Displaying 1-10 of 11 results found.
Numbers k such that k and k+2 are both primitive practical numbers ( A267124).
+20
5
28, 304, 306, 340, 460, 462, 858, 868, 1482, 1768, 1974, 2440, 2728, 2838, 2860, 3318, 3738, 4134, 4264, 4288, 4420, 4422, 5236, 5694, 6100, 6102, 7590, 8814, 9040, 9042, 10218, 11128, 11620, 11778, 12558, 12978, 13110, 14320, 14382, 14670, 15568, 16048, 16110
EXAMPLE
28 is a term since 28 and 28 + 2 = 30 are both primitive practical numbers.
MATHEMATICA
f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[fct_] := (ind = Position[fct[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {}; pracTestQ[fct_, k_] := Module[{f = fct}, f[[k, 2]] -= 1; pracQ[f]]; primPracQ[n_] := Module[{fct = FactorInteger[n]}, pracQ[fct] && AllTrue[Range@Length[fct], fct[[#, 2]] == 1 || ! pracTestQ[fct, #] &]]; Select[Range[2, 16200, 2], primPracQ[#] && primPracQ[# + 2] &]
Primitive practical numbers ( A267124) with a record gap to the next primitive practical number.
+20
2
1, 2, 6, 42, 104, 140, 1036, 1590, 2730, 7900, 10374, 19180, 22660, 23180, 26418, 105868, 114960, 139060, 295780, 403524, 482250, 1294144, 2468944, 4799058, 5379282, 19035500, 20233936, 21803860, 112406992, 789190976, 3520928922
COMMENTS
The record gap values are 1, 4, 14, 24, 36, 64, 74, 82, 84, 104, 106, 112, 120, 132, 154, 188, 204, 224, 236, 246, 258, 308, 326, 360, 418, 440, 452, 508, 674, 804, 846, ...
EXAMPLE
The first 8 primitive practical numbers are 1, 2, 6, 20, 28, 30, 42 and 66. The differences between these terms are 1, 4, 14, 8, 2, 12 and 24. The record gaps are 1, 4, 14 and 24, which occur after the terms 1, 2, 6 and 42.
MATHEMATICA
f[p_, e_] := (p^(e + 1) - 1)/(p - 1); pracQ[fct_] := (ind = Position[fct[[;; , 1]]/(1 + FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)]) == {}; pracTestQ[fct_, k_] := Module[{f = fct}, f[[k, 2]] -= 1; pracQ[f]]; primPracQ[n_] := Module[{fct = FactorInteger[n]}, pracQ[fct] && AllTrue[Range@Length[fct], fct[[#, 2]] == 1 || ! pracTestQ[fct, #] &]]; seq = {1}; m = 2; dm = 1; Do[If[primPracQ[n], d = n - m; If[d > dm, dm = d; AppendTo[seq, m]]; m = n], {n, 4, 10^5, 2}]; seq
Number of primitive practical numbers (PPNs)( A267124) between successive primorial numbers ( A002110) where the PPNs q are in the range A002110(n-1) < q <= A002110(n).
+20
1
1, 1, 3, 8, 108, 1107, 15788, 252603, 5121763
COMMENTS
The sequence of primorial numbers is a subset of the sequence of PPNs. Note that the sequence A002110 has an offset of 0 and A002110(0) = 1.
EXAMPLE
a(4) = 8, because between successive primorials 30 and 210 (that includes 210) is the sequence {42, 66, 78, 88, 104, 140, 204, 210} of PPNs. It contains 8 members.
MATHEMATICA
f[p_, e_] := (p^(e + 1) - 1)/(p - 1);
pracQ[fct_] := (ind=Position[fct[[;; , 1]]/(1+FoldList[Times, 1, f @@@ Most@fct]), _?(# > 1 &)])=={};
pracTestQ[fct_, k_] := Module[{f=fct}, f[[k, 2]]-= 1; pracQ[f]];
primPracQ[n_] := Module[{fct=FactorInteger[n]}, pracQ[fct]&&AllTrue[Range@Length[fct], fct[[#, 2]]==1||!pracTestQ[fct, #] &]];
pri[n_] := Module[{m}, If[n==1, 1, Product[Prime[m], {m, 1, n-1}]]];
plst=Join[{1}, Select[Range[2, 10^9, 2], primPracQ]]; pasc=<||>;
Do[AppendTo[pasc, <|plst[[n]]->n|>], {n, 1, Length@plst}]; Table[pasc[pri[n+1]]-pasc[pri[n]], {n, 1, 9}]
PROG
(PARI)
f(n) = factorback(primes(n)); \\ A002110
Number of decompositions of 2n-1 into sums of a prime and a primitive practical number ( A267124).
+20
1
0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 3, 4, 3, 3, 3, 2, 4, 3, 4, 5, 2, 3, 2, 2, 4, 4, 2, 3, 3, 3, 4, 6, 2, 3, 4, 4, 4, 5, 2, 6, 5, 2, 6, 3, 4, 5, 6, 2, 6, 8, 4, 4, 5, 4, 5, 4, 2, 5, 4, 4, 5, 6, 3, 5, 5, 4, 4, 6, 4, 4, 6, 2, 5, 6, 5, 3, 4, 3, 5, 7, 4, 3, 4, 5, 6, 5, 4, 4, 6
COMMENTS
It has been conjectured in comments for A267124 that every odd number, beginning with 3, is the sum of a prime number and a primitive practical number. That is a(n) > 0 for n >= 2. This sequence is analogous to A045917 for the Goldbach conjecture. The graph of this sequence forms a comet that is analogous to Goldbach's comet. - Frank M Jackson, Sep 18 2024
EXAMPLE
a(5) = 2 because 9 can be decomposed twice as 2+7, 6+3 with 3, 7 prime and 2, 6 primitive practical.
MATHEMATICA
PracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1||(n>1&&OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]];
DivFreeQ[n_] := Module[{plst=First/@Select[FactorInteger[n], #[[2]]>1 &], m, ok=False}, Do[If[!PracticalQ[n/plst[[m]]], ok=True, ok=False; Break[]], {m, 1, Length@plst}]; ok];
PPracticalQ[n_] := PracticalQ[n]&&(SquareFreeQ[n]||DivFreeQ[n]);
f[n_] := Select[2n-1-Prime[Range[PrimePi[2n-1]]], PPracticalQ]; Table[Length@f[n], {n, 1, 200}]
PROG
(PARI) a(n) = my(nn=2*n-1); sum (i=1, nn, isprime(i) && is_ A267124(nn-i)); \\ Michel Marcus, Jun 27 2024
Number of primes between successive primitive practical numbers. That is, number of primes p such that A267124(n) < p <= A267124(n+1).
+20
0
1, 2, 5, 1, 1, 3, 5, 3, 2, 4, 7, 12, 0, 1, 2, 6, 3, 0, 4, 0, 1, 3, 2, 0, 1, 3, 1, 2, 2, 3, 8, 1, 1, 1, 3, 3, 1, 1, 1, 2, 2, 2, 8, 3, 4, 3, 3, 4, 1, 7, 2, 0, 7, 1, 1, 0, 4, 4, 4, 4, 7, 1, 12, 1, 1, 1, 0, 5, 3, 2, 2, 3, 1, 4, 3, 2, 0, 2, 4, 3, 0, 9, 1, 1, 1, 3, 1, 1, 2, 1, 2, 5, 13, 0, 1, 4, 6, 0, 4
EXAMPLE
a(3) = 5 because between the 3rd and 4th primitive practical numbers, namely 6 and 20 there are 5 primes. They are 7, 11, 13, 17 and 19.
MATHEMATICA
lst=Last/@ReadList["https://oeis.org/ A267124/b267124.txt", {Number, Number}]; seq=Table[PrimePi[lst[[n+1]]]-PrimePi[lst[[n]]], {n, 1, Length@lst-1}]; seq[[1;; 100]]
Primorial numbers (first definition): product of first n primes. Sometimes written prime(n)#.
(Formerly M1691 N0668)
+10
1779
1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070, 117288381359406970983270, 7858321551080267055879090
COMMENTS
See A034386 for the second definition of primorial numbers: product of primes in the range 2 to n. a(n) is the least number N with n distinct prime factors (i.e., omega(N) = n, cf. A001221). - Lekraj Beedassy, Feb 15 2002 Smallest number stroked off n times after the n-th sifting process in an Eratosthenes sieve. - Lekraj Beedassy, Mar 31 2005 Apparently each term is a new minimum for phi(x)*sigma(x)/x^2. 6/Pi^2 < sigma(x)*phi(x)/x^2 < 1 for n > 1. - Jud McCranie, Jun 11 2005 Let f be a multiplicative function with f(p) > f(p^k) > 1 (p prime, k > 1), f(p) > f(q) > 1 (p, q prime, p < q). Then the record maxima of f occur at n# for n >= 1. Similarly, if 0 < f(p) < f(p^k) < 1 (p prime, k > 1), 0 < f(p) < f(q) < 1 (p, q prime, p < q), then the record minima of f occur at n# for n >= 1. - David W. Wilson, Oct 23 2006 Wolfe and Hirshberg give ?, ?, ?, ?, ?, 30030, ?, ... as a puzzle.
Records in number of distinct prime divisors. - Artur Jasinski, Apr 06 2008 Where record values occur in A001221. - Melinda Trang (mewithlinda(AT)yahoo.com), Apr 15 2010 It can be proved that there are at least T prime numbers less than N, where the recursive function T is: T = N - N*Sum_{i = 0..T(sqrt(N))} A005867(i)/ A002110(i). This can show for example that at least 0.16*N numbers are primes less than N for 29^2 > N > 23^2. - Ben Paul Thurston, Aug 23 2010 The above comment from Parthasarathy Nambi follows from the observation that digit summing produces a congruent number mod 9, so the digital root of any multiple of 3 is a multiple of 3. prime(n)# is divisible by 3 for n >= 2. - Christian Schulz, Oct 30 2013 The peaks (i.e., local maximums) in a graph of the number of repetitions (i.e., the tally of values) vs. value, as generated by taking the differences of all distinct pairs of odd prime numbers within a contiguous range occur at regular periodic intervals given by the primorial numbers 6 and greater. Larger primorials yield larger (relative) peaks, however the range must be >50% larger than the primorial to be easily observed. Secondary peaks occur at intervals of those "near-primorials" divisible by 6 (e.g., 42). See A259629. Also, periodicity at intervals of 6 and 30 can be observed in the local peaks of all possible sums of two, three or more distinct odd primes within modest contiguous ranges starting from p(2) = 3. - Richard R. Forberg, Jul 01 2015 If a number k and a(n) are coprime and k < (prime(n+1))^b < a(n), where b is an integer, then k has fewer than b prime factors, counting multiplicity (i.e., bigomega(k) < b, cf. A001222). - Isaac Saffold, Dec 03 2017 If n > 0, then a(n) has 2^n unitary divisors ( A034444), and a(n) is a record; i.e., if k < a(n) then k has fewer unitary divisors than a(n) has. - Clark Kimberling, Jun 26 2018 Unitary superabundant numbers: numbers k with a record value of the unitary abundancy index, A034448(k)/k > A034448(m)/m for all m < k. - Amiram Eldar, Apr 20 2019 Psi(n)/n is a new maximum for each primorial (psi = A001615) [proof in link: Patrick Sole and Michel Planat, proposition 1 page 2]; compare with comment 2004: Phi(n)/n is a new minimum for each primorial. - Bernard Schott, May 21 2020 The term "primorial" was coined by Harvey Dubner (1987). - Amiram Eldar, Apr 16 2021
REFERENCES
A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.
P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 4.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wolfe and S. Hirshberg, Underspecified puzzles, in Tribute to A Mathemagician, Peters, 2005, pp. 73-74.
LINKS
C. K. Caldwell, The Prime Glossary, Primorial. G. Caveney, J.-L. Nicolas and J. Sondow, On SA, CA, and GA numbers, arXiv:1112.6010 [math.NT], 2011-2012; Ramanujan J., 29 (2012), 359-384. Eric Weisstein's World of Mathematics, Primorial.
FORMULA
Asymptotic expression for a(n): exp((1 + o(1)) * n * log(n)) where o(1) is the "little o" notation. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001
Binomial transform = A136104: (1, 3, 11, 55, 375, 3731, ...). Equals binomial transform of A121572: (1, 1, 3, 17, 119, 1509, ...). - Gary W. Adamson, Dec 14 2007
EXAMPLE
a(9) = 23# = 2*3*5*7*11*13*17*19*23 = 223092870 divides the difference 5283234035979900 in the arithmetic progression of 26 primes A204189. - Jonathan Sondow, Jan 15 2012
MAPLE
A002110 := n -> mul(ithprime(i), i=1..n);
MATHEMATICA
FoldList[Times, 1, Prime[Range[20]]]
Join[{1}, Denominator[Accumulate[1/Prime[Range[20]]]]] (* Harvey P. Dale, Apr 11 2012 *)
PROG
(Haskell)
a002110 n = product $ take n a000040_list
a002110_list = scanl (*) 1 a000040_list
(Magma) [1] cat [&*[NthPrime(i): i in [1..n]]: n in [1..20]]; // Bruno Berselli, Oct 24 2012 (Magma) [1] cat [&*PrimesUpTo(p): p in PrimesUpTo(60)]; // Bruno Berselli, Feb 08 2015 (PARI) p=1; for (n=0, 100, if (n, p*=prime(n)); write("b002110.txt", n, " ", p) ) \\ Harry J. Smith, Nov 13 2009 (Python)
from sympy import primorial
def a(n): return 1 if n < 1 else primorial(n)
(Scheme) ; with memoization-macro definec
CROSSREFS
A034386 gives the second version of the primorial numbers.
Cf. A001615, A002182, A002201, A003418, A005235, A006862, A034444 (unitary divisors), A034448, A034387, A033188, A035345, A035346, A036691 (compositorial numbers), A049345 (primorial base representation), A057588, A060735 (and integer multiples), A061742 (squares), A072938, A079266, A087315, A094348, A106037, A121572, A053589, A064648, A132120, A260188.
Primitive practical numbers of the form 2^i * prime(k).
+10
5
6, 20, 28, 88, 104, 272, 304, 368, 464, 496, 1184, 1312, 1376, 1504, 1696, 1888, 1952, 4288, 4544, 4672, 5056, 5312, 5696, 6208, 6464, 6592, 6848, 6976, 7232, 8128, 16768, 17536, 17792, 19072, 19328, 20096, 20864, 21376, 22144, 22912, 23168, 24448, 24704, 25216
COMMENTS
a(n) is a number of the form 2^i * prime(k) for i > 0 and A007053(i) < k <= A007053(i+1). Terms are pseudoperfect numbers, A005835 and are also primitive pseudoperfect numbers, A006036.
FORMULA
a(n) = 2^floor(log_2(prime(n+1))) * prime(n+1).
MATHEMATICA
a[n_] := (p = Prime[n+1]) * 2^Floor[Log2[p]]; Array[a, 50] (* Amiram Eldar, Sep 22 2019 *)
For the odd number 2n + 1, the least primitive practical number r such that 2n + 1 = r + p where p is prime.
+10
1
1, 2, 2, 2, 6, 2, 2, 6, 2, 2, 6, 2, 20, 6, 2, 2, 6, 6, 2, 28, 2, 2, 6, 2, 20, 6, 2, 20, 6, 2, 2, 6, 6, 2, 28, 2, 2, 6, 6, 2, 30, 2, 20, 6, 2, 20, 6, 30, 2, 28, 2, 2, 6, 2, 2, 6, 2, 20, 6, 20, 20, 28, 20, 2, 28, 2, 28, 6, 2, 2, 6, 6, 20, 42, 2, 2, 6, 6, 2, 30, 6, 2, 28, 2, 20, 6, 2, 20, 6, 2, 2, 6, 6, 88, 28
COMMENTS
Conjecture: every odd number, beginning with 3, is the sum of a prime number and a primitive practical number. This is a tighter conjecture than that posed by Hal M. Switkay (see comments of A005153).
EXAMPLE
a(61) = 20, because 61st odd number is 123 = {(10+113), (14+109), (16+107), (20+103), ...} and 20 is the least primitive practical number. 10 and 14 are not practical numbers and 16 is practical but not primitive.
MATHEMATICA
PracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1 || (n>1 && OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]];
DivFreeQ[n_] := Module[{plst=First/@Select[FactorInteger[n], #[[2]]>1 &], m, ok=False}, Do[If[! PracticalQ[n/plst[[m]]], ok=True, ok=False; Break[]], {m, 1, Length@plst}]; ok];
PPracticalQ[n_] := PracticalQ[n] && (SquareFreeQ[n] || DivFreeQ[n]);
part[n_, m_] := Module[{p = NextPrime[n, -m], d}, d = n - p; {d, p}];
find[n_] := Module[{m=1}, While[!PPracticalQ[part[n, m][[1]]], m++]; part[n, m]];
Table[find[2 n + 1][[1]], {n, 1, 1000}]
PROG
(Python)
from sympy import prevprime, factorint
def is_primitive(n):
for i in range(0, len(list(factorint(n)))):
if list(factorint(n).values())[i] > 1:
if is_ A005153(n//list(factorint(n))[i]): return False return True
if is_ A005153(n) and is_primitive(n) : return True for odds in range(3, 192, 2):
prime = prevprime(odds)
while not is_ A267124(odds - prime): prime = prevprime(prime)
Least positive integer k with k primitive practical and k*n practical.
+10
0
1, 1, 2, 1, 6, 1, 6, 1, 2, 2, 6, 1, 6, 2, 2, 1, 20, 1, 20, 1, 2, 6, 20, 1, 6, 6, 2, 1, 20, 1, 20, 1, 2, 6, 6, 1, 20, 6, 2, 1, 20, 1, 20, 2, 2, 6, 28, 1, 6, 2, 6, 2, 28, 1, 6, 1, 6, 6, 30, 1, 30, 20, 2, 1, 6, 1, 30, 6, 6, 2, 30, 1, 30, 20, 2, 6, 6, 1, 42, 1, 2, 20, 42, 1, 6, 20, 6, 1, 42, 1, 6, 6, 6, 20, 6, 1, 42, 2, 2, 1
COMMENTS
For all integers n>0 there exists k such that k*n is practical and k is primitive practical. For example, n*prime(f)# is practical where k = prime(f)# = A002110(f) is a primorial number and f is the prime index of the largest prime number in the factorization of n. All primorials are primitive practical numbers. The sequence above gives least k.
EXAMPLE
a(5)=6 since 6*5=30 is practical and 6 is primitive practical. Also 4*5=20 is practical but 4 is not primitive practical.
MATHEMATICA
PracticalQ[n_] := Module[{f, p, e, prod=1, ok=True}, If[n<1||(n>1&&OddQ[n]), False, If[n==1, True, f=FactorInteger[n]; {p, e}=Transpose[f]; Do[If[p[[i]]>1+DivisorSigma[1, prod], ok=False; Break[]]; prod=prod*p[[i]]^e[[i]], {i, Length[p]}]; ok]]];
DivFreeQ[n_] := Module[{plst=First/@Select[FactorInteger[n], #[[2]]>1 &], m, ok=False}, Do[If[!PracticalQ[n/plst[[m]]], ok = True, ok = False; Break[]], {m, 1, Length@plst}]; ok];
PPracticalQ[n_] := PracticalQ[n]&&(SquareFreeQ[n]||DivFreeQ[n]);
lst = {}; Do[m=0; While[!PPracticalQ[m]||(!PracticalQ[m*n]&&m<10000), m++]; AppendTo[lst, m], {n, 1, 500}]; lst
Primitive practical numbers of the form 2 * 3^i * prime(k).
+10
0
30, 42, 66, 78, 306, 342, 414, 522, 558, 666, 2214, 2322, 2538, 2862, 3186, 3294, 3618, 3834, 3942, 4266, 4482, 4806, 5238, 5454, 5562, 5778, 5886, 6102, 20574, 21222, 22194, 22518, 24138, 24462, 25434, 26406, 27054, 28026, 28998, 29322, 30942, 31266, 31914
COMMENTS
This sequence and A308710 are both non-overlapping subsets of A267124. a(n) is a number of the form 2 * 3^i * prime(k) for i > 0 and b(i) < k <= b(i+1) where b(n) = Sum_{m=2..n+1} A233919(m). Terms are pseudoperfect numbers, A005835, but are not primitive pseudoperfect numbers, A006036. Since no term is a square or twice a square, there are no terms k such that sigma(k) is odd. Therefore, according to Proposition 10 by Rao/Peng (see their JNT paper at A083207) all terms are Zumkeller numbers. - Ivan N. Ianakiev, Nov 28 2023
FORMULA
a(n) = 2 * 3^(floor(log_3(2*prime(n+2)))-1) * prime(n+2).
MATHEMATICA
a[n_]:=2*3^(Floor[Log[2*Prime[n+2]]/Log[3]]-1)*Prime[n+2]; Array[a, 43] (* Stefano Spezia, Nov 19 2023 *)
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