A078972 -id:A078972 - cp's OEIS frontend

A084476 Least k such that 10^(2n-1)+k is a brilliant number.

0, 3, 13, 43, 81, 147, 73, 3, 831, 49, 987, 691, 183, 4153, 279, 667, 709, 277, 1687, 997, 1207, 91, 1411, 393, 951, 9793, 2217, 6229, 2317, 213, 399, 19, 2317, 609, 2607, 11901, 10563, 5473, 3, 5923, 17527, 8569, 16701, 11989, 9757, 6489, 3489, 2899

Offset: 1

Robert G. Wilson v, Jun 27 2003

Least brilliant number greater than 10^(2n) is {10^n+A033873(n)}^2. The web site also lists the two prime factors.

			a(3)=13 because 10^5+13 = 100013 = 103*971.

Harry Metrebian, Table of n, a(n) for n = 1..65
Dario Alejandro Alpern, Brilliant numbers

Cf. A078972, A084475.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; LengthBase10[n_] := Floor[ Log[10, n] + 1]; f[n_] := Block[{k = 0}, If[ EvenQ[n] && n > 1, NextPrim[ 10^(n/2)]^2 - 10^(n/2), While[fi = FactorInteger[10^n + k]; Plus @@ Flatten[ Table[ # [[2]], {1}] & /@ fi] != 2 || Length[ Union[ LengthBase10 /@ Flatten[ Table[ # [[1]], {1}] & /@ fi]]] != 1, k++ ]; k]]; Table[ f[2n + 1], {n, 1, 24}]

A087434 Number of brilliant numbers whose prime factors each have n digits.

Original entry on oeis.org

10, 231, 10296, 563391, 34974066, 2374052871, 171745762321, 12989075028126, 1016377282340160, 81690831917887753, 6708792934060150753, 560785267822390134615, 47573053155260626453431

Offset: 1

Ray Chandler, Sep 02 2003

Number of brilliant numbers having 2n or 2n-1 digits.

Charles R Greathouse IV, Table of n, a(n) for n = 1..24

Cf. A006879, A006880, A078972, A086846, A087435.

a(n)=binomial(primepi(10^n)-primepi(10^(n-1))+1,2)
\\ Charles R Greathouse IV, May 03 2012

a(n)=A000217(A006879(n)).

a(14) from Ray Chandler, Jul 21 2005

A121898 Triangular numbers that are sandwiched between two semiprimes; or triangular numbers t such that t-1 and t+1 are both semiprime.

Original entry on oeis.org

120, 300, 528, 780, 2628, 3240, 3828, 5460, 13530, 18528, 19110, 22578, 25878, 31878, 32640, 37128, 49770, 56280, 64980, 72390, 73920, 78210, 103740, 105570, 115440, 137550, 159330, 161028, 277140, 288420, 316410, 335790, 370230, 386760

Offset: 1

Jason Earls, Sep 01 2006

nonn

Why are so many of the t-1's brilliant numbers (A078972)?

Select[Accumulate[Range[1000]],PrimeOmega[#-1]==PrimeOmega[#+1]==2&] (* Harvey P. Dale, Jul 06 2014 *)

A083284 Numbers m such that m and m+2 are both brilliant numbers, where brilliant numbers are semiprimes whose prime factors have an equal number of decimal digits, or whose prime factors are equal.

Original entry on oeis.org

4, 527, 779, 869, 899, 1079, 1157, 1271, 1679, 4187, 6497, 6887, 24287, 24881, 25019, 29591, 35237, 37127, 37769, 38807, 39269, 39911, 41309, 43361, 44831, 45347, 46001, 46127, 47261, 48509, 48929, 51809, 52907, 54389, 55481, 55751, 55961

Offset: 1

Jason Earls, Jun 03 2003

The only consecutive brilliant numbers are {9, 10} and {14, 15}; and for m > 14 there are no brilliant constellations of the form {m, m+(2k+1)} or equivalently {n, 2k+m+1} with k >= 0. Proof: One of m and 2k+m+1 will be even. And there are no even brilliant numbers > 14 since they must have the form 2*p where p is a prime having only one digit.

			a(3) = 779 because 779=19*41 and 781=11*71.

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

Cf. A078972.

A085647 Semiprimes whose prime factors p*q have an equal number of decimal digits and p is not equal to q.

Original entry on oeis.org

6, 10, 14, 15, 21, 35, 143, 187, 209, 221, 247, 253, 299, 319, 323, 341, 377, 391, 403, 407, 437, 451, 473, 481, 493, 517, 527, 533, 551, 559, 583, 589, 611, 629, 649, 667, 671, 689, 697, 703, 713, 731, 737, 767, 779, 781, 793, 799, 803, 817, 851, 869, 871

Offset: 1

Amarnath Murthy, Jason Earls, Jul 11 2003

Chai Wah Wu, Table of n, a(n) for n = 1..10369

Cf. A001358, A078972.

spddQ[n_]:=Module[{fi=FactorInteger[n][[All,1]]},PrimeOmega[n]==2 && fi[[2]]- fi[[1]]>0&&IntegerLength[fi[[1]]]==IntegerLength[fi[[2]]]]; Select[Range[900],spddQ]//Quiet (* Harvey P. Dale, Jul 14 2021 *)

from sympy import sieve
A085647 = []
for n in range(3):
    pr = list(sieve.primerange(10**n,10**(n+1)))
    for i,p in enumerate(pr,start=1):
        for q in pr[i:]:
            A085647.append(p*q)
A085647 = sorted(A085647)
# Chai Wah Wu, Aug 26 2014

A086846 Number of brilliant numbers < 10^n.

Original entry on oeis.org

3, 10, 73, 241, 2504, 10537, 124363, 573928, 7407840, 35547994, 491316166, 2409600865, 34896253009, 174155363186, 2601913448896, 13163230391312, 201431415980418, 1029540512731472

Offset: 1

Jason Earls, Aug 09 2003

Cf. A078972, A087434, A087435.

a(n) = my(N=10^n-1, count=0, L=#digits(sqrtint(N))); for(k=1, L-1, count += binomial(primepi(10^k) - primepi(10^(k-1)) + 1, 2)); my(min = 10^(L-1), max = 10^L-1, pi_min = primepi(min), pi_max = primepi(max), j = 0); forprime(p = min, max, if(p*p <= N, count += if(N >= p*max, pi_max, primepi(N\p)) - pi_min - j; j+=1, break)); count; \\ Daniel Suteu, Apr 09 2022

More terms from Ray Chandler, Aug 31 2003
a(11)-a(14) from Ray Chandler, Jul 21 2005
a(15)-a(16) from Donovan Johnson, May 30 2010
a(17)-a(18) from Daniel Suteu, Apr 09 2022

A087435 Partial sums of A087434.

Original entry on oeis.org

10, 241, 10537, 573928, 35547994, 2409600865, 174155363186, 13163230391312, 1029540512731472, 82720372430619225, 6791513306490769978, 567576781128880904593, 48140629936389507358024

Offset: 1

Ray Chandler, Sep 02 2003

nonn

Number of brilliant numbers <10^2n.

Bisection of A086846.

Cf. A006879, A006880, A078972, A086846, A087434.

a(14) from Ray Chandler, Jul 21 2005

A132435 Composite integers n with two prime factors nearly equidistant from the integer part of the square root of n.

Original entry on oeis.org

4, 6, 9, 10, 14, 22, 25, 35, 49, 55, 65, 77, 85, 91, 119, 121, 143, 169, 187, 209, 221, 247, 253, 289, 299, 319, 323, 361, 377, 391, 407, 437, 493, 527, 529, 551, 589, 629, 667, 697, 703, 713, 841, 851, 899, 943, 961, 989, 1073, 1081, 1147, 1189

Offset: 1

Andrew S. Plewe, Nov 13 2007

nonn

An integer n is included if, for some value y >= 0: n = A007918(A000196(n) + y) * A007918(A000196(n) - y) Or: n = nextprime(sqrtint(n) + y) * nextprime(sqrtint(n) - y) Where "nextprime(x)" is the smallest prime number >= to x and "sqrtint(z)" is the integer part of the square root of z.

Has many terms in common with A078972. - Bill McEachen, Dec 24 2020

			25 = nextprime(5 + 0) * nextprime(5 - 0) = 5 * 5 = 25
35 = nextprime(5 + 1) * nextprime(5 - 1) = 7 * 5 = 35
119 = nextprime(10 + 4) * nextprime(10 - 4) = 17 * 7 = 119

Michel Marcus, Table of n, a(n) for n = 1..3406

Cf. A007918, A000196, A078972.

bal(x,y) = nextprime(sqrtint(x)+y) * nextprime(sqrtint(x)-y);
findbal(x) = local(z,y); z=sqrtint(x); while( 0<=z, y=bal(x,z); if(y==x, print1(x", ");break;); z--;);
for (n=1,1200, findbal(n));

A199009 Composite numbers whose prime factors have an equal number of digits in decimal representation.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 121, 125, 126, 128, 135, 140, 143, 144, 147, 150, 160, 162, 168, 169, 175, 180

Offset: 1

Arkadiusz Wesolowski, Nov 01 2011

			42 = 2*3*7 belongs to this sequence, since its prime factors all have one digit.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Supersequence of A078972 and of A182301.

f[n_] := Length[Union[Length /@ IntegerDigits /@ Transpose[FactorInteger[n]][[1]]]] == 1; Select[Range[2, 180], ! PrimeQ[#] && f[#] &] (* T. D. Noe, Nov 01 2011 *)

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

cp's OEIS Frontend

A084476 Least k such that 10^(2n-1)+k is a brilliant number.

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A087434 Number of brilliant numbers whose prime factors each have n digits.

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A121898 Triangular numbers that are sandwiched between two semiprimes; or triangular numbers t such that t-1 and t+1 are both semiprime.

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A083284 Numbers m such that m and m+2 are both brilliant numbers, where brilliant numbers are semiprimes whose prime factors have an equal number of decimal digits, or whose prime factors are equal.

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A085647 Semiprimes whose prime factors p*q have an equal number of decimal digits and p is not equal to q.

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A086846 Number of brilliant numbers < 10^n.

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A087435 Partial sums of A087434.

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A132435 Composite integers n with two prime factors nearly equidistant from the integer part of the square root of n.

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PARI

A199009 Composite numbers whose prime factors have an equal number of digits in decimal representation.

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Mathematica