A007774 -id:A007774 - cp's OEIS frontend

A304634 Numbers n with prime omicron 2, meaning A304465(n) = 2.

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104

Offset: 1

Gus Wiseman, May 15 2018

nonn

If n > 1 is not a prime number, we have A056239(n) >= Omega(n) >= omega(n) >= A071625(n) >= ... >= omicron(n) > 1 where Omega = A001222, omega = A001221, and omicron = A304465.

			This is a list of normalized factorizations (see A112798) of selected entries:
    4: {1,1}
    6: {1,2}
   12: {1,1,2}
   24: {1,1,1,2}
   36: {1,1,2,2}
   48: {1,1,1,1,2}
   60: {1,1,2,3}
   72: {1,1,1,2,2}
   96: {1,1,1,1,1,2}
  120: {1,1,1,2,3}
  144: {1,1,1,1,2,2}
  180: {1,1,2,2,3}
  192: {1,1,1,1,1,1,2}
  216: {1,1,1,2,2,2}
  240: {1,1,1,1,2,3}
  288: {1,1,1,1,1,2,2}
  384: {1,1,1,1,1,1,1,2}
  420: {1,1,2,3,4}
  432: {1,1,1,1,2,2,2}
  480: {1,1,1,1,1,2,3}
  576: {1,1,1,1,1,1,2,2}
  768: {1,1,1,1,1,1,1,1,2}
  840: {1,1,1,2,3,4}
  864: {1,1,1,1,1,2,2,2}
  960: {1,1,1,1,1,1,2,3}

Cf. A001221, A001222, A001358, A005117, A007774, A007916, A055932, A071625, A112798, A181819, A182850, A182857, A275870, A304464, A304465, A304636, A304647.

Join@@Position[Table[Switch[n,1,0,?PrimeQ,1,,NestWhile[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Length[#]>1&]//First],{n,100}],2]

A325240 Numbers whose minimum prime exponent is 2.

Original entry on oeis.org

4, 9, 25, 36, 49, 72, 100, 108, 121, 144, 169, 196, 200, 225, 288, 289, 324, 361, 392, 400, 441, 484, 500, 529, 576, 675, 676, 784, 800, 841, 900, 961, 968, 972, 1089, 1125, 1152, 1156, 1225, 1323, 1352, 1369, 1372, 1444, 1521, 1568, 1600, 1681, 1764, 1800

Offset: 1

Gus Wiseman, Apr 15 2019

nonn

Or barely powerful numbers, a subset of powerful numbers A001694.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose minimum multiplicity is 2 (counted by A244515).
Powerful numbers (A001694) that are not cubefull (A036966). - Amiram Eldar, Jan 30 2023

			The sequence of terms together with their prime indices begins:
    4: {1,1}
    9: {2,2}
   25: {3,3}
   36: {1,1,2,2}
   49: {4,4}
   72: {1,1,1,2,2}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  121: {5,5}
  144: {1,1,1,1,2,2}
  169: {6,6}
  196: {1,1,4,4}
  200: {1,1,1,3,3}
  225: {2,2,3,3}
  288: {1,1,1,1,1,2,2}
  289: {7,7}
  324: {1,1,2,2,2,2}
  361: {8,8}
  392: {1,1,1,4,4}
  400: {1,1,1,1,3,3}

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

Positions of 2's in A051904.
Maximum instead of minimum gives A067259.
Cf. A001221, A001222, A001358, A001694, A007774, A036966, A051903, A052485, A118914, A244515, A325241.
Cf. A065483, A082695.

Select[Range[1000],Min@@FactorInteger[#][[All,2]]==2&]

is(n)={my(e=factor(n)[,2]); n>1 && vecmin(e) == 2; } \\ Amiram Eldar, Jan 30 2023

from math import isqrt, gcd
from sympy import integer_nthroot, factorint, mobius
def A325240(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    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):
        c, l = n+x, 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        c -= squarefreepi(integer_nthroot(x,3)[0])-l
        for w in range(1,integer_nthroot(x,5)[0]+1):
            if all(d<=1 for d in factorint(w).values()):
                for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1):
                    if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
                        c += integer_nthroot(z//y**4,3)[0]
        return c
    return bisection(f,n,n**2) # Chai Wah Wu, Oct 02 2024

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Product_{p prime} (1 + 1/(p^2*(p-1))) = A082695 - A065483 = 0.6038122832... . - Amiram Eldar, Jan 30 2023

A084227 Numbers of the form p*q^k with distinct primes p and q, k>0.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 74, 75, 76, 77, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 104, 106, 111, 112, 115, 116, 117, 118, 119, 122, 123, 124, 129

Offset: 1

Reinhard Zumkeller, May 20 2003

nonn

A001221(a(n)) = 2 AND A001222(a(n)) = A051903(a(n)) + 1. [Clarified by N. J. A. Sloane, Aug 22 2021]

See A007774 for the numbers with omega(n) = A001221(n) = 2. - N. J. A. Sloane, Aug 22 2021

			80 = 5*2^4, therefore 80 is a term.

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

Cf. A000961, A006881, A007774.

doit[{p_,q_}]:=Table[{p q^k,q p^k},{k,10}]; Take[Union[Flatten[ doit/@ Subsets[Prime[Range[20]],{2}]]],70] (* Harvey P. Dale, May 09 2012 *)

is(n)=my(f=factor(n)[,2]); #f==2 && vecmin(f)==1 \\ Charles R Greathouse IV, Oct 19 2015

a(n) ~ n log n/log log n. - Charles R Greathouse IV, Oct 19 2015

A136151 Composites n with exactly two distinct prime divisors and of the form n=1+(any prime).

Original entry on oeis.org

6, 12, 14, 18, 20, 24, 38, 44, 48, 54, 62, 68, 72, 74, 80, 98, 104, 108, 152, 158, 164, 192, 194, 200, 212, 224, 242, 272, 278, 284, 314, 332, 338, 368, 384, 398, 422, 432, 458, 464, 488, 500, 524, 542, 548, 578, 608, 614, 632, 648, 662, 674, 692, 734, 752, 758

Offset: 1

Enoch Haga, Dec 16 2007

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

Cf. A136152, A136153, A136154, A136155.

isA136151 := proc(n) if isprime(n-1) then if nops(numtheory[factorset](n)) =2 then true; else false ; fi ; else false ; fi ; end: for i from 1 to 200 do n := ithprime(i)+1 ; if isA136151( n) then printf("%d, ",n) ; fi ; od: # R. J. Mathar, Feb 01 2008

Select[Range[800],PrimeNu[#]==2&&PrimeQ[#-1]&] (* Harvey P. Dale, Jun 22 2018 *)

A008864 INTERSECT A007774. - R. J. Mathar, Feb 01 2008

Edited by R. J. Mathar, Feb 01 2008

A143202 Numbers having exactly two distinct prime factors p, q with q = p+2.

Original entry on oeis.org

15, 35, 45, 75, 135, 143, 175, 225, 245, 323, 375, 405, 675, 875, 899, 1125, 1215, 1225, 1573, 1715, 1763, 1859, 1875, 2025, 3375, 3599, 3645, 4375, 5183, 5491, 5625, 6075, 6125, 6137, 8575, 9375, 10125, 10403, 10935, 11663, 12005, 16875, 17303, 18225

Offset: 1

Reinhard Zumkeller, Aug 12 2008

nonn

Subsequence of A007774.

A037074 is a subsequence.

			a(1) = 15 = 3 * 5 = A001359(1) * A006512(1).
a(2) = 35 = 5 * 7 = A001359(2) * A006512(2).
a(3) = 45 = 3^2 * 5 = A001359(1)^2 * A006512(1).
a(4) = 75 = 3 * 5^2 = A001359(1) * A006512(1)^2.
a(5) = 135 = 3^3 * 5 = A001359(1)^3 * A006512(1).
a(6) = 143 = 11 * 13 = A001359(3) * A006512(3).
a(7) = 175 = 5^2 * 7 = A001359(2)^2 * A006512(2).
a(8) = 225 = 3^2 * 5^2 = A001359(1)^2 * A006512(1)^2.
a(9) = 245 = 5 * 7^2 = A001359(2) * A006512(2)^2.
a(10) = 323 = 17 * 19 = A001359(4) * A006512(4).
a(11) = 375 = 3 * 5^3 = A001359(1) * A006512(1)^3.
a(12) = 405 = 3^4 * 5 = A001359(1)^4 * A006512(1).

Reinhard Zumkeller, Table of n, a(n) for n = 1..250
Eric Weisstein's World of Mathematics, Twin Primes.
Index entries for primes, gaps between.

Cf. A001221, A001359, A006512, A006530, A007774, A020639, A037074, A143201.

a143202 n = a143202_list !! (n-1)
a143202_list = filter (\x -> a006530 x - a020639 x == 2) [1,3..]
-- Reinhard Zumkeller, Sep 13 2011

tdpfQ[n_]:=Module[{fi=FactorInteger[n][[;;,1]]},Length[fi]==2&&fi[[2]]-fi[[1]]==2]; Select[Range[20000],tdpfQ] (* Harvey P. Dale, Mar 04 2023 *)

A143201(a(n)) = 3.
A020639(a(n)) in A001359 and A006530(a(n)) in A006512.
A001221(a(n)) = 2.
Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/(A001359(n)^2-1) = 0.1812568234997... . - Amiram Eldar, Oct 26 2024

A143203 Numbers having exactly two distinct prime factors p, q with q = p+4.

Original entry on oeis.org

21, 63, 77, 147, 189, 221, 437, 441, 539, 567, 847, 1029, 1323, 1517, 1701, 2021, 2873, 3087, 3757, 3773, 3969, 4757, 5103, 5929, 6557, 7203, 8303, 9261, 9317, 9797, 10051, 11021, 11907, 12317, 15309, 16637, 21609

Offset: 1

Reinhard Zumkeller, Aug 12 2008

nonn

Subsequence of A007774.
A033850 is a subsequence.
Subsequence of A195106. - Reinhard Zumkeller, Sep 13 2011

			a(1) = 21 = 3 * 7 = A023200(1) * A046132(1).
a(2) = 63 = 3^2 * 7 = A023200(1)^2 * A046132(1).
a(3) = 77 = 7 * 11 = A023200(2) * A046132(2).
a(4) = 147 = 3 * 7^2 = A023200(1) * A046132(1)^2.
a(5) = 189 = 3*3 * 7 = A023200(1)^3 * A046132(1).
a(6) = 221 = 13 * 17 = A023200(3) * A046132(3).
a(7) = 437 = 19 * 23 = A023200(4) * A046132(4).
a(8) = 441 = 3^2 * 7^2 = A023200(1)^2 * A046132(1)^2.
a(9) = 539 = 7^2 * 11 = A023200(2)^2 * A046132(2).
a(10) = 567 = 3^4 * 7 = A023200(1)^4 * A046132(1).

Reinhard Zumkeller, Table of n, a(n) for n = 1..250
Eric Weisstein's World of Mathematics, Cousin Primes.
Index entries for primes, gaps between.

Cf. A001221, A006530, A007774, A020639, A023200, A027748, A033850, A046132, A143201, A195106.

a143203 n = a143203_list !! (n-1)
a143203_list = filter f [1,3..] where
   f x = length pfs == 2 && last pfs - head pfs == 4 where
       pfs = a027748_row x
-- Reinhard Zumkeller, Sep 13 2011

dpf2Q[n_]:=Module[{fi=FactorInteger[n][[;;,1]]},Length[fi]==2&&fi[[2]]-fi[[1]]==4]; Select[Range[22000],dpf2Q] (* Harvey P. Dale, Mar 18 2023 *)

A143201(a(n)) = 5.
A020639(a(n)) in A023200 and A006530(a(n)) in A046132.
A001221(a(n)) = 2.
Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A023200(n)+1)^2-4) = 0.109882433872... . - Amiram Eldar, Oct 26 2024

A267114 Numbers n for which A001222(n) = A267115(n) + A267116(n).

Original entry on oeis.org

1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104, 106, 108, 111, 112, 115, 116, 117, 118, 119, 122, 123, 124, 129, 133, 134, 135, 136, 141, 142, 143, 144

Offset: 1

Antti Karttunen, Feb 03 2016

nonn

			6 = 2^1 * 3^1 is included as bitwise-or of its exponents is 1 and likewise bitwise-and(1,1) = 1 and 1+1 = A001222(6) = 2, the number of the prime factors of 6 when counted with multiplicity.
12 = 2^2 * 3^1 is included as bitwise-or of its exponents ("10" and "01" in binary) is 3 ("11"), bitwise-and(1,2) = 0 and 3+0 = A001222(12).
60 = 2^2 * 3^1 * 5^1 is NOT included as bitwise-or(2,1,1) = 3, bitwise-and(2,1,1) = 0 and 3+0 < 4 = A001222(60).

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001222, A267115, A267116, A267117.
Differs from A030231 for the first time at n=118, where A030231(118) = 210 (= 2*3*5*7), which term is missing from this sequence.
a(n+1) differs from A007774 for the first time at n=375, as a(376) = 720 = 2^4 * 3^2 * 5^1.
Cf. A007774 (subsequence).

{1}~Join~Select[Range@ 144, Function[n, PrimeOmega@ n == BitAnd @@ # + BitOr @@ # &@ Map[Last, FactorInteger@ n]]] (* Michael De Vlieger, Feb 07 2016 *)

is(n)=if(n>1, my(f=factor(n)[,2]); fold(bitand, f) + fold(bitor, f) == vecsum(f), 1) \\ Charles R Greathouse IV, Aug 04 2016

Erroneous claim corrected by Antti Karttunen, Feb 07 2016

A323916 Numbers k with exactly two distinct prime divisors and such that cototient(k) is a square.

Original entry on oeis.org

6, 21, 24, 28, 54, 68, 69, 96, 112, 124, 133, 141, 189, 216, 237, 272, 284, 301, 384, 388, 448, 481, 486, 496, 501, 508, 589, 621, 657, 669, 781, 796, 864, 964, 1025, 1029, 1077, 1088, 1136, 1141, 1269, 1317, 1348, 1357, 1372, 1417, 1536, 1537, 1552, 1701, 1792, 1796

Offset: 1

Bernard Schott, Feb 09 2019

nonn

The integers with only one prime factor and whose cototient is a square are in A246551.
This sequence is the intersection of A007774 and A063752.
There are exactly two different families of integers which realize a partition of this sequence (A323917 and A323918); there is also another family with the even perfect numbers of A000396 which is a subsequence of this sequence.
See the file "Subfamilies of terms" (& II) in A063752 for more details, proofs with data, comments, formulas and examples.

			1st family: 189 = 3^3 * 7 and cototient(189) = 9^2;
2nd family: 272 = 2^4 * 17 and cototient(272) = 12^2;
3rd family: 8128 = 2^6 * 127 and cototient(8128) = 64^2.

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

Subsequence of A063752.
Cf. A051953.
Cf. A000396, A323917, A323918

Select[Range[1800], 2 == Length@ FactorInteger@ # == 2 && IntegerQ@ Sqrt[# - EulerPhi@ #] &] (* Giovanni Resta, Feb 27 2019 *)
Select[Range[2000],PrimeNu[#]==2&&IntegerQ[Sqrt[#-EulerPhi[#]]]&] (* Harvey P. Dale, Jan 06 2022 *)

isok(n) = (omega(n)==2) && issquare(n - eulerphi(n)); \\ Michel Marcus, Feb 10 2019

[n for n in (1..2500) if len([1 for d in divisors(n) if is_prime(d)])==2 and is_square(n - euler_phi(n))] # G. C. Greubel, Mar 01 2019

1st family (A323917): if k = p^(2s+1) * q^(2t+1) with s,t >=0, p, q primes and p + q - 1 = M^2, then cototient(k) = (p^r * q^s * M)^2. The primitive terms are p*q with cototient(p*q) = p+q-1 = M^2
2nd family (A323918): if k = p^(2s) * q^(2t+1) with s>=1, t>=0, p, q primes, p < q and such that p*(p+q-1)= M^2, then cototient(k) = (p^(s-1) * q^t * M)^2. The primitive terms are p^2 *q with cototient(p^2 * q) = p * (p+q-1) = M^2
3rd family (A000396): the even perfect Numbers, if 2^p - 1 is a Mersenne prime, then cototient(2^(p-1) * (2^p - 1)) = (2^(p-1))^2.

A064040 Integers whose number of distinct prime divisors is prime.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105

Offset: 1

Lior Manor, Aug 23 2001

For all terms below 210 this sequence and A024619 are identical.

			210 = 2*3*5*7 has 4 prime factors, hence it is not here, but it is part of A024619.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A001221, A024619, A000977, A033992, A007774, A033993, A051270, A063989.

q:= n-> isprime(nops(ifactors(n)[2])):
select(q, [$1..210])[];  # Alois P. Heinz, Apr 18 2024

Select[Range[200], PrimeQ[PrimeNu[#]] &] (* Paolo Xausa, Mar 28 2024 *)

n=0; for (m=1, 10^9, if (isprime(omega(m)), write("b064040.txt", n++, " ", m); if (n==1000, break))) \\ Harry J. Smith, Sep 06 2009

is(n)=isprime(omega(n)) \\ Charles R Greathouse IV, Sep 18 2015

Edited by Charles R Greathouse IV, Mar 18 2010

Name edited by Michel Marcus, Oct 16 2023

A143205 Numbers having exactly two distinct prime factors p, q with q = p+6.

Original entry on oeis.org

55, 91, 187, 247, 275, 391, 605, 637, 667, 1147, 1183, 1375, 1591, 1927, 2057, 2491, 3025, 3127, 3179, 3211, 4087, 4459, 4693, 4891, 5767, 6647, 6655, 6875, 7387, 8281, 8993, 9991, 10807, 11227, 12091, 15125, 15341, 15379, 17947, 19343, 22627, 23707

Offset: 1

Reinhard Zumkeller, Jul 30 2008

nonn

Subsequence of A007774.
A111192 is a subsequence.
Subsequence of A195118. - Reinhard Zumkeller, Sep 13 2011

			a(1) = 55 = 5 * 11 = A023201(1) * A046117(1).
a(2) = 91 = 7 * 13 = A023201(2) * A046117(2).
a(3) = 187 = 11 * 17 = A023201(3) * A046117(3).
a(4) = 247 = 13 * 19 = A023201(4) * A046117(4).
a(5) = 275 = 5^2 * 11 = A023201(1)^2 * A046117(1).
a(6) = 391 = 17 * 23 = A023201(5) * A046117(5).
a(7) = 605 = 5 * 11^2 = A023201(1) * A046117(1)^2.
a(8) = 637 = 7^2 * 13 = A023201(2)^2 * A046117(2).
a(9) = 667 = 23 * 29 = A023201(6) * A046117(6).
a(10) = 1147 = 31 * 37 = A023201(7) * A046117(7).

Reinhard Zumkeller, Table of n, a(n) for n = 1..250
Eric Weisstein's World of Mathematics, Sexy Primes. [The definition in this webpage is unsatisfactory, because it defines a "sexy prime" as a pair of primes.- _N. J. A. Sloane_, Mar 07 2021]
Index entries for primes, gaps between.

Cf. A001221, A006530, A007774, A020639, A023201, A027748, A046117, A111192, A143201, A195118.

a143205 n = a143205_list !! (n-1)
a143205_list = filter f [1,3..] where
   f x = length pfs == 2 && last pfs - head pfs == 6 where
       pfs = a027748_row x
-- Reinhard Zumkeller, Sep 13 2011

okQ[n_]:=Module[{fi=Transpose[FactorInteger[n]][[1]]},Length[fi]==2 && Last[fi]-First[fi]==6]; Select[Range[25000],okQ]  (* Harvey P. Dale, Apr 18 2011 *)

A143201(a(n)) = 7.
A020639(a(n)) in A023201 and A006530(a(n)) in A046117.
A001221(a(n)) = 2.
Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/((A023201(n)+2)^2-9) = 0.058842810164... . - Amiram Eldar, Oct 26 2024

cp's OEIS Frontend

A304634 Numbers n with prime omicron 2, meaning A304465(n) = 2.

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A325240 Numbers whose minimum prime exponent is 2.

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A084227 Numbers of the form p*q^k with distinct primes p and q, k>0.

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A136151 Composites n with exactly two distinct prime divisors and of the form n=1+(any prime).

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A143202 Numbers having exactly two distinct prime factors p, q with q = p+2.

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A143203 Numbers having exactly two distinct prime factors p, q with q = p+4.

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A267114 Numbers n for which A001222(n) = A267115(n) + A267116(n).

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