037442989, 42191467826, 311911160465, 2434014941905, 20007995450483, 172911791611798, 1568190042677867

Offset: 0

Jonathan Vos Post, Apr 29 2005

nonn

Semiprime equivalent of R. G. Wilson's primeth recurrence: A007097.

			a(1) = A001358(1) = 4,
a(2) = A001358(a(1)) = A001358(4) = 10,
a(3) = A001358(a(2)) = A001358(10) = 26.

Cf. A001358, A007097, A091022, A105997, A105998.

SemiPrimePi[n_] := Sum[PrimePi[n/Prime@i] - i + 1, {i, PrimePi@ Sqrt@n}]; SemiPrime[n_] := Block[{e = Floor[Log[2, n] + 1], a, b}, a = 2^e; Do[b = 2^p; While[SemiPrimePi@a < n, a = a + b]; a = a - b/2, {p, e, 0, -1}]; a + b/2]; NestList[SemiPrime@# &, 1, 18] (* Robert G. Wilson v, May 31 2006 *)

a(5)-a(15) from Robert G. Wilson v, Apr 30 2005
a(16)-a(20) from Robert G. Wilson v, May 31 2006
a(21)-a(22) from Donovan Johnson, Sep 24 2010

A115709 Pentagonal numbers (A000326) whose digit reversal is a semiprime (A001358).

Original entry on oeis.org

12, 22, 51, 176, 330, 532, 590, 715, 782, 925, 1162, 1247, 1335, 1426, 1717, 3151, 3290, 4187, 5551, 7107, 7315, 7957, 10542, 10795, 11051, 11837, 12376, 14950, 15251, 15555, 15862, 16172, 17120, 19097, 19780, 20126, 22265, 24130, 24512, 26467, 26867, 30175

Offset: 1

Giovanni Resta, Jan 31 2006

			532 is the 19th pentagonal number and 235=5*47 is semiprime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000326, A001358, A115708.

Select[PolygonalNumber[5,Range[200]],PrimeOmega[IntegerReverse[#]]==2&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 27 2019 *)

from sympy import factorint
def ok(p): return sum(e for e in factorint(int(str(p)[::-1])).values()) == 2
print([p for p in (n*(3*n-1)//2 for n in range(143)) if ok(p)]) # Michael S. Branicky, Dec 22 2021

A065728 Partition numbers (A000041) that are semiprimes (A001358).

Original entry on oeis.org

15, 22, 77, 1255, 2012558, 2679689, 9289091, 18004327, 38887673, 56634173, 72533807, 82010177, 104651419, 2056148051, 2552338241, 20390982757, 27517052599, 118159068427, 749474411781, 5134205287973, 18028182516671

Offset: 1

Patrick De Geest, Nov 18 2001

nonn

Enoch Haga asks if this is a finite sequence. The larger these numbers get, the more opportunity for more factors.

			E.g., the 808th partition number 8151756509675604512522473567 = 5963320232189 * 1366982853893003.

Harry J. Smith, Table of n, a(n) for n=1,...,100
Hisanori Mishima, Factorization results
Eric Weisstein's World of Mathematics, Semiprime.

Intersection of A001358 and A000041.

Cf. A049575, A065729.

Select[PartitionsP[Range[0,450]],PrimeOmega[#]==2&] (* Harvey P. Dale, Sep 19 2016 *)

{ n=0; for (m=1, 10^9, p=numbpart(m); if (bigomega(p) == 2, write("b065728.txt", n++, " ", p); if (n==100, return)) ) } \\ Harry J. Smith, Oct 28 2009

A064911(a(n))*A167392(a(n)) = 1. [From Reinhard Zumkeller, Nov 03 2009]

OFFSET changed from 0,1 to 1,1 by Harry J. Smith, Oct 28 2009

A152447 Decimal expansion of the sum_q 1/(q*(q-1)) over the semiprimes q = A001358.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 04 2008

The semiprime analog of A136141. To obtain the (smaller) sum over the squarefree semiprimes A006881, subtract the prime zeta functions of 4 ( A085964 ), 6, 8 etc. from this constant here. The first term in the representation as the geometric series in powers 1/q^s is in A117543 .

R. J. Mathar, Series of reciprocal powers of k-almost primes, constant B_{2,1} in table 8.

Equals 0.17105189297999663662220256437237421399124661203550059749107997... = 1/(4*3)+1/(6*5)+1/(9*8)+1/(10*9)+...

A259677 Octagonal numbers (A000567) that are semiprimes (A001358).

Original entry on oeis.org

21, 65, 133, 341, 481, 1541, 4033, 5461, 6533, 8321, 11041, 13333, 14981, 31621, 38081, 48133, 56033, 79381, 83333, 97921, 109061, 111361, 133141, 188501, 197633, 206981, 219781, 229633, 256961, 282133, 293281, 328021, 340033, 360533, 416641, 481601, 556421

Offset: 1

Colin Barker, Jul 03 2015

nonn

			The octagonal number 21 is in the sequence because 21 = 3 * 7.

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

Cf. A129521, A245365, A259676.

IsSemiprime:=func; [s: n in [2..500] | IsSemiprime(s) where s is n*(3*n-2) ]; // Vincenzo Librandi, Jul 04 2015

a={}; Do[If[PrimeOmega[n (3 n - 2)]==2, AppendTo[a, n(3 n - 2)]], {n, 1, 200}]; a (* Vincenzo Librandi, Jul 04 2015 *)
Select[PolygonalNumber[8,Range[500]],PrimeOmega[#]==2&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 15 2019 *)

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(8, n)))

Equals A000567 intersect A001358.

A076343 A076340(A001358(n)), real part of semiprimes mapped as defined in A076340, A076341.

Original entry on oeis.org

4, 8, 15, 8, 16, 17, 31, 24, 15, 24, 47, 32, 33, 40, 49, 48, 63, 65, 49, 79, 56, 64, 47, 95, 72, 95, 80, 63, 88, 113, 97, 127, 96, 81, 104, 145, 97, 120, 129, 143, 120, 161, 175, 159, 136, 191, 144, 145, 111, 144, 129, 160, 209, 191, 168, 143, 239, 176, 241, 143

Offset: 1

Reinhard Zumkeller, Oct 08 2002

nonn

			Applying the map as defined in A076340, A076341:
A001358(15)=39=3*13=(4-1)*(12+1) -> (4,-1)*(12,1) = (4*12+1,4-12) = (49,-8), therefore a(15)=49 and A076344(120)=-8.

Imaginary part = A076344, A076342, A076347.

A076344 A076341(A001358(n)), imaginary part of semiprimes mapped as defined in A076340, A076341.

Original entry on oeis.org

0, -2, -8, 2, -2, 0, -12, -2, 8, 2, -16, 2, 4, -2, -8, -2, -16, -12, 8, -24, 2, -2, 16, -28, 2, -20, 2, 20, -2, -24, -4, -36, -2, 16, 2, -32, 20, -2, -8, -24, 2, -36, -48, -28, -2, -52, -2, 0, 32, 2, 28, -2, -48, -32, -2, 24, -64, 2, -56, 40, -4, 2, -72, 2, -20, 44, -2, -32, -76, -2

Offset: 1

Reinhard Zumkeller, Oct 08 2002

sign

			Applying the map as defined in A076340, A076341:
A001358(15)=39=3*13=(4-1)*(12+1) -> (4,-1)*(12,1) = (4*12+1,4-12) = (49,-8), therefore a(15)=-8 and A076343(15)=49.

Real part = A076343, A070750, A076348.

A104012 Indices of centered dodecahedral numbers (A005904) which are semiprimes (A001358).

Original entry on oeis.org

1, 2, 3, 5, 6, 11, 14, 15, 21, 26, 30, 35, 36, 44, 54, 63, 69, 74, 81, 114, 128, 131, 135, 138, 153, 165, 168, 191, 194, 209, 216, 224, 228, 231, 239, 261, 270, 303, 315, 321, 323, 326, 330, 336, 345, 363, 380, 384, 398, 404, 410, 411, 414, 429, 440, 443, 455, 468, 470

Offset: 1

Jonathan Vos Post, Feb 24 2005

Because the cubic polynomial for centered dodecahedral numbers factors into n time an irreducible quadratic, the dodecahedral numbers can never be prime, but can be semiprime iff (2*n+1) is prime and (5*n^2+5*n+1) is prime. Centered dodecahedral numbers (A005904) are not to be confused with dodecahedral numbers (A006566) = n(3n-1)(3n-2)/2, nor with rhombic dodecahedral numbers (A005917).

Intersection of A005097 and A090563. - Michel Marcus, Apr 30 2016

			a(1) = 1 because A005904(1) = 33 = 3 * 11, which is semiprime.
a(2) = 2 because A005904(2) = 155 = 5 * 31, which is semiprime.
a(3) = 3 because A005904(3) = 427 = 7 * 61, which is semiprime.
a(4) = 5 because A005904(5) = 1661 = 11 * 151.
194 is in this sequence because A005904(194) = 73579739 = 389 * 189151, which is semiprime.

Zak Seidov, Table of n, a(n) for n = 1..1000
B. K. Teo and N. J. A. Sloane, Magic numbers in polygonal and polyhedral clusters, Inorgan. Chem. 24 (1985),4545-4558.

Cf. A001222, A001358, A005904, A090563, A104011.

isok(n) = isprime(2*n+1) && isprime(5*n^2+5*n+1); \\ Michel Marcus, Apr 30 2016

n such that A001222(A005904(n)) = 2. n such that Bigomega((2*n+1)*(5*n^2 + 5*n + 1)) is 2. n such that A104011(n) = 2.

A113173 Ascending descending base exponent transform of semiprimes (A001358).

Original entry on oeis.org

256, 5392, 315361, 11667713, 717360537, 83932270482, 27775696582531, 22260761742531649, 109563850113131234720, 2013390472722146301196, 1899501614194512059559835, 85600281199526209989968735

Offset: 1

Jonathan Vos Post, Jan 07 2006

A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. a(7) is itself semiprime. The smallest primes in this sequence are a(3) = 315361 and a(4) = 11667713. What is the next prime?

			a(1) = 256 because semiprime(1)^semiprime(1) = 4^4 = 256.
a(2) = 5392 because prime(1)^prime(2) + prime(2)^prime(1) = 4^6 + 6^4 = 5392.
a(3) = 315361 because 4^9 + 6^6 + 9^4 = 315361.
a(4) = 11667713 = 4^10 + 6^9 + 9^6 + 10^4.
a(5) = 717360537 = 4^14 + 6^10 + 9^9 + 10^6 + 14^4.
a(6) = 83932270482 = 4^15 + 6^14 + 9^10 + 10^9 + 14^6 + 15^4.
a(7) = 27775696582531 = 4^21 + 6^15 + 9^14 + 10^10 + 14^9 + 15^6 + 21^4.
a(8) = 22260761742531649 = 4^22 + 6^21 + 9^15 + 10^14 + 14^10 + 15^9 + 21^6 + 22^4.
a(9) = 109563850113131234720 = 4^25 + 6^22 + 9^21 + 10^15 + 14^14 + 15^10 + 21^9 + 22^6 + 25^4.

G. C. Greubel, Table of n, a(n) for n = 1..100

Cf. A001358, A005408, A113122, A113153, A113154.

A001358[A001358%5Bk%5D%5B%5Bk%5D%5D)%5E((A001358%5Bn%20-%20k%20+%201%5D%5B%5Bn%20-%20k%20+%201%5D%5D)),%20%7Bk,%201,%20n%7D%5D,%20%7Bn,%201,%2010%7D%5D%20(*%20_G.%20C.%20Greubel">] := Select[Range[100], PrimeOmega[#] == 2 &]; Table[Sum[(A001358[k][[k]])^((A001358[n - k + 1][[n - k + 1]])), {k, 1, n}], {n, 1, 10}] (* _G. C. Greubel, May 19 2017 *)

a(n) = Sum_{i=1..n} (semiprime(i))^(semiprime(n-i+1)).

a(n) = Sum_{i=1..n} (A001358(i))^(A001358(n-i+1)).

cp's OEIS Frontend

A105998 Semiprime function n -> A001358(n) applied four times to n.

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A105999 Semiprimeth recurrence: a(0) = 1, a(n+1) = semiprime(a(n)) = A001358(a(n)).

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A115709 Pentagonal numbers (A000326) whose digit reversal is a semiprime (A001358).

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A065728 Partition numbers (A000041) that are semiprimes (A001358).

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A152447 Decimal expansion of the sum_q 1/(q*(q-1)) over the semiprimes q = A001358.

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A259677 Octagonal numbers (A000567) that are semiprimes (A001358).

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Magma

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A076343 A076340(A001358(n)), real part of semiprimes mapped as defined in A076340, A076341.

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A076344 A076341(A001358(n)), imaginary part of semiprimes mapped as defined in A076340, A076341.

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A104012 Indices of centered dodecahedral numbers (A005904) which are semiprimes (A001358).

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