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Search: a252459 -id:a252459
Displaying 1-7 of 7 results found. page 1
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A284263 a(n) = A252459(2*A000040(n)), a(0) = 0 by convention. +20
2
0, 0, 0, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..11000
FORMULA
a(0) = 0, for n >= 1, a(n) = A252459(2*A000040(n)).
a(n) = A252459(A002110(n)).
MATHEMATICA
a[n_] := If[n<1, 0, Block[{k=1}, While[Prime[n + k - 1] > Prime[k]^2, k++]; k - 1]]; Table[a[n], {n, 0, 130}] (* Indranil Ghosh, Mar 24 2017 *)
PROG
(PARI) A284263(n) = { my(k=1); if(0==n, 0, while(prime(n+k-1) > (prime(k)^2), k = k+1); (k-1)); };
(Scheme) (define (A284263 n) (if (zero? n) n (A252459 (* 2 (A000040 n)))))
(Python)
from sympy import prime
def a(n):
if n<1: return 0
k=1
while prime(n + k - 1)>prime(k)**2:k+=1
return k - 1 # Indranil Ghosh, Mar 24 2017
CROSSREFS
Cf. A000040, A002110, A251719, A252459, A284262.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Mar 24 2017
STATUS
approved
A251726 Numbers n > 1 for which gpf(n) < lpf(n)^2, where lpf and gpf (least and greatest prime factor of n) are given by A020639(n) and A006530(n). +10
17
2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 54, 55, 59, 61, 63, 64, 65, 67, 71, 72, 73, 75, 77, 79, 81, 83, 85, 89, 91, 95, 96, 97, 101, 103, 105, 107, 108, 109, 113, 115, 119, 121, 125, 127, 128, 131, 133, 135, 137, 139, 143, 144 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Numbers n > 1 for which there exists r <= gpf(n) such that r^k <= lpf(n) and gpf(n) < r^(k+1) for some k >= 0, where lpf and gpf (least and greatest prime factor of n) are given by A020639(n) and A006530(n) (the original, equivalent definition of the sequence).
Numbers n > 1 such that A252375(n) < 1 + A006530(n). Equally, one can substitute A251725 for A252375.
These are numbers n all of whose prime factors "fit between" two consecutive powers of some positive integer which itself is <= the largest prime factor of n.
Conjecture: If any n is in the sequence, then so is A003961(n).
Note: if Legendre's or Brocard's conjecture is true, then the above conjecture is true as well. See my comments at A251728. - Antti Karttunen, Jan 01 2015
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
Ratio A251726(n)/A251727(n) (plotted with OEIS-server's Plot2-utility)
FORMULA
Other identities. For all n >= 1:
A252373(a(n)) = n. [A252373 works as an inverse or ranking function for this sequence.]
EXAMPLE
For 35 = 5*7, 7 is less than 5^2, thus 35 is included.
For 90 = 2*3*3*5, 5 is not less than 2^2, thus 90 is NOT included.
For 105 = 3*5*7, 7 is less than 3^2, thus 105 is included.
MATHEMATICA
pfQ[n_]:=Module[{f=FactorInteger[n]}, f[[-1, 1]]<f[[1, 1]]^2]; Select[ Range[ 200], pfQ] (* Harvey P. Dale, May 01 2015 *)
PROG
(Scheme with Antti Karttunen's IntSeq-library, three alternative versions)
(define A251726 (MATCHING-POS 1 2 (lambda (n) (< (A006530 n) (A000290 (A020639 n))))))
(define A251726 (MATCHING-POS 1 2 (lambda (n) (< (A251725 n) (+ 1 (A006530 n))))))
(define A251726 (MATCHING-POS 1 2 (lambda (n) (< (A252375 n) (+ 1 (A006530 n))))))
(PARI) for(n=2, 150, if(vecmax(factor(n)[, 1]) < vecmin(factor(n)[, 1])^2, print1(n, ", "))) \\ Indranil Ghosh, Mar 24 2017
(Python)
from sympy import primefactors
print([n for n in range(2, 150) if max(primefactors(n))<min(primefactors(n))**2]) # Indranil Ghosh, Mar 24 2017
CROSSREFS
Complement: A251727. Subsequences: A251728, A000961 (after 1).
Characteristic function: A252372. Inverse function: A252373.
Gives the positions of zeros in A252459 (after its initial zero), cf. also A284261.
Cf. A252370 (gives the difference between the prime indices of gpf and lpf for each a(n)).
Sequence gives all n > 1 for which A284252(n) (equally: A284254) is 1, and A284256(n) (equally A284258) is 0, and also n > 1 such that A284260(n) = A006530(n).
Cf. A003961, A006530, A020639, A252375, A251725, A253784.
Related permutations: A252757-A252758.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Dec 17 2014
EXTENSIONS
A new simpler definition found Jan 01 2015 and the original definition moved to the Comments section.
STATUS
approved
A336835 Number of iterations of x -> A003961(x) needed before the result is deficient (sigma(x) < 2x), when starting from x=n. +10
15
0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,120
COMMENTS
It holds that a(n) <= A336836(n) for all n, because sigma(n) <= A003961(n) for all n (see A286385 for a proof).
The first 3 occurs at n = 19399380, the first 4 at n = 195534950863140268380. See A336389.
If x and y are relatively prime (i.e., gcd(x,y) = 1), then a(x*y) >= max(a(x),a(y)). Compare to a similar comment in A336915.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization
Index entries for sequences related to sigma(n)
FORMULA
If A294934(n) = 1, a(n) = 0, otherwise a(n) = 1 + a(A003961(n)).
From Antti Karttunen, Aug 21-Sep 01 2020: (Start)
For all n >= 1,
a(A046523(n)) >= a(n).
a(A071364(n)) >= a(n).
a(A108951(n)) = A337474(n).
a(A025487(n)) = A337475(n).
(End)
EXAMPLE
For n = 120, sigma(120) = 360 >= 2*120, thus 120 is not deficient, and we get the next number by applying the prime shift, A003961(120) = 945, and sigma(945) = 1920 >= 945*2, so neither 945 is deficient, so we prime shift once again, and A003961(945) = 9625, which is deficient, as sigma(9625) = 14976 < 2*9625. Thus after two iteration steps we encounter a deficient number, and therefore a(120) = 2.
MATHEMATICA
Array[-1 + Length@ NestWhileList[If[# == 1, 1, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}]] &, #, DivisorSigma[1, #] >= 2 # &] &, 120] (* Michael De Vlieger, Aug 27 2020 *)
PROG
(PARI)
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A336835(n) = { my(i=0); while(sigma(n) >= (n+n), i++; n = A003961(n)); (i); };
CROSSREFS
Cf. A003961, A005100, A005940, A023196, A046523, A047802, A071364, A108951, A286385, A294934, A336834, A337474, A337475.
Cf. A336389 (position of the first occurrence of a term >= n).
Differs from A294936 for the first time at n=120.
Cf. also A246271, A252459, A336836 and A336915 for similar iterations.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Aug 07 2020
STATUS
approved
A252372 Characteristic function for A251726: a(n) = 1 if n > 1 and gpf(n) < spf(n)^2, otherwise 0; here spf(n) and gpf(n) (smallest and greatest prime factor of n) are sequences A020639(n) and A006530(n). +10
5
0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET
1
COMMENTS
a(n) = 1 if n > 1 and there exists r <= A006530(n) such that r^k <= A020639(n) and A006530(n) < r^(k+1) for some k >= 0, otherwise 0 (the original definition).
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10000
FORMULA
Other identities. For all n >= 1:
a(n) = a(A066048(n)). [The result depends only on the smallest and the largest prime factor of n.]
PROG
(Scheme) (define (A252372 n) (if (< (A252375 n) (+ 1 (A006530 n))) 1 0))
CROSSREFS
Characteristic function of A251726.
A252373 gives the partial sums.
Cf. A006530, A020639, A066048, A252459, A252757.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Dec 17 2014. A new simpler definition found Jan 04 2015 and the original definition moved to the Comments section.
STATUS
approved
A252373 Partial sums of A252372, inverse function for A251726. +10
4
0, 1, 2, 3, 4, 5, 6, 7, 8, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 16, 17, 17, 18, 19, 20, 20, 21, 21, 22, 22, 23, 24, 24, 24, 25, 26, 27, 27, 27, 27, 28, 28, 29, 29, 30, 30, 31, 32, 33, 33, 33, 33, 34, 35, 36, 36, 36, 36, 37, 37, 38, 38, 39, 40, 41, 41, 42, 42, 42, 42, 43, 44, 45, 45, 46, 46, 47, 47, 48, 48, 49, 49, 50 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10001
FORMULA
a(1) = 0, a(n) = A252372(n) + a(n-1).
Other identities. For all n >= 1:
a(A251726(n)) = n. [This works as an inverse or ranking function for the injection A251726.]
PROG
(Scheme, with memoization-macro definec)
(definec (A252373 n) (if (<= n 1) 0 (+ (A252372 n) (A252373 (- n 1)))))
CROSSREFS
Cf. A251726, A252372, A252459.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Dec 17 2014
STATUS
approved
A336836 Number of iterations of x -> A003961(x) needed before A003961(x) < 2x, when starting from x=n, or -1 if such a number is never reached. +10
4
0, 0, 0, 2, 0, 3, 0, 4, 1, 2, 0, 4, 0, 1, 2, 4, 0, 5, 0, 4, 1, 0, 0, 6, 0, 0, 3, 4, 0, 4, 0, 6, 0, 0, 1, 6, 0, 0, 1, 4, 0, 4, 0, 4, 3, 0, 0, 6, 1, 2, 0, 4, 0, 5, 0, 4, 1, 0, 0, 6, 0, 0, 3, 6, 0, 4, 0, 4, 1, 4, 0, 9, 0, 0, 4, 4, 0, 4, 0, 6, 3, 0, 0, 6, 0, 0, 0, 6, 0, 5, 1, 4, 0, 0, 0, 9, 0, 3, 3, 4, 0, 3, 0, 4, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Starting from n, the number of prime shifts needed before a term of A246281 is reached.
It holds that a(n) >= A336835(n) for all n, because sigma(n) <= A003961(n) for all n (see A286385 for a proof).
Note that in contrast to abundancy used in A336835, the condition [A003961(x) > 2x] (= A252742) is not monotonic when iterating with A003961. For example, we have A003961(9) = 25 > 2*9, A003961(25) = 49 < 2*25, and then again A003961(49) = 121 > 2*49.
Question: Is the escape clause necessary in the definition?
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization
PROG
(PARI)
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336836(n) = for(i=0, oo, my(n2 = n+n); n = A003961(n); if(n < n2, return(i)));
CROSSREFS
Cf. A246281 (positions of zeros, numbers k for which A003961(k) < 2*k).
Cf. A005940, A252742, A252743, A286385.
Cf. also A246271, A252459, A336835 for similar iterations.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Aug 07 2020
STATUS
approved
A284261 a(n) = A284258(n) - A284258(A003961(n)). +10
3
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,429
COMMENTS
The first 2 occurs at n = 429 = 3*11*13. A003961(429) = 1105 = 5*13*17.
The first 3 occurs at n = 7293 = 3*11*13*17. A003961(7293) = 20995 = 5*13*17*19.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..10001
FORMULA
a(n) = A284258(n) - A284258(A003961(n)).
a(n) = A001221(A284254(n)) - A001221(A284254(A003961(n))).
MATHEMATICA
Table[If[n == 1, 0, Subtract @@ Map[Count[#, d_ /; d > First[#]^2] &@ FactorInteger[#][[All, 1]] &, {n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[n == 1]}]], {n, 120}] (* Michael De Vlieger, Mar 24 2017 *)
PROG
(Scheme) (define (A284261 n) (- (A284258 n) (A284258 (A003961 n))))
CROSSREFS
Cf. A001221, A003961, A284254, A284258, A252459.
KEYWORD
nonn
AUTHOR
Antti Karttunen, Mar 24 2017
EXTENSIONS
Secondary offset corrected by Antti Karttunen, Dec 15 2017
STATUS
approved
page 1

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Last modified August 29 06:09 EDT 2024. Contains 375510 sequences. (Running on oeis4.)