A001358 -id:A001358 - cp's OEIS frontend

A123665 a(n) = Sum_{k=1..21} n^A001358(k).

22, 471260364628084305, 6457022669043550542502557676, 105149403852520725445003265581519105, 41911381174488637014293971538580334000626

Offset: 1

Jonathan Vos Post, Oct 04 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A001358, A002523, A091940, A123177, A123656-A123665.

[1 + n^4 + n^6 + n^9 + n^10 + n^14 + n^15 + n^21 + n^22 + n^25 +
n^26 + n^33 + n^34 + n^35 + n^38 + n^39 + n^46 + n^49 + n^51 + n^55 + n^57 + n^58: n in [1..50]]; // G. C. Greubel, Oct 26 2017

Table[1 + n^4 + n^6 + n^9 + n^10 + n^14 + n^15 + n^21 + n^22 + n^25 +
  n^26 + n^33 + n^34 + n^35 + n^38 + n^39 + n^46 + n^49 + n^51 +
  n^55 + n^57 + n^58, {n, 1, 50}] (* G. C. Greubel, Oct 26 2017 *)

for(n=1,50, print1(1 + n^4 + n^6 + n^9 + n^10 + n^14 + n^15 + n^21 + n^22 + n^25 + n^26 + n^33 + n^34 + n^35 + n^38 + n^39 + n^46 + n^49 + n^51 + n^55 + n^57 + n^58, ", ")) \\ G. C. Greubel, Oct 26 2017

a(n) = 1 +n^4 +n^6 +n^9 +n^10 +n^14 +n^15 +n^21 +n^22 +n^25 +n^26 + n^33 +n^34 +n^35 +n^38 +n^39 +n^46 +n^49 +n^51 +n^55 +n^57 +n^58.

Better name from Joerg Arndt, May 23 2021

A218172 Centered 12-gonal numbers which are semiprimes, intersection of A003154 and A001358.

Original entry on oeis.org

121, 253, 793, 1261, 1441, 1633, 1837, 2773, 3601, 3901, 4213, 4537, 4873, 5221, 7141, 9841, 11881, 14113, 14701, 16537, 17173, 17821, 19153, 19837, 21241, 22693, 23437, 24193, 24961, 28153, 28981, 29821, 30673, 34201, 37921, 38881, 39853, 40837, 41833, 43861, 45937, 48061, 49141, 50233, 53581, 55873

Offset: 1

Zak Seidov, Oct 22 2012

nonn

Might also be called 'semiprime star numbers'.

A083749 and A006061 are subsequences.

			a(1) = 121 = 11^2 = A001358(40) = A003154(5) = A083749(1) = A006061(1) = A078972(11).
a(2) = 253 = 11*23 = A001358(81) = A003154(7) = A083749(2) = A078972(18).

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A003154, A001358, A006061, A078972, A083749.

Select[Table[6n(n-1)+1,{n,100}],PrimeOmega[#]==2&] (* Harvey P. Dale, Sep 01 2014 *)

lista(nn) = {for (n = 1, nn, if (bigomega(v = 6*n*(n-1) + 1) == 2, print1(v, ", ")););} \\ Michel Marcus, Nov 09 2013

A228170 The least semiprime (A001358) such that between it and the next n semiprimes, but not the next n+1 semiprimes, there are no primes.

Original entry on oeis.org

9, 33, 91, 141, 115, 213, 1382, 1639, 1133, 2558, 2973, 1329, 15685, 16143, 9974, 35678, 34063, 43333, 19613, 107381, 162145, 44294, 404599, 461722, 838259, 155923, 535403, 492117, 396737, 2181739, 370262, 1468279, 6034249, 3933601, 1671783, 25180174, 1357203

Offset: 1

Jack Brennen, Jonathan Vos Post, Zak Seidov, and Robert G. Wilson v, Aug 14 2013

nonn

If prime_omega(n) as defined as A001222 and a set of values becomes a string, then the 'just' means that its string is not a substring of some larger string. See the example below.
Yet another way to think of this is that between any two consecutive primes there are 'just' n semiprimes with the first one being cited above.
a(91) > 1.8*10^12. - Giovanni Resta, Aug 15 2013

			a(1) = 9 because between 9 and 10 there are no primes;
a(2) = 33 because between 33 and 35 (the second semiprime past 33) there are no primes;
a(3) = 91 because between 91 and 95 (the third semiprime past 91 with 93 & 94 also semiprimes) there are no primes;
a(4) = 141 because between 141 and 146 (the fourth semiprime past 141 with 142, 143 & 145 also being semiprimes) there are no primes;
the reason a(4) is not 115 is because although there are no primes between 115 and 121, the string "2, 3, 3, 2, 2, 5, 2, 2" is a substring of the string generated by 115 through 123. See the next line.
a(5) = 115 because between 115 and 123 (the fifth semiprime past 115 with 118, 119, 121, and 122 also being semiprimes) there are no primes;

Giovanni Resta, Table of n, a(n) for n = 1..90

Cf. A001358, A133478, A112888, A228171.

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]]; t = Table[0, {100}]; p=3; While[p < 3100000000, q = NextPrime[p]; a = Count[ PrimeOmega[ Range[p, q]], 2]; If[ t[[a]] == 0, t[[a]] = p; Print[{p, a}]]; p = q]; NextSemiPrime@# & /@ t

a(n) is the next semiprime after A228171(n+1).

A330477 Semiprimes (A001358) pq such that pq+p+q is also a semiprime.

Original entry on oeis.org

9, 22, 25, 39, 62, 69, 77, 87, 91, 94, 95, 106, 115, 119, 121, 122, 133, 134, 142, 146, 159, 183, 187, 202, 213, 214, 218, 219, 226, 235, 237, 249, 253, 259, 262, 265, 274, 287, 289, 291, 299, 303, 305, 309, 314, 335, 362, 381, 386, 393, 403, 411, 417, 422, 446, 458, 469, 473, 489, 501, 502, 505

Offset: 1

J. M. Bergot and Robert Israel, Dec 15 2019

nonn

			a(3) = 25 is a member because 25 = 5*5 and 25+5+5 = 5*7 is also a semiprime.

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

Cf. A001358.

Contains A108570.

N:= 1000:
Primes:= select(isprime, [2,seq(i,i=3..N)]):
SP:= sort([seq(seq([p,q],q=select(t -> t >= p and p*t<=N, Primes)),p=Primes)],(a,b) -> a[1]*a[2] t[1]*t[2], select(t -> numtheory:-bigomega(t[1]*t[2]+t[1]+t[2])=2, SP));

Select[Union@ Apply[Join, Table[Flatten@{p #, Sort[{p, #}]} & /@ Prime@ Range@ PrimePi@ Floor[Max[#]/p], {p, #}]] &@ Prime@ Range@ 97, PrimeOmega[Total@ #] == 2 &][[All, 1]] (* Michael De Vlieger, Dec 15 2019 *)

issemi(n)=bigomega(n)==2
list(lim)=my(v=List()); forprime(p=2, sqrtint(lim\=1), forprime(q=p, lim\p, if(issemi(p*q+p+q), listput(v,p*q)))); Set(v) \\ Charles R Greathouse IV, Dec 16 2019

from sympy import factorint
def is_semiprime(n): return sum(e for e in factorint(n).values()) == 2
def ok(n):
    f = factorint(n, multiple=True)
    if len(f) != 2: return False
    p, q = f
    return len(factorint(p*q + p + q, multiple=True)) == 2
print(list(filter(ok, range(506)))) # Michael S. Branicky, Sep 22 2021

A383468 Semiprimes s = A001358(k) such that k, s - k and s + k are also semiprimes.

Original entry on oeis.org

10, 15, 141, 166, 274, 298, 299, 687, 995, 1115, 1227, 1299, 1345, 1891, 1945, 2194, 2661, 2998, 3093, 3287, 3566, 3781, 3902, 4174, 4262, 4497, 4798, 5378, 5414, 5758, 6609, 7094, 7666, 8354, 8434, 9566, 10041, 10342, 11051, 11091, 11486, 11582, 11702, 12279, 12574, 13154, 13346, 13387, 13466

Offset: 1

Zak Seidov and Robert Israel, Apr 27 2025

nonn

Except for a(1) = 10 = A001358(4), s and k always have different parities.

			a(3) = 141 is a term because 141 = 3 * 47 = A001358(46) is a semiprime and 46 = 2 * 23, 141 - 46 = 95 = 5 * 19 and 141 + 46 = 187 = 11 * 17 are all semiprimes.

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

Cf. A001358. A383469.

k:= 0: R:= NULL: count:= 0:
for s from 1 while count < 100 do
  if numtheory:-bigomega(s) = 2 then
    k:= k+1;
    if andmap(t -> numtheory:-bigomega(t) = 2, [k, s-k, s+k]) then
      R:= R, s; count:= count+1;
    fi
  fi;
od:
R;

a(n) = A001358(A383469(n)).

A383469 Semiprimes k such that A001358(k) + k and A001358(k) - k are also semiprimes.

Original entry on oeis.org

4, 6, 46, 55, 87, 93, 94, 206, 298, 326, 362, 382, 394, 542, 562, 629, 758, 841, 866, 926, 993, 1046, 1079, 1147, 1167, 1234, 1317, 1469, 1477, 1561, 1774, 1895, 2047, 2227, 2245, 2515, 2638, 2705, 2894, 2902, 3007, 3031, 3063, 3202, 3269, 3409, 3453, 3466, 3487, 3809, 3891, 4058, 4174, 4207

Offset: 1

Zak Seidov and Robert Israel, Apr 27 2025

nonn

			a(3) = 46 is a term because 46 = 2 * 23 is a semiprime, A001358(46) = 141, and 141 - 46 = 95 = 5 * 19 and 141 + 46 = 187 = 11 * 17 are semiprimes.

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

Cf. A001358, A383468.

k:= 0: K:= NULL: count:= 0:
for s from 1 while count < 100 do
  if numtheory:-bigomega(s) = 2 then
    k:= k+1;
    if andmap(t -> numtheory:-bigomega(t) = 2, [k, s-k, s+k]) then
      K:= K, k; count:= count+1;
    fi
  fi;
od:
K;

A001358(a(k)) = A383468(k).

A072165 Values of Moebius function of the products of two (not necessarily distinct) primes (semiprimes or 2-almost primes, A001358).

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1

Offset: 1

Jani Melik, Jun 28 2002

Characteristic function for numbers n such that A001358(n) is not a square. - Antti Karttunen, Oct 04 2017

			For n = 2995, A001358(2995) = 11449 = 107^2, and as Moebius mu is zero for squares, we have a(2995) = 0. - _Antti Karttunen_, Oct 04 2017

Antti Karttunen, Table of n, a(n) for n = 1..3001
Kimberly Schneider, Moebius function.
Index entries for characteristic functions

Cf. A001358, A008683.

semiprimes := proc(d_n) local a,i; a := [ ]; for i from 1 to d_n do if((tau(i) = 3) or ((mobius(i) <> 0) and (tau(i) = 4))) then a := [ op(a), mobius(i) ]; fi; od: RETURN(a); end;

a(n) = A008683(A001358(n)). - Antti Karttunen, Oct 04 2017

More terms from Antti Karttunen, Oct 04 2017

A073411 cototient(x) - 1 where x are the odd semiprimes (i.e., x are odd terms in A001358).

Original entry on oeis.org

2, 6, 8, 4, 12, 10, 14, 6, 18, 14, 20, 16, 24, 16, 20, 30, 18, 32, 22, 38, 26, 22, 10, 42, 44, 24, 48, 22, 32, 34, 54, 28, 12, 60, 62, 40, 26, 68, 34, 44, 28, 72, 46, 36, 74, 28, 50, 80, 30, 84, 32, 42, 56, 90, 46, 16, 98, 62, 34, 48, 102, 64, 104, 38, 108, 34, 110, 52, 70

Offset: 1

Benoit Cloitre, Aug 23 2002

If the Goldbach conjecture is true, all even numbers appear at least once in the sequence since pq - phi(pq) - 1 = p + q.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001358 (semiprimes), A046315 (odd semiprimes), A051953 (cototient).

Map[# - EulerPhi@ # - 1 &, Select[Range[1, 335, 2], PrimeOmega # == 2 &]] (* Michael De Vlieger, Jul 30 2017 *)

lista(nn) = forstep(n=1, nn, 2, if (bigomega(n)==2, print1(n-eulerphi(n)-1, ", "))); \\ Michel Marcus, Jul 30 2017

a(n) = A051953(A046315(n)) - 1. - Michel Marcus, Jul 30 2017; corrected Dec 29 2020

A099980 Bisection of A001358.

Original entry on oeis.org

4, 9, 14, 21, 25, 33, 35, 39, 49, 55, 58, 65, 74, 82, 86, 91, 94, 106, 115, 119, 122, 129, 134, 142, 145, 155, 159, 166, 177, 183, 187, 201, 203, 206, 213, 215, 218, 221, 235, 247, 253, 259, 265, 274, 287, 291, 298, 301, 303, 309, 319, 323, 327, 334, 339, 346

Offset: 0

N. J. A. Sloane, Nov 19 2004

Cf. A001358.

P:=[seq(ithprime(n),n=1..100)]: B:={seq(seq(P[i]*P[j],j=1..100),i=1..100)}:C:={seq(B[k],k=1..140)}: seq(C[2*j-1],j=1..70); # Emeric Deutsch, Dec 14 2004

from math import isqrt
from sympy import primepi, primerange
def A099980(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int((n<<1)+1+x+((t:=primepi(s:=isqrt(x)))*(t-1)>>1)-sum(primepi(x//p) for p in primerange(s+1)))
    return bisection(f,(n<<1)+1,(n<<1)+1) # Chai Wah Wu, Oct 23 2024

More terms from Emeric Deutsch, Dec 14 2004

A099981 Bisection of A001358.

Original entry on oeis.org

6, 10, 15, 22, 26, 34, 38, 46, 51, 57, 62, 69, 77, 85, 87, 93, 95, 111, 118, 121, 123, 133, 141, 143, 146, 158, 161, 169, 178, 185, 194, 202, 205, 209, 214, 217, 219, 226, 237, 249, 254, 262, 267, 278, 289, 295, 299, 302, 305, 314, 321, 326, 329, 335, 341, 355

Offset: 0

N. J. A. Sloane, Nov 19 2004

P:=[seq(ithprime(n),n=1..100)]: B:={seq(seq(P[i]*P[j],j=1..100),i=1..100)}:C:={seq(B[k],k=1..140)}: seq(C[2*j],j=1..70); # Emeric Deutsch, Dec 14 2004

More terms from Emeric Deutsch, Dec 14 2004

cp's OEIS Frontend

A123665 a(n) = Sum_{k=1..21} n^A001358(k).

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A218172 Centered 12-gonal numbers which are semiprimes, intersection of A003154 and A001358.

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A228170 The least semiprime (A001358) such that between it and the next n semiprimes, but not the next n+1 semiprimes, there are no primes.

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A330477 Semiprimes (A001358) p*q such that p*q+p+q is also a semiprime.

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A383468 Semiprimes s = A001358(k) such that k, s - k and s + k are also semiprimes.

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A383469 Semiprimes k such that A001358(k) + k and A001358(k) - k are also semiprimes.

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A072165 Values of Moebius function of the products of two (not necessarily distinct) primes (semiprimes or 2-almost primes, A001358).

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A330477 Semiprimes (A001358) pq such that pq+p+q is also a semiprime.