cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A110934 Difference between 3-almostprime(n) and 3-almostprime(n+2).

Original entry on oeis.org

10, 8, 9, 8, 3, 14, 14, 3, 6, 7, 13, 14, 5, 4, 7, 6, 3, 16, 20, 7, 4, 6, 8, 9, 6, 3, 8, 8, 6, 13, 17, 10, 6, 6, 11, 11, 6, 6, 2, 3, 3, 8, 11, 6, 4, 7, 17, 17, 15, 18, 9, 6, 7, 6, 6, 3, 2, 10, 12, 6, 8, 7, 7, 7, 6, 7, 5, 3, 2, 5, 6, 20, 24, 8, 6, 7, 10, 8, 6, 10, 7
Offset: 1

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Author

Jonathan Vos Post, Jan 21 2006

Keywords

Comments

This is the 3-almost prime analog of what A113784 is for semiprimes and what A031131 is for primes. The minimum values in the sequence are 2 because we have, for example, the 3 consecutive 3-almost primes 170, 171, 172, so a(39) = A014612(41) - A014612(39) = 172 - 170 = 2. Equivalently, there are 2 consecutive 1 values of A114403 (3-almost prime gaps; first differences of A014612). This happens for elements of A113789 (numbers n such that n, n+1 and n+2 are 3-almost primes).

Examples

			a(1) = 10 because the difference between the first and third 3-almost primes is A014612(3) - A014612(1) = 18 - 8 = 10.
a(2) = A014612(4) - A014612(2) = 20 - 12 = 8.
a(3) = A014612(5) - A014612(3) = 27 - 18 = 9.

Crossrefs

Cf. A014612, A031131, A067813, A113784, A113789, A114403.

Formula

a(n) = A014612(n+2) - A014612(n).

Extensions

a(28) corrected by R. J. Mathar, Dec 22 2010

A349262 a(n) is the start of the least run of exactly n consecutive numbers with the same value of A349258.

Original entry on oeis.org

1, 14, 20, 2, 91, 6850, 2302, 141, 56014, 184171, 2800171, 27805034, 35297611, 8313366182, 1791416073, 3618621410
Offset: 1

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Author

Amiram Eldar, Nov 12 2021

Keywords

Comments

a(17) > 10^11, if it exists.

Examples

			a(2) = 14 since A349258(14) = A349258(15) = 2, but A349258(13) != 2 and A349258(16) != 2.

Crossrefs

Cf. A349258.
Similar sequences: A006558, A045983, A048932, A067813, A077657, A318166.

Programs

  • Mathematica
    f[p_, e_] := 2^DigitCount[e, 2, 1] - 1; d[1] = 0; d[n_] := Plus @@ f @@@ FactorInteger[n]; seq[len_, nmax_] := Module[{s = Table[0, {len}], dprev = 0, n = 2, c = 1, k = 1}, s[[1]] = 1; While[k < len && n < nmax, d1 = d[n]; If[d1 == dprev, c++, If[c > 0 && c <= len && s[[c]] == 0, k++; s[[c]] = n - c]; c = 1]; n++; dprev = d1]; TakeWhile[s, # > 0 &]]; seq[8, 10^4]

A349305 a(n) is the start of the least run of exactly n consecutive numbers with the same number of nonunitary divisors.

Original entry on oeis.org

4, 10, 1, 19940, 54584, 204323, 2789143044, 27092041443
Offset: 1

Views

Author

Amiram Eldar, Nov 14 2021

Keywords

Comments

a(9) > 10^11, if it exists.

Examples

			a(2) = 10 since A048105(10) = A048105(11) = 0, and A048105(9) != 0 and A048105(12) != 0.

Crossrefs

Cf. A048105, A344315.
Similar sequences: A006558, A045983, A048932, A067813, A077657, A318166.

Programs

  • Mathematica
    d[n_] := DivisorSigma[0, n] - 2^PrimeNu[n]; seq[len_, nmax_] := Module[{s = Table[0, {len}], dprev = -1, n = 1, c = 0, k = 0}, While[k < len && n < nmax, d1 = d[n]; If[d1 == dprev, c++, If[c > 0 && c <= len && s[[c]] == 0, k++; s[[c]] = n - c]; c = 1]; n++; dprev = d1]; TakeWhile[s, # > 0 &]]; seq[6, 10^6]
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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