A014963 -id:A014963 - cp's OEIS frontend

A100995 If n is a prime power p^m, m >= 1, then m, otherwise 0.

0, 1, 1, 2, 1, 0, 1, 3, 2, 0, 1, 0, 1, 0, 0, 4, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 5, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 6, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 4, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Reinhard Zumkeller, Nov 26 2004

nonn

Calculate matrix powers: (A175992^1)/1 - (A175992^2)/2 + (A175992^3)/3 - (A175992^4)/4 + ... Then the nonzero values of a(n) are found as reciprocals in the first column. Compare this to the Taylor series for log(1+x) = (x)/1 - (x^2)/2 + (x^3)/3 - (x^4)/4 + ... Therefore it is natural to write 0, 1/1, 1/1, 1/2, 1/1, 0, 1/1, 1/3, 1/2, 0, 1/1, ... Raising n to a such power gives A014963. - Mats Granvik, Gary W. Adamson, Apr 04 2011
The Dirichlet series that generates the reciprocals of this sequence is the logarithm of the Riemann zeta function. - Mats Granvik, Gary W. Adamson, Apr 04 2011
Number of automorphisms of the finite field with n elements, or 0 if the field does not exist. For n=p^k where p is a prime and k is an integer, the automorphism group of the finite field with n elements is a cyclic group of order k generated by the Frobenius endomorphism. - Yancheng Lu, Jan 11 2021

Daniel Forgues, Table of n, a(n) for n = 1..100000

Cf. A028233, A069513, A010055.

a100995 n = f 0 n where
   f e 1 = e
   f e x = if r > 0 then 0 else f (e + 1) x'
           where (x', r) = divMod x p
   p = a020639 n
-- Reinhard Zumkeller, Mar 19 2013

f:= proc(n) local F;
    F:= ifactors(n)[2];
    if nops(F) = 1 then F[1][2]
    else 0
    fi
end proc:
map(f, [$1..100]); # Robert Israel, Jun 09 2015

ppm[n_]:=If[PrimePowerQ[n],FactorInteger[n][[1,2]],0]; Array[ppm,110] (* Harvey P. Dale, Mar 03 2014 *)
a=Table[Limit[Sum[If[Mod[n, k] == 0, MoebiusMu[n/k]/(n/k)^(s - 1)/(1 - 1/n^(s - 1)), 0], {k, 1, n}], s -> 1], {n, 1, 105}];
Numerator[a]*Denominator[a] (* Mats Granvik, Jun 09 2015 *)
a = FullSimplify[Table[MangoldtLambda[n]/Log[n], {n, 1, 105}]]
Numerator[a]*Denominator[a] (* Mats Granvik, Jun 09 2015 *)

{a(n) = my(t); if( n<1, 0, t = factor(n); if( [1,2] == matsize(t), t[1,2], 0))} /* Michael Somos, Aug 15 2012 */

{a(n) = my(t); if( n<1, 0, if( t = isprimepower(n), t))} /* Michael Somos, Aug 15 2012 */

A100994(n) = A014963(n)^a(n);
a(A000961(n)) = A025474(n).
a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(n/d) * bigomega(d). - Ilya Gutkovskiy, Apr 15 2021

Edited by Daniel Forgues and N. J. A. Sloane, Aug 18 2009

A004094 Powers of 2 written backwards.

Original entry on oeis.org

1, 2, 4, 8, 61, 23, 46, 821, 652, 215, 4201, 8402, 6904, 2918, 48361, 86723, 63556, 270131, 441262, 882425, 6758401, 2517902, 4034914, 8068838, 61277761, 23445533, 46880176, 827712431, 654534862, 219078635, 4281473701, 8463847412, 6927694924, 2954399858, 48196897171

Offset: 0

N. J. A. Sloane

Freeman Dyson believes that A014963(a(n)) <> 5 is true but cannot be proved, see link. - Reinhard Zumkeller, Jan 05 2005

T. D. Noe, Table of n, a(n) for n = 0..1000
Edge Foundation, Annual Question 2005
Richard Lipton, More on testing Dyson's conjecture (2014)
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.

Cf. A014963, A102382, A102383, A102384, A102385.
The following are parallel families: A000079 (2^n), A004094 (2^n reversed), A028909 (2^n sorted up), A028910 (2^n sorted down), A036447 (double and reverse), A057615 (double and sort up), A263451 (double and sort down); A000244 (3^n), A004167 (3^n reversed), A321540 (3^n sorted up), A321539 (3^n sorted down), A163632 (triple and reverse), A321542 (triple and sort up), A321541 (triple and sort down).
Cf. A004086 (read n backwards).
For indices of primes see A057708.

a004094 = a004086 . a000079  -- Reinhard Zumkeller, Apr 02 2014

[Seqint(Reverse(Intseq(2^n))): n in [0..35]]; // Vincenzo Librandi, Jan 22 2020

a:= n-> (s-> parse(cat(s[-i]$i=1..length(s))))(""||(2^n)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 21 2020

Table[FromDigits[Reverse[IntegerDigits[2^n]]], {n, 0, 35}] (* Vincenzo Librandi, Jan 22 2020 *)

rev(n)=subst(Polrev(digits(n)),'x,10)
a(n)=rev(2^n) \\ Charles R Greathouse IV, Oct 20 2014

apply( {A004094(n)=fromdigits(Vecrev(digits(2^n)))}, [0..44]) \\ M. F. Hasler, Feb 18 2021

def A004094(n):
    return int(str(2**n)[::-1]) # Chai Wah Wu, Feb 19 2021

a(n) = A004086(A000079(n)). - Reinhard Zumkeller, Apr 02 2014

More terms from Reinhard Zumkeller, Jan 05 2005

A069513 Characteristic function of the prime powers p^k, k >= 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Benoit Cloitre, Apr 15 2002

Also, number of Galois fields of order n. - Charles R Greathouse IV, Mar 12 2008

Also, number of abelian indecomposable groups of order n. - Kevin Lamoreau, Mar 13 2023

Daniel Forgues, Table of n, a(n) for n = 1..100000.

Cf. A246655, A010055, A001221, A001222, A008683, A090751, A000688, A361414.
The partial sums of this sequence give A025528. - Daniel Forgues, Mar 02 2009
Cf. A014963, A003418. - Enrique Pérez Herrero, Jun 01 2011

a069513 1 = 0
a069513 n = a010055 n  -- Reinhard Zumkeller, Mar 19 2013

A069513 := proc(n)
    if n = 1 then
        0 ;
    elif A001221(n) > 1 then
        0;
    else
        1 ;
    end if ;
end proc:
seq(A069513(n),n=1..80) ; # R. J. Mathar, Nov 02 2016

A069513[n_]:=Boole[PrimeNu[n]==1]; A069513/@Range[20] (* Enrique Pérez Herrero, May 30 2011 *)

PARI
```
for(n=1,120,print1(omega(n)==1,","))
```

from sympy import primefactors
def A069513(n): return int(len(primefactors(n)) == 1) # Chai Wah Wu, Mar 31 2023

If n >= 2, a(n) = A010055(n).
a(n) = Sum_{d|n} bigomega(d)*mu(n/d); equivalently, Sum_{d|n} a(d) = bigomega(n); equivalently, Möbius transform of bigomega(n).
Dirichlet g.f.: ppzeta(s). Here ppzeta(s) = Sum_{p prime} Sum_{k>=1} 1/(p^k)^s. Note that ppzeta(s) = Sum_{p prime} 1/(p^s - 1) = Sum_{k>=1} primezeta(k*s). - Franklin T. Adams-Watters, Sep 11 2005
a(n) = floor(1/A001221(n)), for n > 1. - Enrique Pérez Herrero, Jun 01 2011
a(n) = - Sum_{d|n} mu(d)*bigomega(d), where bigomega = A001222. - Ridouane Oudra, Oct 29 2024
a(n) = - Sum_{d|n} mu(d)*omega(d), where omega = A001221. - Ridouane Oudra, Jul 30 2025

Moved original definition to formula line. Used comment (that I previously added) as definition. - Daniel Forgues, Mar 08 2009

Edited by Franklin T. Adams-Watters, Nov 02 2009

A025473 a(1) = 1; for n > 1, a(n) = prime root of n-th prime power (A000961).

Original entry on oeis.org

1, 2, 3, 2, 5, 7, 2, 3, 11, 13, 2, 17, 19, 23, 5, 3, 29, 31, 2, 37, 41, 43, 47, 7, 53, 59, 61, 2, 67, 71, 73, 79, 3, 83, 89, 97, 101, 103, 107, 109, 113, 11, 5, 127, 2, 131, 137, 139, 149, 151, 157, 163, 167, 13, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

David W. Wilson, Dec 11 1999

This sequence is related to the cyclotomic sequences A013595 and A020500, leading to the procedure used in the Mathematica program. - Roger L. Bagula, Jul 08 2008
"LCM numeral system": a(n+1) is radix for index n, n >= 0; a(-n+1) is 1/radix for index n, n < 0. - Daniel Forgues, May 03 2014
This is the LCM-transform of A000961; same as A014963 with all 1's (except a(1)) removed. - David James Sycamore, Jan 11 2024

Paul J. McCarthy, Algebraic Extensions of Fields, Dover books, 1976, pages 40, 69

David Wasserman, Table of n, a(n) for n = 1..1000
OEIS Wiki, LCM numeral system
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A013595, A020500, A025476.

Cf. A000961, A014963, A051451.

a025473 = a020639 . a000961 -- Reinhard Zumkeller, Aug 14 2013

cvm := proc(n, level) local f,opf; if n < 2 then RETURN() fi;
f := ifactors(n); opf := op(1,op(2,f)); if nops(op(2,f)) > 1 or
op(2,opf) <= level then RETURN() fi; op(1,opf) end:
A025473_list := n -> [1,seq(cvm(i,0),i=1..n)];
A025473_list(240); # Peter Luschny, Sep 21 2011

a = Join[{1}, Flatten[Table[If[PrimeQ[Apply[Plus, CoefficientList[Cyclotomic[n, x], x]]], Apply[Plus, CoefficientList[Cyclotomic[n, x], x]], {}], {n, 1, 1000}]]] (* Roger L. Bagula, Jul 08 2008 *)
Join[{1}, First@ First@# & /@ FactorInteger@ Select[Range@ 240, PrimePowerQ]] (* Robert G. Wilson v, Aug 17 2017 *)

print1(1); for(n=2,1e3, if(isprimepower(n,&p), print1(", "p))) \\ Charles R Greathouse IV, Apr 28 2014

from sympy import primepi, integer_nthroot, primefactors
def A025473(n):
    if n == 1: return 1
    def f(x): return int(n+x-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 primefactors(m)[0] # Chai Wah Wu, Aug 15 2024

def A025473_list(n) :
    R = [1]
    for i in (2..n) :
        if i.is_prime_power() :
            R.append(prime_divisors(i)[0])
    return R
A025473_list(239) # Peter Luschny, Feb 07 2012

a(n) = A006530(A000961(n)) = A020639(A000961(n)). - David Wasserman, Feb 16 2006
From Reinhard Zumkeller, Jun 26 2011: (Start)
A000961(n) = a(n)^A025474(n).
A056798(n) = a(n)^(2*A025474(n)).
A192015(n) = A025474(n)*a(n)^(A025474(n)-1). (End)
a(1) = A051451(1) ; for n > 1, a(n) = A051451(n)/A051451(n-1). - Peter Munn, Aug 11 2024

Offset corrected by David Wasserman, Dec 22 2008

A135506 a(n) = x(n+1)/x(n) - 1 where x(1)=1 and x(k) = x(k-1) + lcm(x(k-1),k). Here x(n) = A135504(n).

Original entry on oeis.org

2, 1, 2, 5, 1, 7, 1, 1, 5, 11, 1, 13, 1, 5, 1, 17, 1, 19, 1, 1, 11, 23, 1, 5, 13, 1, 1, 29, 1, 31, 1, 11, 17, 1, 1, 37, 1, 13, 1, 41, 1, 43, 1, 1, 23, 47, 1, 1, 1, 17, 13, 53, 1, 1, 1, 1, 29, 59, 1, 61, 1, 1, 1, 13, 1, 67, 1, 23, 1, 71, 1, 73, 1, 1, 1, 1, 13, 79, 1, 1, 41, 83, 1, 1, 43, 29, 1, 89

Offset: 1

Benoit Cloitre, Feb 09 2008

nonn

This sequence has properties related to primes. For instance: terms consist of 1's or primes only; if 3 never occurs, any prime p occurs finitely many times.
All prime numbers 'p' from the sequence A014963(n), which equals A003418(n+1)/A003418(n), are in a(n-1) = p. - Eric Desbiaux, Jan 11 2015
For any prime p > 3, a(p-1) = p. Also a(n) is not 3 for any n. All terms but a(1) and a(3) are odd, and probably all of them are not composite numbers; this is strongly related to a strong version of Linnik's Theorem (see Ruiz-Cabello link). - Serafín Ruiz-Cabello, Sep 30 2015
Per the prior comment, the distinct prime terms correspond to A045344. This is every prime except for 3. - Bill McEachen, Sep 12 2022

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 10000 terms from Robert Israel)
Eric S. Rowland, Prime-Generating Recurrences and a Tale of Logarithmic Scale, YouTube video, 2023. (See especially the last section beginning at 20:08).
Serafín Ruiz-Cabello, On the use of the lowest common multiple to build a prime-generating recurrence, arXiv:1504.05041 [math.CO], 2015.

Cf. A045344, A135504, A361460, A361461 (positions of 1's), A361462 [= a(n) mod 4], A361463, A361464, A361470.

Cf. also A106108.

x[1]:= 1;
for n from 2 to 101 do
  x[n]:= x[n-1] + ilcm(x[n-1],n);
  a[n-1]:= x[n]/x[n-1]-1;
od:
seq(a[n],n=1..100); # Robert Israel, Jan 11 2015

a[n_] := x[n+1]/x[n] - 1; x[1] = 1; x[k_] := x[k] = x[k-1] + LCM[x[k-1], k]; Table[a[n], {n, 1, 88}] (* Jean-François Alcover, Jan 08 2013 *)

x1=1;for(n=2,40,x2=x1+lcm(x1,n);t=x1;x1=x2;print1(x2/t-1,","))

from itertools import count, islice
from math import lcm
def A135506_gen(): # generator of terms
    x = 1
    for n in count(2):
        y = x+lcm(x,n)
        yield y//x-1
        x = y
A135506_list = list(islice(A135506_gen(),20)) # Chai Wah Wu, Mar 13 2023

a(n) = A135504(n+1)/A135504(n) - 1.

a(n) = (n+1) / A361470(n). - Antti Karttunen, Mar 26 2023

References to A135504 added by Antti Karttunen, Mar 07 2023

A089026 a(n) = n if n is a prime, otherwise a(n) = 1.

Original entry on oeis.org

1, 2, 3, 1, 5, 1, 7, 1, 1, 1, 11, 1, 13, 1, 1, 1, 17, 1, 19, 1, 1, 1, 23, 1, 1, 1, 1, 1, 29, 1, 31, 1, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 1, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 1, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 1, 1, 83, 1, 1, 1, 1, 1, 89, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Roger L. Bagula, Nov 12 2003

nonn

This sequence was the subject of the 1st problem of the 9th Irish Mathematical Olympiad 1996 with gcd((n + 1)!, n! + 1) = a(n+1) for n >= 0 (see formula Jan 23 2009 and link). - Bernard Schott, Jul 22 2020

For sequence A with terms a(1), a(2), a(3),... , let R(0) = 1 and for k >= 1 let R(k) = rad(a(1)*a(2)*...*a(k)). Define the Rad-transform of A to be R(n)/R(n-1); n >= 1, where rad is A007947. Then this sequence is the Rad transform of the positive integers, A = A000027. - David James Sycamore, Apr 19 2024

			From Larry Tesler (tesler(AT)pobox.com), Nov 08 2010: (Start)
a(9) = (8*9*10)/(2^((5+2+1)-(3+1+0))*3^((3+1)-(2+0))*5^((2)-(1))*7^((1)-(1))) = 1 [composite].
a(10) = (8*9*10)/(2^((5+2+1)-(3+1+0))*3^((3+1)-(2+0))*5^((2)-(1))*7^((1)-(1))) = 1 [composite].
a(11) = (8*9*10*11*12)/(2^((6+3+1)-(3+1+0))*3^((4+1)-(2+0))*5^((2)-(1))*7^((1)-(1))) = 11 [prime]. (End)

Paulo Ribenboim, The little book of big primes, Springer 1991, p. 106.
L. Tesler, "Factorials and Primes", Math. Bulletin of the Bronx H.S. of Science (1961), 5-10. [From Larry Tesler (tesler(AT)pobox.com), Nov 08 2010]

G. C. Greubel, Table of n, a(n) for n = 1..5000
The IMO Compendium, Problem 1, 9th Irish Mathematical Olympiad 1996.
Index to sequences related to Olympiads.

Differs from A080305 at n=30.
Cf. A090585, A000217, A069268, A090586, A007619, A007406.
Cf. A061397, A135683.
Cf. A000027, A007947.

a = [1:96]; a(isprime(a) == false) = 1; % Thomas Scheuerle, Oct 06 2022

[IsPrime(n) select n else 1: n in [1..96]]; // Marius A. Burtea, Aug 02 2019

digits=200; a=Table[If[PrimePi[n]-PrimePi[n-1]>0, n, 1], {n, 1, digits}]; Table[Numerator[(n/2)/(n-1)! ] + Floor[2/n] - 2*Floor[1/n], {n,1,200}] (* Alexander Adamchuk, May 20 2006 *)
Range@ 120 /. k_ /; CompositeQ@ k -> 1 (* or *)
Table[n Boole@ PrimeQ@ n, {n, 120}] /. 0 -> 1 (* or *)
Table[If[PrimeQ@ n, n, 1], {n, 120}] (* Michael De Vlieger, Jul 02 2016 *)

a(n) = n^isprime(n) \\ David A. Corneth, Oct 06 2022

from sympy import isprime
def a(n): return n if isprime(n) else 1
print([a(n) for n in range(1, 97)]) # Michael S. Branicky, Oct 06 2022

def A089026(n):
    if n == 4: return 1
    f = factorial(n-1)
    return (f + 1) - n*(f//n)
[A089026(n) for n in (1..96)]   # Peter Luschny, Oct 16 2013

From Peter Luschny, Nov 29 2003: (Start)
a(n) = denominator(n! * Sum_{m=0..n} (-1)^m*m!*Stirling2(n+1, m+1)/(m+1)).
a(n) = denominator(n! * Sum_{m=0..n} (-1)^m*m!*Stirling2(n, m)/(m+1)). (End)
From Alexander Adamchuk, May 20 2006: (Start)
a(n) = numerator((n/2)/(n-1)!) + floor(2/n) - 2*floor(1/n).
a(n) = A090585(n-1) = A000217(n-1)/A069268(n-1) for n>2. (End)
a(n) = gcd(n,(n-1)!+1). - Jaume Oliver Lafont, Jul 17 2008, Jan 23 2009
a(1) = 1, a(2) = 2, then a(n) = 1 or a(n) = n = prime(m) = (Product q+k, k = 1 .. 2*floor(n/2+1)-q) / (Product prime(i)^(Sum (floor((n+1)/(prime(i)^w)) - floor(q/(prime(i)^w)) ), w = 1 .. floor(log[base prime(i)] n+1) ), i = 2 .. m-1) where q = prime(m-1). -  Larry Tesler (tesler(AT)pobox.com), Nov 08 2010
a(n) = (n!*HarmonicNumber(n) mod n)+1, n != 4. - Gary Detlefs, Dec 03 2011
a(n) = denominator of (n!)/n^(3/2). - Arkadiusz Wesolowski, Dec 04 2011
a(n) = A034386(n+1)/A034386(n). - Eric Desbiaux, May 10 2013
a(n) = n^c(n), where c = A010051. - Wesley Ivan Hurt, Jun 16 2013
a(n) = A014963(n)^(-A008683(n)). - Mats Granvik, Jul 02 2016
Conjecture: for n > 3, a(n) = gcd(n, A007406(n-1)). - Thomas Ordowski, Aug 02 2019
a(n) = 1 + c(n)*(n-1), where c = A010051. - Wesley Ivan Hurt, Jun 21 2025

A373671 Length of the n-th maximal antirun of prime-powers.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 7, 26, 27, 1007, 5558, 5734, 31209

Offset: 1

Gus Wiseman, Jun 14 2024

An antirun of a sequence (in this case A000961 without 1) is an interval of positions at which consecutive terms differ by more than one.

			The maximal antiruns of prime-powers begin:
   2
   3
   4
   5   7
   8
   9  11  13  16
  17  19  23  25  27  29  31

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

For prime antiruns we have A027833.
For nonsquarefree runs we have A053797, firsts A373199.
For non-prime-powers runs we have A110969, firsts A373669, sorted A373670.
For squarefree runs we have A120992.
For prime-power runs we have A174965.
For prime runs we have A175632.
For composite runs we have A176246, firsts A073051, sorted A373400.
For squarefree antiruns we have A373127, firsts A373128.
For composite antiruns we have A373403.
For antiruns of prime-powers:
- length A373671 (this sequence)
- min A120430
- max A006549
For antiruns of non-prime-powers:
- length A373672
- min A373575
- max A255346
A000961 lists the powers of primes (including 1).
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists the non-prime-powers (not including 1 A024619).
Cf. A000040, A001359, A008864, A014963, A038664, A054265, A067774, A251092, A373401.

Length/@Split[Select[Range[100],PrimePowerQ[#]&],#1+1!=#2&]//Most

Partial sums are A025528(A006549(n)).

A373672 Length of the n-th maximal antirun of non-prime-powers.

Original entry on oeis.org

5, 3, 1, 6, 1, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 3, 2, 2, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 1, 3, 1, 2, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 1

Gus Wiseman, Jun 14 2024

nonn

An antirun of a sequence (in this case A361102 or A024619 with 1) is an interval of positions at which consecutive terms differ by more than one.

			The maximal antiruns of non-prime-powers begin:
   1   6  10  12  14
  15  18  20
  21
  22  24  26  28  30  33
  34
  35
  36  38
  39
  40  42  44
  45
  46  48  50

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

For prime antiruns we have A027833.
For nonsquarefree runs we have A053797, firsts A373199.
For non-prime-powers runs we have A110969, firsts A373669, sorted A373670.
For squarefree runs we have A120992.
For prime-power runs we have A174965.
For prime runs we have A175632.
For composite runs we have A176246, firsts A073051, sorted A373400.
For squarefree antiruns we have A373127, firsts A373128.
For composite antiruns we have A373403.
For antiruns of prime-powers:
- length A373671
- min A120430
- max A006549
For antiruns of non-prime-powers:
- length A373672 (this sequence), firsts (3,7,2,25,1,4)
- min A373575
- max A255346
A000961 lists all powers of primes. A246655 lists just prime-powers.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A356068 counts non-prime-powers up to n.
A361102 lists all non-prime-powers (A024619 if not including 1).
Cf. A001359, A008864, A014963, A038664, A054265, A067774, A251092, A373401.

Length/@Split[Select[Range[100],!PrimePowerQ[#]&],#1+1!=#2&]//Most

Partial sums are A356068(A255346(n)).

A373576 Sums of maximal antiruns of prime-powers.

Original entry on oeis.org

2, 3, 4, 12, 8, 49, 171, 2032, 5157, 3997521, 199713082, 561678378, 10122001905, 109934112352390774

Offset: 1

Gus Wiseman, Jun 17 2024

An antirun of a sequence (in this case A246655) is an interval of positions at which consecutive terms differ by more than one.

			The maximal antiruns of powers of primes begin:
   2
   3
   4
   5   7
   8
   9  11  13  16
  17  19  23  25  27  29  31

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

See link for composite, prime, nonsquarefree, and squarefree runs/antiruns.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Non-prime-power runs: A373678, min A373676, max A373677, length A110969.
Prime-power antiruns: A373576 (this sequence), min A120430, max A006549, length A373671.
Non-prime-power antiruns: A373679, min A373575, max A255346, length A373672.
A000040 lists the primes, differences A001223.
A000961 lists all powers of primes. A246655 lists just prime-powers.
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A356068 counts non-prime-powers up to n.
A361102 lists all non-prime-powers (A024619 if not including 1).
Cf. A001359, A014963, A027833, A029707, A038664, A067774, A251092, A293697, A371201, A373401, A373669.

Total/@Split[Select[Range[1000],PrimePowerQ],#1+1!=#2&]//Most

a(14) from Giorgos Kalogeropoulos, Jun 18 2024

A373675 Sums of maximal runs of powers of primes A000961.

Original entry on oeis.org

15, 24, 11, 13, 33, 19, 23, 25, 27, 29, 63, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 255, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Gus Wiseman, Jun 16 2024

nonn

A run of a sequence (in this case A000961) is an interval of positions at which consecutive terms differ by one.

			The maximal runs of powers of primes begin:
   1   2   3   4   5
   7   8   9
  11
  13
  16  17
  19
  23
  25
  27
  29
  31  32
  37
  41
  43
  47
  49

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

A000040 lists the primes, differences A001223.
A000961 lists all powers of primes (A246655 if not including 1).
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists all non-prime-powers (A024619 if not including 1).
See link for composite, prime, nonsquarefree, and squarefree runs.
Prime-power runs: A373675, min A373673, max A373674, length A174965.
Non-prime-power runs: A373678, min A373676, max A373677, length A110969.
Prime-power antiruns: A373576, min A120430, max A006549, length A373671.
Non-prime-power antiruns: A373679, min A373575, max A255346, length A373672.
Cf. A005117, A014963, A029707, A038664, A046933, A054265, A065855, A071148, A293697, A371201.

pripow[n_]:=n==1||PrimePowerQ[n];
Total/@Split[Select[Range[nn],pripow],#1+1==#2&]//Most

cp's OEIS Frontend

A100995 If n is a prime power p^m, m >= 1, then m, otherwise 0.

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A004094 Powers of 2 written backwards.

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A069513 Characteristic function of the prime powers p^k, k >= 1.

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A025473 a(1) = 1; for n > 1, a(n) = prime root of n-th prime power (A000961).

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A135506 a(n) = x(n+1)/x(n) - 1 where x(1)=1 and x(k) = x(k-1) + lcm(x(k-1),k). Here x(n) = A135504(n).

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A089026 a(n) = n if n is a prime, otherwise a(n) = 1.