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 A104264 Number of n-digit squares with no zero digits. 11
 3, 6, 19, 44, 136, 376, 1061, 2985, 8431, 24009, 67983, 193359, 549697, 1563545, 4446173, 12650545, 35999714, 102439796, 291532841, 829634988, 2360947327, 6719171580, 19122499510, 54423038535, 154888366195 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Comments from David W. Wilson, Feb 26 2005: (Start) "There are approximately s(d) = (10^d)^(1/2) - (10^(d-1))^(1/2) d-digit squares. A random d-digit number has the probability p(d) = (9/10)^(d-1) of being zeroless (exponent d-1 as opposed to d because the first digit is not zero). So we expect p(d)s(d) zeroless d-digit squares. "For d = 1 through 12, we get (truncating): 1, 5, 15, 44, 127, 363, 1034, 2943, 8377, 23841, 67854, 193117, ... The elements grow approximately geometrically with limit ratio (9/10)*10^(1/2) = 2.846+. "The same naive estimate can easily be generalize to k-th powers, giving the estimate s(d) = (10^d)^(1/k) - (10^(d-1))^(1/k) for d-digit k-th powers. p(d) remains the same. The resulting estimates have ratio (9/10)*10^(1/k). "We should expect an infinite number of zeroless k-th powers when this ratio is >= 1, which it is for k <= 21. For k >= 22, the ratio is < 1 and we should expect a finite number of zeroless k-th powers." (End) LINKS Table of n, a(n) for n=1..25. EXAMPLE a(3) = #{121, 144, 169, 196, 225, 256, 289, 324, 361, 441, 484, 529, 576, 625, 676, 729, 784, 841, 961} = 19. PROG (Python) def aupton(terms): c, k, kk = [0 for i in range(terms)], 1, 1 while kk < 10**terms: s = str(kk) c[len(s)-1], k, kk = c[len(s)-1] + (s.count('0')==0), k+1, kk + 2*k + 1 return c print(aupton(14)) # Michael S. Branicky, Mar 06 2021 CROSSREFS Cf. A052041, A104265, A104266, A075415, A102807. Sequence in context: A019097 A347685 A219286 * A007098 A226322 A148566 Adjacent sequences: A104261 A104262 A104263 * A104265 A104266 A104267 KEYWORD nonn,base,more AUTHOR Reinhard Zumkeller and Ron Knott, Feb 26 2005 EXTENSIONS a(14)-a(18) from Donovan Johnson, Nov 05 2009 a(19)-a(21) from Donovan Johnson, Mar 23 2011 a(22)-a(25) from Donovan Johnson, Jan 29 2013 STATUS approved

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Last modified September 23 02:06 EDT 2023. Contains 365532 sequences. (Running on oeis4.)