Displaying 11-20 of 352 results found.
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First differences of non-prime-powers (inclusive).
+0
6
4, 2, 2, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 2, 1, 1
COMMENTS
Inclusive means 1 is a prime-power but not a non-prime-power.
Non-prime-powers (inclusive) are listed by A024619.
EXAMPLE
The 5th non-prime-power (inclusive) is 15, and the 6th is 18, so a(5) = 3.
MATHEMATICA
Differences[Select[Range[2, 100], !PrimePowerQ[#]&]]
PROG
(Python)
from itertools import count
from sympy import primepi, integer_nthroot, primefactors
def f(x): return int(n+1+sum(primepi(integer_nthroot(x, k)[0]) for k in range(1, x.bit_length())))
m, k = n, f(n)
while m != k: m, k = k, f(k)
return next(i for i in count(m+1) if len(primefactors(i))>1)-m # Chai Wah Wu, Sep 10 2024
CROSSREFS
Essentially the same as the exclusive version, A375708. Positions of 1's are A375713(n) - 1. For runs of non-prime-powers:
A000961 lists prime-powers (inclusive).
A246655 lists prime-powers (exclusive).
Maximum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.
+0
6
2, 5, 6, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 28, 29, 30, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88
COMMENTS
Non-perfect-powers ( A007916) are numbers with no proper integer roots. An anti-run of a sequence is an interval of positions at which consecutive terms differ by more than one.
Also non-perfect-powers x such that x + 1 is also a non-perfect-power.
EXAMPLE
The initial anti-runs are the following, whose maxima are a(n):
(2)
(3,5)
(6)
(7,10)
(11)
(12)
(13)
(14)
(15,17)
(18)
(19)
(20)
(21)
(22)
(23)
(24,26,28)
MATHEMATICA
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Max/@Split[Select[Range[100], radQ], #1+1!=#2&]//Most
- or -
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Select[Range[100], radQ[#]&&radQ[#+1]&]
CROSSREFS
For nonsquarefree numbers we have A068781, runs A072284 minus 1 and shifted. For anti-runs of non-perfect-powers:
For runs of non-perfect-powers:
E.g.f. satisfies A(x) = (exp(x / (1 - A(x))^3) - 1) / (1 - A(x)).
+0
6
0, 1, 9, 190, 6435, 301126, 18007161, 1311752590, 112703870439, 11158543451926, 1250964512674533, 156642117419304958, 21668625406445359227, 3281750147124057118966, 540094007004476783547825, 95975344500184607391266734, 18314947854834472094038237647
FORMULA
a(n) = Sum_{k=1..n} (3*n+2*k-2)!/(3*n+k-1)! * Stirling2(n,k).
E.g.f.: Series_Reversion( (1 - x)^3 * log(1 + x * (1 - x)) ).
PROG
(PARI) a(n) = sum(k=1, n, (3*n+2*k-2)!/(3*n+k-1)!*stirling(n, k, 2));
E.g.f. satisfies A(x) = (-log(1 - x / (1 - A(x))^3)) / (1 - A(x)).
+0
6
0, 1, 9, 191, 6496, 305164, 18317390, 1339293822, 115492112640, 11476262240520, 1291250885222592, 162271449317302632, 22528350072978189600, 3424249337820235241472, 565573503590604522245136, 100864333223422171393303488, 19317041144591537348567168256
FORMULA
a(n) = Sum_{k=1..n} (3*n+2*k-2)!/(3*n+k-1)! * |Stirling1(n,k)|.
E.g.f.: Series_Reversion( (1 - x)^3 * (1 - exp(-x * (1 - x))) ).
PROG
(PARI) a(n) = sum(k=1, n, (3*n+2*k-2)!/(3*n+k-1)!*abs(stirling(n, k, 1)));
Expansion of g.f. A(x) satisfying 0 = Sum_{k=0..n} (-1)^k * binomial(n,k)^2 * ([x^k] A(x)^n) for n >= 1.
+0
5
1, 1, 3, 21, 264, 5100, 138595, 5021209, 233863116, 13628372628, 972514037307, 83479400425677, 8490972592164813, 1010263560882000981, 139051185192340895094, 21926159523172792097194, 3927328317712845680689864, 793059545751159815604109176, 179339266160209677707004583560
COMMENTS
Note that 0 = Sum_{k=0..n} (-1)^k * binomial(n,k) * ([x^k] G(x)^n) is satisfied by G(x) = 1/(1-x) for n >= 1.
FORMULA
a(n) ~ c * d^n * n!^2, where d = 0.691660276122579707675... = 4/BesselJZero(0,1)^2 = 4/ A115368^2 and c = 3.8999463598998648630203... - Vaclav Kotesovec, Sep 10 2024
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 264*x^4 + 5100*x^5 + 138595*x^6 + 5021209*x^7 + 233863116*x^8 + ...
The table of coefficients of x^k in A(x)^n begins:
n=1: [1, 1, 3, 21, 264, 5100, 138595, ...];
n=2: [1, 2, 7, 48, 579, 10854, 289415, ...];
n=3: [1, 3, 12, 82, 954, 17352, 453657, ...];
n=4: [1, 4, 18, 124, 1399, 24696, 632656, ...];
n=5: [1, 5, 25, 175, 1925, 33001, 827900, ...];
n=6: [1, 6, 33, 236, 2544, 42396, 1041046, ...];
...
from which we may illustrate the defining property given by
0 = Sum_{k=0..n} (-1)^k * binomial(n,k)^2 * ([x^k] A(x)^n).
Using the coefficients in the table above, we see that
n=1: 0 = 1*1 - 1*1;
n=2: 0 = 1*1 - 4*2 + 1*7;
n=3: 0 = 1*1 - 9*3 + 9*12 - 1*82;
n=4: 0 = 1*1 - 16*4 + 36*18 - 16*124 + 1*1399;
n=5: 0 = 1*1 - 25*5 + 100*25 - 100*175 + 25*1925 - 1*33001;
n=6: 0 = 1*1 - 36*6 + 225*33 - 400*236 + 225*2544 - 36*42396 + 1*1041046;
...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = sum(k=0, #A-1, (-1)^(#A-k) * binomial(#A-1, k)^2 * polcoef(Ser(A)^(#A-1), k) )/(#A-1) ); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
Numbers k such that A013929(k+1) - A013929(k) = 2. In other words, the k-th nonsquarefree number is 2 less than the next nonsquarefree number.
+0
5
5, 6, 9, 19, 20, 21, 33, 34, 36, 49, 57, 58, 62, 63, 66, 76, 77, 88, 89, 91, 96, 97, 103, 104, 113, 114, 118, 119, 130, 131, 132, 136, 142, 149, 150, 161, 162, 174, 175, 187, 188, 189, 190, 201, 202, 206, 215, 217, 218, 225, 226, 231, 232, 245, 246, 249, 253
COMMENTS
The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
EXAMPLE
The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 2 after the fifth and sixth terms.
MATHEMATICA
Join@@Position[Differences[Select[Range[1000], !SquareFreeQ[#]&]], 2]
CROSSREFS
For nonprime numbers we appear to have A014689. A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A373409, A373573, A375927.
Numbers k such that A013929(k+1) - A013929(k) = 3. In other words, the k-th nonsquarefree number is 3 less than the next nonsquarefree number.
+0
5
3, 16, 23, 27, 31, 44, 46, 51, 55, 60, 68, 74, 79, 86, 95, 101, 105, 107, 112, 116, 121, 126, 129, 146, 147, 152, 159, 164, 167, 172, 177, 182, 185, 191, 195, 199, 204, 209, 220, 223, 229, 234, 237, 242, 244, 257, 262, 270, 275, 285, 286, 291, 299, 305, 312
COMMENTS
The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
EXAMPLE
The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 3 after the third term.
MATHEMATICA
Join@@Position[Differences[Select[Range[1000], !SquareFreeQ[#]&]], 3]
CROSSREFS
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A014689, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A373409, A375927.
Numbers k such that A013929(k+1) - A013929(k) = 4. In other words, the k-th nonsquarefree number is 4 less than the next nonsquarefree number.
+0
5
1, 4, 7, 11, 12, 13, 14, 22, 25, 26, 29, 32, 35, 39, 40, 41, 42, 50, 53, 54, 61, 64, 70, 71, 72, 75, 78, 81, 82, 83, 84, 87, 90, 98, 99, 102, 109, 110, 117, 120, 123, 124, 127, 135, 139, 140, 144, 151, 154, 155, 156, 157, 160, 163, 168, 169, 170, 173, 176, 179
COMMENTS
The difference of consecutive nonsquarefree numbers is at least 1 and at most 4, so there are four disjoint sequences of this type:
EXAMPLE
The initial nonsquarefree numbers are 4, 8, 9, 12, 16, 18, 20, 24, 25, which first increase by 4 after the first, fourth, and seventh terms.
MATHEMATICA
Join@@Position[Differences[Select[Range[100], !SquareFreeQ[#]&]], 4]
CROSSREFS
A053797 gives lengths of runs of nonsquarefree numbers, firsts A373199.
A375707 counts squarefree numbers between consecutive nonsquarefree numbers.
Cf. A007674, A014689, A029707, A049094, A061399, A068781, A072284, A110969, A120992, A294242, A375927.
Minimum of the n-th maximal anti-run of adjacent (increasing by more than one at a time) non-perfect-powers.
+0
5
2, 3, 6, 7, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 29, 30, 31, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 83, 84, 85, 86, 87, 88
COMMENTS
Non-perfect-powers ( A007916) are numbers with no proper integer roots. An anti-run of a sequence is an interval of positions at which consecutive terms differ by more than one.
EXAMPLE
The initial anti-runs are the following, whose minima are a(n):
(2)
(3,5)
(6)
(7,10)
(11)
(12)
(13)
(14)
(15,17)
(18)
(19)
(20)
(21)
(22)
(23)
(24,26,28)
MATHEMATICA
radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Min/@Split[Select[Range[100], radQ], #1+1!=#2&]//Most
CROSSREFS
For composite numbers we have A005381, runs A008864 (except first term). For anti-runs of non-perfect-powers:
For runs of non-perfect-powers:
Numbers k such that A002808(k+1) = A002808(k) + 1. In other words, the k-th composite number is 1 less than the next.
+0
5
3, 4, 7, 8, 11, 12, 14, 15, 16, 17, 20, 21, 22, 23, 25, 26, 29, 30, 32, 33, 34, 35, 37, 38, 39, 40, 43, 44, 45, 46, 48, 49, 52, 53, 54, 55, 57, 58, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 72, 73, 76, 77, 80, 81, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94
EXAMPLE
The composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, ... which increase by 1 after positions 3, 4, 7, 8, ...
MATHEMATICA
Join@@Position[Differences[Select[Range[100], CompositeQ]], 1]
PROG
(Python)
from sympy import primepi
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 f(x): return n+bisection(lambda y:primepi(x+2+y))-2
(Python) # faster for initial segment of sequence
from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
pic, prevc = 0, -1
for i in count(4):
if not isprime(i):
if i == prevc + 1:
yield pic
pic, prevc = pic+1, i
CROSSREFS
First differences are A373403 (except first). The version for non-perfect-powers is A375740. The version for nonprime numbers is A375926. A046933 counts composite numbers between primes.
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Last modified September 21 04:06 EDT 2024. Contains 376079 sequences. (Running on oeis4.)
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