A061987 -id:A061987 - cp's OEIS frontend

A061985 Values which can occur in sequence b(k) = 1 + b(floor(k/2)) + b(floor(k/3)) with b(0) = 0, i.e., in A061984.

0, 1, 2, 3, 4, 6, 7, 8, 11, 12, 15, 19, 20, 21, 27, 32, 36, 37, 47, 48, 54, 64, 65, 80, 85, 92, 112, 113, 114, 135, 150, 158, 193, 199, 200, 228, 263, 264, 273, 329, 350, 351, 387, 457, 464, 474, 558, 614, 615, 616, 661, 787, 815, 826, 946, 1072, 1073, 1081, 1136

Offset: 0

Henry Bottomley, May 24 2001

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..100

Cf. A061984, A061986, A061987.

a061985 n = a061985_list !! n
a061985_list = f (-1) a061984_list where
   f u (v:vs) = if v == u then f u vs else v : f v vs
-- Reinhard Zumkeller, Jan 11 2014

a(n) = a(n-1) + C(A022328(n) + A022329(n), A022328(n)). - David Wasserman, Nov 17 2005

A160519 Range and record values of A088468.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 12, 13, 16, 20, 21, 22, 28, 33, 37, 38, 48, 49, 55, 65, 66, 81, 86, 93, 113, 114, 115, 136, 151, 159, 194, 200, 201, 229, 264, 265, 274, 330, 351, 352, 388, 458, 465, 475, 559, 615, 616, 617, 662, 788, 816, 827, 947, 1073, 1074, 1082, 1137

Offset: 1

Reinhard Zumkeller, May 16 2009

nonn

A061987(n-1) = number of times a(n) is repeated in A088468.

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

Cf. A003586, A061987, A088468.

f[0] = 1; f[n_] := f[n] = f[Floor[n/2]] + f[Floor[n/3]]; With[{max = 10^4}, f /@ Join[{0}, Sort@ Flatten@ Table[2^i*3^j, {i, 0, Log2[max]}, {j, 0, Log[3, max/2^i]}]]] (* Amiram Eldar, Jul 13 2023 *)

a(1) = A088468(0); for n>1: a(n) = A088468(A003586(n)) and a(m) < A088468(m) for m < A003586(n).

A236210 Pairs of "harmonic numbers" 2^m * 3^n that differ by 1.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 8, 9

Offset: 1

Jonathan Sondow, Jan 20 2014

Philippe de Vitry (1291-1361), a musician from Vitry-en-Artois in France, called numbers of the form 2^m * 3^n "harmonic numbers". He asked if all powers of 2 and 3 differ by more than 1 except the pairs 1 and 2, 2 and 3, 3 and 4, 8 and 9 (which correspond to musically significant ratios, representing an octave, fifth, fourth, and whole tone). Levi Ben Gerson (1288-1344) answered yes by proving that 3^n +- 1 is not a power of 2 if n > 2; see A235365, A235366.

			8 + 1 = 2^3 + 1 = 3^2 = 9, so the pair 8 and 9 is in the sequence.

L. E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea, NY 1992; see p. 731.

A. Herschfeld, The equation 2^x - 3^y = d, Bull. Amer. Math. Soc., 42 (1936), 231-234.
H. Lenstra, Harmonic Numbers, MSRI, 1998.
J. J. O'Connor and E. F. Robertson, Levi ben Gerson, The MacTutor History of Mathematics archive, 2009.
I. Peterson, Medieval Harmony, Math Trek, MAA, 2012.
Wikipedia, Gersonides
Wikipedia, Philippe de Vitry

Cf. A003586, A006899, A061987, A108906, A235365, A235366.

A061986 Number of times n appears in sequence b(k) = 1 + b(floor(k/2)) + b(floor(k/3)) with b(0) = 0, i.e., in A061984.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 2, 1, 3, 0, 0, 4, 2, 0, 0, 6, 0, 0, 0, 3, 5, 4, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 6, 0, 0, 0, 10, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 15, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0

Offset: 0

Henry Bottomley, May 24 2001

nonn

If n is not in A061985 then a(n)=0, otherwise if n=A061985(m) then a(n) = A061987(m).

A086247 Differences of successive 7-smooth numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 1, 3, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 3, 1, 1, 4, 2, 4, 3, 1, 6, 2, 3, 5, 1, 3, 6, 6, 2, 2, 5, 3, 4, 8, 5, 1, 2, 7, 5, 4, 3, 3, 10, 2, 6, 7, 5, 9, 3, 4, 4, 10, 6, 8, 1, 15, 3, 2, 5, 2, 4, 14, 10, 8, 6, 6, 15, 5, 4, 12, 7, 7, 10, 15, 3, 6, 8, 8, 5, 15, 12

Offset: 1

Reinhard Zumkeller, Jul 13 2003

A002473(n) is a term of A085153 iff a(n)=1.

			a(23) = 3 as A002473(23 + 1) - A002473(23) = 35 - 32 = 3. - _David A. Corneth_, Mar 31 2021

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smooth Number

Cf. A002473, A061987, A085153.

smooth7Q[n_] := n == Times@@({2, 3, 5, 7}^IntegerExponent[n, {2, 3, 5, 7}]);
Select[Range[1000], smooth7Q] // Differences (* Jean-François Alcover, Oct 17 2021 *)

a(n) = A002473(n+1) - A002473(n).

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A061985 Values which can occur in sequence b(k) = 1 + b(floor(k/2)) + b(floor(k/3)) with b(0) = 0, i.e., in A061984.

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A160519 Range and record values of A088468.

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A236210 Pairs of "harmonic numbers" 2^m * 3^n that differ by 1.

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A061986 Number of times n appears in sequence b(k) = 1 + b(floor(k/2)) + b(floor(k/3)) with b(0) = 0, i.e., in A061984.

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A086247 Differences of successive 7-smooth numbers.

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