A080196 -id:A080196 - cp's OEIS frontend

A080197 13-smooth numbers: numbers whose prime divisors are all <= 13.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 52, 54, 55, 56, 60, 63, 64, 65, 66, 70, 72, 75, 77, 78, 80, 81, 84, 88, 90, 91, 96, 98, 99, 100, 104, 105, 108, 110, 112, 117, 120

Offset: 1

Klaus Brockhaus, Feb 10 2003

Numbers of the form 2^r*3^s*5^t*7^u*11^v*13^w with r, s, t, u, v, w >= 0.

			33 = 3*11 and 39 = 3*13 are terms but 34 = 2*17 is not.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000079, A080196. For p-smooth numbers with other values of p, see A003586, A051037, A002473, A051038, A080681, A080682, A080683.

[n: n in [1..150] | PrimeDivisors(n) subset PrimesUpTo(13)]; // Bruno Berselli, Sep 24 2012

mx = 120; Sort@ Flatten@ Table[ 2^i*3^j*5^k*7^l*11^m*13^n, {i, 0, Log[2, mx]}, {j, 0, Log[3, mx/2^i]}, {k, 0, Log[5, mx/(2^i*3^j)]}, {l, 0, Log[7, mx/(2^i*3^j*5^k)]}, {m, 0, Log[11, mx/(2^i*3^j*5^k*7^l)]}, {n, 0, Log[13, mx/(2^i*3^j*5^k*7^l*11^m)]}] (* Robert G. Wilson v, Aug 17 2012 *)

test(n)=m=n; forprime(p=2,13, while(m%p==0,m=m/p)); return(m==1)
for(n=1,200,if(test(n),print1(n",")))

is_A080197(n,p=13)=n<=p||vecmax(factor(n,p+1)[,1])<=p \\ M. F. Hasler, Jan 16 2015

list(lim,p=13)=if(p==2, return(powers(2, logint(lim\1,2)))); my(v=[],q=precprime(p-1),t=1); for(e=0,logint(lim\=1,p), v=concat(v, list(lim\t,q)*t); t*=p); Set(v) \\ Charles R Greathouse IV, Apr 16 2020

import heapq
from itertools import islice
from sympy import primerange
def agen(p=13): # generate all p-smooth terms
    v, oldv, h, psmooth_primes, = 1, 0, [1], list(primerange(1, p+1))
    while True:
        v = heapq.heappop(h)
        if v != oldv:
            yield v
            oldv = v
            for p in psmooth_primes:
                heapq.heappush(h, v*p)
print(list(islice(agen(), 69))) # Michael S. Branicky, Nov 20 2022

from sympy import integer_log, prevprime
def A080197(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x,m): return sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1)) if m==3 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    def f(x): return n+x-g(x,13)
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Sum_{n>=1} 1/a(n) = Product_{primes p <= 13} p/(p-1) = (2*3*5*7*11*13)/(1*2*4*6*10*12) = 1001/192. - Amiram Eldar, Sep 22 2020

A125624 Array read by antidiagonals: n-th row contains the positive integers with their largest prime factor equal to the n-th prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 7, 10, 9, 16, 11, 14, 15, 12, 32, 13, 22, 21, 20, 18, 64, 17, 26, 33, 28, 25, 24, 128, 19, 34, 39, 44, 35, 30, 27, 256, 23, 38, 51, 52, 55, 42, 40, 36, 512, 29, 46, 57, 68, 65, 66, 49, 45, 48, 1024, 31, 58, 69, 76, 85, 78, 77, 56, 50, 54, 2048, 37, 62, 87, 92

Offset: 1

Leroy Quet, Jan 27 2007

This sequence is a permutation of the integers >= 2.
Since the table has been entered by rising instead of falling antidiagonals, the sequence represents the transpose, with columns instead of rows: cf. the "table" link, section "infinite square array". - M. F. Hasler, Oct 22 2019
Start with table headed by primes in the first row, then list beneath each prime(k) the ordered prime(k)-smooth numbers. Read the table by falling antidiagonals to get the terms of this sequence. - David James Sycamore, Jun 23 2024

			Array begins: (rows here appear as columns in the "table" display of the sequence)
   2,  4,  8, 16, 32, 64, 128, 256, 512, ... (A000079)
   3,  6,  9, 12, 18, 24,  27,  36,  48, ... (A065119)
   5, 10, 15, 20, 25, 30,  40,  45,  50, ... (A080193)
   7, 14, 21, 28, 35, 42,  49,  56,  63, ... (A080194)
  11, 22, 33, 44, 55, 66,  77,  88,  99, ... (A080195)
  13, 26, 39, 52, 65, 78,  91, 104, 117, ... (A080196)
The 3rd row, for example, contains the positive integers where the 3rd prime, 5, is the largest prime divisor. That is, each integer in this row is divisible by 5 and may be divisible by 2 or 3 as well, but none of the integers in this row are divisible by primes larger than 5. (So for example, 35 = 5*7 is excluded from the 3rd row.)

Ivan Neretin, Table of n, a(n) for n = 1..5050

Cf. A083140, A006530, A000040 (1st col), A033286 (main diag), A077320.

Cf. A000079, A003586, A051037, A002473, A051038, A080197, A080681.

lpf[n_] := FactorInteger[n][[ -1, 1]];f[n_, m_] := f[n, m] = Block[{k},k = If[m == 1, Prime[n], f[n, m - 1] + 1];While[lpf[k] != Prime[n], k++ ];k];Table[f[ d - m + 1, m], {d, 12}, {m, d}] // Flatten (* Ray Chandler, Feb 09 2007 *)

T=List(); r=c=1; for(n=1,99, #TT[r][1], ); print1(T[r][c]","); r-- && c++ || r=c+c=1) \\ M. F. Hasler, Oct 22 2019

Extended by Ray Chandler, Feb 09 2007

A080188 Primes p such that 13 is the largest of all prime factors of the numbers between p and the next prime (cf. A052248).

Original entry on oeis.org

23, 311, 349, 857, 1091, 1871, 1949, 2027, 2339, 2729, 3119, 3821, 5849, 6551, 7487, 9437, 10139, 10529, 11699, 15287, 18251, 21059, 21839, 38609, 42899, 49919, 51479, 57329, 61151, 65519, 69497, 70199, 70979, 81899, 97499, 108107, 109199, 114659

Offset: 1

Klaus Brockhaus, Feb 10 2003

nonn

The sequence appears to consist of 23, 349 and the lesser of twin primes q (A001359) such that q+1 is 13-smooth (A080197) but not 11-smooth (A051038, A080196).

			349 is a term since 350 = 2*5^2*7, 351 = 3^3*13, 352 = 2^5*11 are the numbers between 349 and the next prime 353; 857 is a term since 858 = 2*3*11*13 is the only number between 857 and the next prime 859.

Amiram Eldar, Table of n, a(n) for n = 1..1092 (terms below 10^11)

Cf. A052248, A001359, A080197, A051038, A080196.

maxPrime[n1_, n2_] := FactorInteger[#][[-1, 1]] & /@ Range[n1, n2]; Select[Range[120000], PrimeQ[#] && Max[maxPrime[# + 1, NextPrime[#] - 1]] == 13 &] (* Amiram Eldar, Feb 08 2020 *)

{forprime(p=2,120000,q=nextprime(p+1); m=0; j=p+1; while(j

cp's OEIS Frontend

A080197 13-smooth numbers: numbers whose prime divisors are all <= 13.

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A125624 Array read by antidiagonals: n-th row contains the positive integers with their largest prime factor equal to the n-th prime.

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A080188 Primes p such that 13 is the largest of all prime factors of the numbers between p and the next prime (cf. A052248).

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI