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Search: id:A050376
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| A050376 |
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Numbers of the form p^(2^k) where p is prime and k >= 0. |
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24
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| 2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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Every number is a product of a unique subset of these numbers.
Or, a(1) = 2; for n>1, a(n) = smallest number which can not be obtained as the product of previous terms. This is evident from the unique factorization theorem and the fact that every number can be expressed as the sum of powers of 2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jan 09 2002
Except for the first term, same as A084400. - David Wasserman (wasserma(AT)spawar.navy.mil), Dec 22 2004
The least number having 2^n divisors (=A037992(n)) is the product of the first n terms of this sequence according to Ramanujan.
d(n)=trivial prime (2 or 3). [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Dec 01 2009]
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REFERENCES
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S. Ramanujan, Highly Composite Numbers, Collected Papers of Srinivasa Ramanujan, p. 125, Ed. G. H. Hardy et al., AMS Chelsea 2000.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
S. R. Finch, Unitarism and infinitarism.
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EXAMPLE
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6=2*3, 8=2*4, 24=2*3*4. 120 is not a member since it is divisible by two different primes.
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PROGRAM
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(PARI) {a(n)= local(m, c, k, p); if(n<=0, 2*(n==0), c=0; m=2; while( c<n, m++; if( isprime(m)| ( (k=ispower(m, (null), &p))&isprime(p)& k ==2^valuation(k, 2) ), c++)); m)} /* Michael Somos Apr 15 2005 */
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CROSSREFS
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Cf. A000040 (primes), A050377-A050380, A026416, A000028, A066724, A026477, A084400.
Cf. A000005 (number of divisors of n), A168546 (the nontrivial primes). [From Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Dec 01 2009]
Sequence in context: A026477 A079852 A084400 this_sequence A050198 A158923 A008740
Adjacent sequences: A050373 A050374 A050375 this_sequence A050377 A050378 A050379
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KEYWORD
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nonn,easy,nice,new
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AUTHOR
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Christian G. Bower (bowerc(AT)usa.net), Nov 15 1999.
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