A113855 Numbers whose prime factors are raised to the powers of each other.

72, 800, 6272, 30375, 247808, 750141, 1384448, 37879808, 189267968, 235782657, 1313046875, 1749600000, 3502727631, 4437573632, 338751673344, 451508436992, 634465620819, 2063731785728, 7863818359375, 7971951402153, 188153927303168

Offset: 1

Author

Cino Hilliard, Jan 25 2006

Keywords

• nonn

Comments

More precisely, n is a term iff n = prod(p_i^(sopf(n)-p_i)), where n has at least two distinct prime factors p_i and sopf(n) = A008472(n). - Rick L. Shepherd, Feb 02 2006

Examples

			72 = 8*9 = 2^3*3^2. So primes 2 and 3 are raised to the power of each other.
800 = 2^5*5^2 = 2 to the power 5 times 5 to the power 2.
		

Crossrefs

Cf. A082949 (numbers of the form p^q * q^p, p, q distinct primes), A008472 (sum of distinct prime factors of n).

Programs

PARI

allpwrfact(n) = { local(x, a, b); a = vector(50); a[1] = 2^3*3^2; a[2] = 2^5*5^2; a[3] = 2^7*7^2; a[4] = 2^11*11^2; a[5] = 2^13*13^2; a[6] = 2^17*17^2; a[7] = 2^19*19^2; a[8] = 2^23*23^2; a[9] = 2^29*29^2; a[10]= 2^31*31^2; a[11]= 2^37*37^2; a[12]= 2^41*41^2; a[13]= 3^5*5^3; a[14]= 3^7*7^3; a[15]= 3^11*11^3; a[16]= 3^13*13^3; a[17]= 3^17*17^3; a[18]= 3^19*19^3; a[19]= 3^23*23^3; a[20]= 3^29*29^3; a[21]= 3^31*31^3; a[22]= 3^37*37^3; a[23]= 2^3*2^5*3^2*3^5*5^2*5^3; a[24]= 2^3*2^7*3^2*3^7*7^2*7^3; a[25]= 2^5*2^7*5^2*5^7*7^2*7^5; a[26]= 2^5*2^11*5^2*5^11*11^2*11^5; a[27]= 3^5*3^7*5^3*5^7*7^3*7^5; a[28]=5^7*7^5; a[29]=5^11*11^5; b= vecsort(a); for(x=1, 42, if(b[x]<>0, print1(b[x]", "))) } (Shepherd)

Extensions

Corrected by Rick L. Shepherd, Feb 02 2006