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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A180126 a(n) is the least k such that for numbers x >= k, PrimePi(n,x) > PrimePi(n-1,x), where PrimePi(n,x) is the number of n-almost-primes <= x.

Original entry on oeis.org

3, 34, 15530, 151165607042
Offset: 1

Views

Author

T. D. Noe, Aug 11 2010

Keywords

Comments

Note that a(n) is an n-almost-prime. An n-almost-prime is a number having exactly n prime factors (counted with multiplicity). So 1 is the only 0-almost-prime; 1-almost-primes are the usual prime numbers; 2-almost-primes are also called semiprimes. The first three terms are mentioned in A125149.
For 2 <= n <= 4, the values for a(n)/a(n-1) (11.3, 456.8, 9733780.2) are each a little larger than A281889(n), "the median n-th least prime factor of the integers". - Peter Munn, Jan 04 2023

Crossrefs

Cf. A078840, A125149, A281889.

Extensions

Name edited by Peter Munn, Jan 04 2023

A186000 Consider the list s(1), s(2), ... of numbers that are products of exactly n primes; a(n) is the smallest s(j) whose decimal expansion ends in j.

Original entry on oeis.org

1, 17, 291, 12, 56, 78645, 1350, 192, 896, 7936, 36096, 3072, 14336, 250880, 1247232, 49152, 229376, 4014080, 6718464, 786432, 3670016, 64225280, 45203456000, 12582912, 58720256, 622854144, 219792015360, 201326592, 939524096, 8321499136, 37849399296, 3221225472, 15032385536, 263066746880, 2924872728576, 51539607552, 240518168576, 4209067950080, 7044820107264, 824633720832, 3848290697216
Offset: 0

Views

Author

Jonathan Vos Post and Robert G. Wilson v, Feb 24 2011

Keywords

Comments

For n = 0, 1, 2, ..., the index j corresponding to a(n) is 1, 7, 91, 2, 6, 8645, 50, 2, 6, 36, 96, 2, 6, 80, 232, 2, 6, 80, 64, 2, 6, 80, >6136, 2, 6, 44, > 564, 2, 6, 36, 96, 2, 6, 80, >286, 2, 6, 80, 64, 2, 6, ..., .

Examples

			a(0) = 1 because 1 is the first and only positive integer (A000027) which is not a prime, a semiprime, a triprime, etc.;
a(1) = 17 because 17 is the seventh term of A000040 (it is also the first term of A046883);
a(2) = 291 because 291 is the 91st term of A001358;
a(3) = 12 because 12 is the second term of A014612;
a(4) = 56 because 56 is the sixth term of A014613; etc.

Links

Crossrefs

Cf. A000040, A001358, A046883, A014612, A014613, A078840, A185999.

Programs

  • Mathematica
    nextKthAlmostPrime[n_, k_] := Block[{m = n + 1}, While[ Plus @@ Last /@ FactorInteger@ m != k, m++]; m] (* Eric W. Weisstein, Feb 07 2006 *); f[n_] := Block[{c = 1, kp = 2^n}, While[ Mod[kp, 10^Floor[1 + Log10@ c]] != c, c++; kp = nextKthAlmostPrime[kp, n]]; kp]
(* These terms can be crosschecked by: *)
AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]] (* Eric W. Weisstein, Feb 07 2006 *)

Extensions

Edited by N. J. A. Sloane, Mar 04 2011
The missing values, a(22), a(26) & a(34), were supplied to me via email dtd Mar 03 2011 from Donovan Johnson. - Robert G. Wilson v, Mar 22 2011

A125288 a(n) = least integer k such that for all integers m greater than k, 2*Pi(n,m) is greater than Pi(n,2*m).

Original entry on oeis.org

10, 297, 49650, 180701087317
Offset: 1

Views

Author

Jonathan Vos Post and Robert G. Wilson v, Jan 22 2007

Keywords

Comments

Pi(n, m) is the number of integers <= m that have n prime factors counting multiplicity, also known as n-almost-primes (A078840).

Examples

			a(1) = 10 since the first term relates to 1-almost-primes, which are the primes themselves; and there are 4 primes <= 10, and 2*4 = 8 primes <= 2*10 = 20; but for m = 11 and all larger integers, the number of primes <= 2*m is less than twice the number of primes <= m. - _Peter Munn_, Dec 23 2022

Crossrefs

Cf. A078840, A092097, A117526, A126279, A125149.

Programs

  • Mathematica
    AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[ PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]] ]]]; (* Eric W. Weisstein, Feb 07 2006 *)

Extensions

a(4) from Donovan Johnson, Nov 13 2010
Edited by Peter Munn, Jan 05 2023

A214364 Irregular triangle read by rows n lists the values k of all decompositions of 2n into unordered sums of two k-almost primes.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 3, 2, 1, 1, 2, 1, 2, 1, 3, 2, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 3, 1, 2, 2, 1, 2, 1, 3, 1, 2, 1, 2, 2, 3, 1, 4, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 3, 2, 1, 3, 3, 1, 1, 2, 1, 3, 2, 2, 4, 1, 3, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 2, 1, 2
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jul 23 2012

Keywords

Comments

Row n has 1+A214154(n) entries, if n <> 1, where the '1' means the splitting 2*n=n+n into non-distinct k-almost primes is also registered here, but not in A214154. Therefore the last entry in row n is A001222(n) for n > 3.
Row sums are 0, 1, 1, 3, 4, 3, 4, 7, 6, 9, 3, 10, 8, 9, 12, 13, 6, 16, 10, 19, 12, 11, 9, 25, 16, 16, 15, 19, 14, 31, 14, 30, 14, 15, 24,..

Examples

			If written as a triangle:
0,
1,
1,
1, 2,
1, 2, 1,
1, 2,
1, 2, 1,
1, 1, 2, 3,
2, 1, 1, 2,
1, 2, 1, 3, 2,
1, 1, 1,
1, 1, 2, 2, 1, 3,
1, 2, 1, 3, 1,
1, 2, 3, 1, 2,
2, 1, 2, 1, 3, 1, 2,
1, 2, 2, 3, 1, 4,
1, 1, 2, 1, 1,
1, 1, 3, 2, 1, 2, 2, 1, 3.

Crossrefs

Cf. A078840. - N. J. A. Sloane, Jul 29 2012

A321590 Smallest number m that is a product of exactly n primes and is such that m-1 and m+1 are products of exactly n-1 primes.

Original entry on oeis.org

4, 50, 189, 1863, 10449, 447849, 4449249, 5745249, 3606422049, 16554218751, 105265530369, 1957645712385
Offset: 2

Views

Author

Zak Seidov, Nov 13 2018

Keywords

Comments

From Jon E. Schoenfield, Nov 15 2018: (Start)
If a(11) is odd, it is 16554218751.
If a(12) is odd, it is 105265530369.
If a(13) is odd, it is 1957645712385. (End)
a(11), a(12), and a(13) are indeed odd. - Giovanni Resta, Jan 04 2019
10^13 < a(14) <= 240455334218751, a(15) <= 2992278212890624. - Giovanni Resta, Jan 06 2019

Examples

			For n = 3, 50 = 2*5*5, and the numbers before and after 50 are 49 = 7*7 and 51 = 3*17.

Crossrefs

Cf. A078840.
Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), A014613 (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275(r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20).

Programs

  • Mathematica
    a[n_] := Module[{o={0,0,0}, k=1}, While[o!={n-1,n,n-1}, o=Rest[AppendTo[o,PrimeOmega[k]]]; k++]; k-2]; Array[a,7,2] (* Amiram Eldar, Nov 14 2018 *)
  • PARI
    {for(n=2,10,for(k=2^n,10^12,if(n==bigomega(k) &&
n-1==bigomega(k-1) && n-1==bigomega(k+1),print1(k", ");break())))}

Extensions

a(10) from Jon E. Schoenfield, Nov 14 2018
a(11)-a(13) from Giovanni Resta, Jan 04 2019

A335737 a(n) is the smallest prime p such that 2p+3q and 3p+2q are n-almost primes, where q is next prime after p.

Original entry on oeis.org

5, 47, 139, 2521, 77269, 631459, 6758117, 33059357, 7607209367, 173030234371, 152129921851
Offset: 1

Views

Author

Zak Seidov, Jun 20 2020

Keywords

Examples

			Both 2*5+3*7 = 31 and 3*5 +2*7 = 29 are primes.
Both 2*47+3*53 = 253 = 11*23 and 3*47+2*53 = 247 = 13*19 are semiprimes.

Crossrefs

Cf. A000040, A001358, A014612, A014613, A014614, A078840.

Programs

  • Mathematica
    m = 6; s = Table[0, {m}]; p = 2; c = 0; While[c < m, q = NextPrime[p]; If[(o = PrimeOmega[2 p + 3 q]) ==  PrimeOmega[ 3 p + 2 q] && o <= m && s[[o]] == 0, c++; s[[o]] = p]; p = q]; s (* Amiram Eldar, Jun 23 2020 *)
  • PARI
    for(n=1,8,my(p=2); forprime(q=3, oo, my(pq=2*p+3*q, qp=3*p+2*q); if(bigomega(pq)==n, if(bigomega(qp)==n, print1(p, ", "); break));p=q)) \\ Hugo Pfoertner, Jun 24 2020

Extensions

a(10)-a(11) from Amiram Eldar, Jun 23 2020

A374231 a(n) is the minimum number of distinct numbers with exactly n prime factors (counted with multiplicity) whose sum of reciprocals exceeds 1.

Original entry on oeis.org

3, 13, 96, 1772, 108336, 35181993
Offset: 1

Views

Author

Amiram Eldar, Jul 01 2024

Keywords

Examples

			a(1) = 3 since Sum_{k=1..2} 1/prime(k) = 1/2 + 1/3 = 5/6 < 1 and Sum_{k=1..3} 1/prime(k) = 1/2 + 1/3 + 1/5 = 31/30 > 1.
a(2) = 13 since Sum_{k=1..12} 1/A001358(k) = 1/4 + 1/6 + 1/9 + 1/10 + 1/14 + 1/15 + 1/21 + 1/22 + 1/25 + 1/26 + 1/33 + 1/34 = 15271237/15315300 < 1 and Sum_{k=1..13} 1/A001358(k) = 1/4 + 1/6 + ... + 1/35 = 15708817/15315300 > 1.

Crossrefs

Cf. A078840.
Cf. A000040, A001358, A014612, A014613, A014614, A046306, A046308, A046310, A046312, A046314.

Programs

  • Mathematica
    next[p_, n_] := Module[{k = p + 1}, While[PrimeOmega[k] != n, k++]; k]; a[n_] := Module[{k = 0, sum = 0, p = 0}, While[sum <= 1, p = next[p, n]; sum += 1/p; k++]; k]; Array[a, 5]
  • PARI
    nextnum(p, n) = {my(k = p + 1); while(bigomega(k) != n, k++); k;}
a(n) = {my(k = 0, sum = 0, p = 0); while(sum <= 1, p = nextnum(p, n); sum += 1/p; k++); k;}
Previous Showing 31-37 of 37 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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