A190472 -id:A190472 - cp's OEIS frontend

A190473 Numbers with prime factorization pqrs^7.

13440, 21120, 24960, 29568, 32640, 34944, 36480, 44160, 45696, 49280, 51072, 54912, 55680, 58240, 59520, 61824, 71040, 71808, 76160, 77952, 78720, 80256, 82560, 83328, 84864, 85120, 90240, 91520, 94848, 97152, 99456, 101760, 103040, 110208

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 10 2011

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures

Cf. A190470, A190471, A190472.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,1,7}; Select[Range[150000],f]

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtnint(lim\30, 7), t1=p^7; forprime(q=2,lim\(6*t1), if(q==p, next); t2=q*t1; forprime(r=2,lim\(2*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\t3, if(s==p || s==q || s==r, next); listput(v, t3*s))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

A190474 Numbers with prime factorization pqr^2s^6.

Original entry on oeis.org

20160, 31680, 33600, 37440, 44352, 47040, 48960, 52416, 52800, 54720, 62400, 66240, 68544, 76608, 81600, 82368, 83520, 89280, 91200, 92736, 102060, 103488, 106560, 107712, 110400, 116160, 116928, 118080, 120384, 122304, 123200, 123840

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 10 2011

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures

Cf. A190472, A190473.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,2,6}; Select[Range[300000],f]

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtnint(lim\60, 6), t1=p^6; forprime(q=2,sqrtint(lim\(6*t1)), if(q==p, next); t2=q^2*t1; forprime(r=2,lim\(2*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\t3, if(s==p || s==q || s==r, next); listput(v, t3*s))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

A190475 Numbers with prime factorization pqr^3s^5.

Original entry on oeis.org

30240, 47520, 56160, 66528, 68040, 73440, 78624, 82080, 84000, 99360, 102816, 106920, 114912, 123552, 125280, 126360, 132000, 133920, 139104, 149688, 156000, 159840, 161568, 164640, 165240, 175392, 176904, 177120, 180576, 184680, 185760

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 10 2011

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures

Cf. A190472, A190473, A190474.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,3,5}; Select[Range[400000],f]

list(lim)=my(v=List(),t1,t2,t3); forprime(p=2,sqrtnint(lim\120, 5), t1=p^5; forprime(q=2,sqrtnint(lim\(6*t1), 3), if(q==p, next); t2=q^3*t1; forprime(r=2,lim\(2*t2), if(r==p || r==q, next); t3=r*t2; forprime(s=2,lim\t3, if(s==p || s==q || s==r, next); listput(v, t3*s))))); Set(v) \\ Charles R Greathouse IV, Aug 25 2016

A375143 Numbers whose prime factorization has a minimum exponent that is larger than 1 and is 1 less than the maximum exponent.

Original entry on oeis.org

72, 108, 200, 392, 432, 500, 648, 675, 968, 1125, 1323, 1352, 1372, 1800, 2000, 2312, 2592, 2700, 2888, 3087, 3267, 3528, 3888, 4232, 4500, 4563, 5000, 5292, 5324, 5400, 5488, 6125, 6728, 7688, 7803, 8575, 8712, 8788, 9000, 9747, 9800, 10125, 10584, 10952, 11979

Offset: 1

Amiram Eldar, Aug 01 2024

Numbers k such that 2 <= A051904(k) = A051903(k) - 1.

Numbers that are product of two coprime nonsquarefree powers of squarefree numbers (A072777) with consecutive exponents.

			72 = 2^3 * 3^2 is a term since A051904(72) = 2 is larger than 1 and is 1 less than A051903(72) = 3.

Cf. A051903, A051904, A072777.
Subsequence of A001694.
Subsequences: A143610, A167747 \ {1, 2, 12}, A093136 \ {1, 2, 20}, A179666, A179702, A190472, A375073.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 2 <= Min[e] == Max[e] - 1]; Select[Range[12000], q]

is(k) = {my(e = factor(k)[,2]); k > 1 && 2 <= vecmin(e) && vecmin(e) + 1 == vecmax(e);}

Sum_{n>=1} 1/a(n) = Sum_{k>=2} f(k) = 0.053695635500385312854..., where f(k) = Product_{p prime} (1 + 1/p^k + 1/p^(k+1)) - zeta(k)/zeta(2*k) - zeta(k+1)/zeta(2*k+2) + 1 is the sum of reciprocals of the subset of numbers m with A051904(m) = k.

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A190473 Numbers with prime factorization pqrs^7.

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A190474 Numbers with prime factorization pqr^2s^6.

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A190475 Numbers with prime factorization pqr^3s^5.

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A375143 Numbers whose prime factorization has a minimum exponent that is larger than 1 and is 1 less than the maximum exponent.

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