A000961 -id:A000961 - cp's OEIS frontend

A068315 For numbers k such that A025474(k) > 1 and A025474(k+1) > 1, sequence gives A000961(k).

8, 25, 121, 2187, 32761

Offset: 1

Naohiro Nomoto, Mar 08 2002

Equivalently, prime powers (either A000961 or A246655) q such that q and the next prime power are both composite numbers. - Paolo Xausa, Oct 25 2023

			The interval (121,122,123,124,125) contains no primes, so 121 is in the sequence. - _Gus Wiseman_, Dec 24 2024

Bisection of A068435.
For perfect powers instead of prime powers we have A116086, indices A274605.
The position of a(k) in the prime powers A246655 is A379156(k).
For just one prime we have A379157, indices A379155.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A031218 gives the greatest prime power <= n.
A046933 gives run-lengths of composites between primes.
A065514 gives the greatest prime power < prime(n), difference A377289.
A246655 lists the prime powers, differences A057820.
A366833 counts prime powers between primes, see A053607, A304521.
A366835 counts primes between prime powers.
Cf. A025474, A067871, A080101, A080769, A175106, A345531, A377281, A377287, A378368.

With[{upto=33000},Map[First,Select[Partition[Select[Range[upto],PrimePowerQ],2,1],NoneTrue[#,PrimeQ]&]]] (* Paolo Xausa, Oct 25 2023 *)

a(n) = A246655(A379156(n)). - Gus Wiseman, Dec 24 2024

Definition corrected by Jinyuan Wang, Sep 05 2020

A097621 In canonical prime factorization of n replace p^e with its index in A000961.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 7, 8, 10, 9, 12, 10, 12, 15, 11, 12, 16, 13, 20, 18, 18, 14, 21, 15, 20, 16, 24, 17, 30, 18, 19, 27, 24, 30, 32, 20, 26, 30, 35, 21, 36, 22, 36, 40, 28, 23, 33, 24, 30, 36, 40, 25, 32, 45, 42, 39, 34, 26, 60, 27, 36, 48, 28, 50, 54, 29, 48, 42, 60, 30, 56, 31

Offset: 1

Reinhard Zumkeller, Aug 17 2004

The definition of the sequence has been corrected, given that it uses A095874, the indices in the list A000961 of "powers of primes" starting with A000961(1) = 1, rather than A322981, index of p^e in the list of prime powers A246655, as written in the original definition. See A333235 for the variant of this sequence which uses A322981 and A246655 instead, maybe the more natural choice given that the factorization of integers consists of prime powers > 1. - M. F. Hasler, Jun 15 2021

			n=600 = 2^3 * 3 * 5^2 -> A095874(8)*A095874(3)*A095874(25) = 7 * 3 * 15 = 315.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000961 (powers of primes), A246655 (prime powers), A003963, A018266, A095874 (index of n = p^e in A000961).
Cf. A097622, A097623, A097624.
Cf. A322981 (index of n = p^e in A246655), A333235 (variant of this sequence).

N:= 1000: # to get a(1) to a(N)
Primes:= select(isprime,[2, seq(2*i+1,i=1..(N-1)/2)]):
PP:= sort([1,seq(seq(p^k, k=1..floor(log[p](N))),p=Primes)]):
for n from 1 to nops(PP) do B[PP[n]]:= n od:
seq(mul(B[f[1]^f[2]],f=ifactors(n)[2]),n=1..N); # Robert Israel, Sep 02 2015

pp = Select[Range@100, Length[FactorInteger[#]] == 1 &]; a = Table[Times @@ (Position[pp, #][[1, 1]] & /@ (#[[1]]^#[[2]] & /@ FactorInteger[n])), {n, 73}] (* Ivan Neretin, Sep 02 2015 *)

f(n) = if(isprimepower(n), sum(i=1, logint(n, 2), primepi(sqrtnint(n, i)))+1, n==1); \\ A095874
a(n) = my(fr=factor(n)); for (k=1, #fr~, fr[k,1] = f(fr[k,1]^fr[k,2]); fr[k,2] = 1); factorback(fr); \\ Michel Marcus, May 29 2021
A097621(n)=vecprod([A095874(f[1]^f[2])|f<-factor(n)~]) \\ M. F. Hasler, Jun 15 2021

from math import prod
from sympy import primepi, integer_nthroot, factorint
def A097621(n): return prod(1+int(primepi(m:=p**e)+sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length()))) for p,e in factorint(n).items()) # Chai Wah Wu, Jan 19 2025

Multiplicative with: p^e -> A095874(p^e), p prime.
a(A000961(n)) = n; a(a(n)) = A097622(n); a(a(a(n))) = A097623(n);
a(n) <= n; a(n) = n iff 60 mod n = 0: a(A018266(n)) = A018266(n);
a(A097624(n)) = n and a(m) <> n for n < A097624(n).

Definition corrected by M. F. Hasler, Jun 16 2021

Example corrected by Ray Chandler, Jun 30 2021

A192015 Arithmetic derivative of prime powers: a(n) = A003415(A000961(n)).

Original entry on oeis.org

0, 1, 1, 4, 1, 1, 12, 6, 1, 1, 32, 1, 1, 1, 10, 27, 1, 1, 80, 1, 1, 1, 1, 14, 1, 1, 1, 192, 1, 1, 1, 1, 108, 1, 1, 1, 1, 1, 1, 1, 1, 22, 75, 1, 448, 1, 1, 1, 1, 1, 1, 1, 1, 26, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 405, 1, 1024, 1, 1, 1, 1, 1, 1, 1, 34

Offset: 1

Reinhard Zumkeller, Jun 26 2011

nonn

a(A000040(n)) = 1; a(A002808(n)) > 1;
A001787, A027471, A100484, A079705 and A051674 are subsequences;
A001787 and A024622 give record values and where they occur;
A192016(n) = A003415(a(n)).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Power

a192015 = a003415 . a000961  -- Reinhard Zumkeller, Apr 16 2014

Join[{0}, Reap[For[n = 1, n <= 300, n++, f = FactorInteger[n]; If[Length[f] == 1, Sow[n*Total[Apply[#2/#1&, f, {1}]]]]]][[2, 1]]] (* Jean-François Alcover, Feb 21 2014 *)

from sympy import primepi, integer_nthroot, factorint
def A192015(n):
    if n == 1: return 0
    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 sum((m*e//p for p,e in factorint(m).items())) # Chai Wah Wu, Aug 15 2024

a(n) = A025474(n) * A025473(n)^(A025474(n) - 1).

A024620 Positions of primes among the powers of primes (A000961).

Original entry on oeis.org

2, 3, 5, 6, 9, 10, 12, 13, 14, 17, 18, 20, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 85, 86, 88, 89, 90, 91, 93, 94

Offset: 1

Clark Kimberling

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Complement of A024621.
Cf. A001222 (bigomega), A025474, A056604, A027883.
Cf. A025528, A065515, A000040.

a024620 n = a024620_list !! (n-1)
a024620_list = filter ((== 1) . a025474) [1..]
-- Reinhard Zumkeller, May 01 2015

a[n_] := PrimeOmega[LCM @@ Range@Prime@n] + 1; Array[a, 100] (* Amiram Eldar, Dec 02 2018 *)

lista(nn) = my(powpr = select((i->((omega(i)==1) || (i==1))), [1..nn])); for (i = 1, #powpr, if (isprime(powpr[i]), print1(i, ", ")); ); \\ Michel Marcus, Jun 03 2021

from sympy import prime, primepi, integer_nthroot
def A024620(n):
    x = prime(n)
    return n+1+sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())) # Chai Wah Wu, Nov 05 2024

A025474(a(n)) = 1. - Reinhard Zumkeller, May 01 2015
a(n) = A001222(A056604(n)) + 1. - Eric Desbiaux, Dec 02 2018
From Ridouane Oudra, Oct 18 2020: (Start)
a(n) = A027883(n) + 1;
a(n) = A025528(A000040(n)) + 1;
a(n) = A065515(A000040(n)). (End)

A024622 Position of 2^n among the powers of primes (A000961).

Original entry on oeis.org

1, 2, 4, 7, 11, 19, 28, 45, 71, 118, 199, 341, 605, 1079, 1962, 3591, 6636, 12371, 23151, 43580, 82268, 155922, 296348, 564689, 1078556, 2064590, 3959000, 7605135, 14632961, 28195587, 54403836, 105102702, 203287170, 393625232, 762951923, 1480223717, 2874422304

Offset: 0

Clark Kimberling

nonn

Number of prime powers <= 2^n. - Jon E. Schoenfield, Nov 06 2016

A000961(a(n)) = A000079(n); also position of record values in A192015: A001787(n) = A192015(a(n)). - Reinhard Zumkeller, Jun 26 2011

Ray Chandler, Table of n, a(n) for n = 0..92 (using b-file from A007053, first 61 terms from Hiroaki Yamanouchi)

Cf. A000079, A000720, A000961, A001787, A007053, A025528, A182908, A192015.

{1}~Join~Flatten[1 + Position[Select[Range[10^6], PrimePowerQ], k_ /; IntegerQ@ Log2@ k ]] (* Michael De Vlieger, Nov 14 2016 *)

lista(nn) = {v = vector(2^nn, i, i); vpp = select(x->ispp(x), v); print1(1, ", "); for (i=1, #vpp, if ((vpp[i] % 2) == 0, print1(i, ", ")););} \\ Michel Marcus, Nov 17 2014

a(n)=sum(k=1,n,primepi(sqrtnint(2^n,k)))+1 \\ Charles R Greathouse IV, Nov 21 2014

a(n)=my(s=0);for(i=1, 2^n, isprimepower(i) && s++);s+1 \\ Dana Jacobsen, Mar 23 2021

use ntheory ":all"; for my $n (0..20) { my $s=1; is_prime_power($) && $s++ for 1..2**$n; print "$n $s\n" } # _Dana Jacobsen, Mar 23 2021

use ntheory ":all"; for my $n (0..64) { my $s = ($n < 1) ? 1 : vecsum(map{prime_count(rootint(powint(2,$n)-1,$))}1..$n)+2; print "$n $s\n"; } # _Dana Jacobsen, Mar 23 2021

# with b-file for pi(2^n)
perl -Mntheory=:all -nE 'my($n,$pc)=split; say "$n ", addint($pc,vecsum( map{prime_count(rootint(powint(2,$n),$))} 2..$n )+1);'  b007053.txt  # _Dana Jacobsen, Mar 23 2021

from sympy import primepi, integer_nthroot
def A024622(n):
    x = 1<Chai Wah Wu, Nov 05 2024

def a(n): return sum(prime_pi(ZZ(2^n).nth_root(k+1,truncate_mode=1)[0]) for k in range(n))+1 # Dana Jacobsen, Mar 23 2021

From Ridouane Oudra, Oct 26 2020: (Start)
a(n) = 1 + Sum_{i=1..n} pi(floor(2^(n/i))), where pi(n) = A000720(n);
a(n) = 1 + A182908(n). (End)
a(n) = A025528(2^n)+1. - Pontus von Brömssen, Sep 28 2024

a(28)-a(36) from Hiroaki Yamanouchi, Nov 21 2014

a(46)-a(53) corrected by Hiroaki Yamanouchi, Nov 15 2016

A335866 Number of classes of simple difference sets of the Singer type (m^2 + m + 1, m + 1, 1) with m = m(n) = A000961(n), for n >= 1.

Original entry on oeis.org

1, 2, 4, 2, 10, 12, 8, 12, 36, 40, 12, 102, 84, 156, 60, 84, 264, 220, 60, 264, 574, 420, 720, 252, 816, 1180, 768, 144, 840, 1704, 1200, 1176, 432, 2196, 2670, 2112, 3434, 2380, 3024, 2280, 3960, 1296, 1656, 3612, 672, 5764, 5184, 3984, 6120, 4368, 5512, 4752, 9352, 3120, 10034, 9204, 7176, 9360, 7128

Offset: 1

Wolfdieter Lang, Jul 26 2020

For details on these simple difference sets see A333852, with references, and a W. Lang link.
The formula given below was conjectured by Singer for n >= 2 on p. 383. See also the table on p. 384.
This conjecture was later proved by Berman.

			n = 2, m(2) = 2 = 2^1, a(2) = phi(7)/(3*1) = 6/3 = 2. There are two classes of type (7,3,1) (Fano plane), with representatives {0, 1, 3} and {0, 1, 5}. The two equivalence classes (by elementwise addition of 1, 2, ..., 6 modulo 7) are Dev({0, 1, 3}) = {{0, 1, 3}, {0, 2, 6}, {0, 4, 5}, {1, 2, 4}, {1, 5, 6}, {2, 3, 5}, {3, 4, 6}, and Dev({0, 1, 5}) = {{0, 1, 5}, {0, 2, 3}, {0, 4, 6}, {1, 2, 6}, {1, 3, 4}, {2, 4, 5}, {3, 5, 6}}.

Martin Becker, Table of n, a(n) for n = 1..20000
Gerald Berman, Finite projective plane geometries and difference sets, Trans. AMS, 74 (1953) 492-499.
James Singer, A Theorem in Finite Projective Geometry and Some Applications to Number Theory, Trans. AMS, 43 (1938) 377-385, Table on p. 384.

Cf. A000010, A025474, A000961, A333852, A335865.

print1(1); for(q=2, 193 , if(n=isprimepower(q), print1(", ", eulerphi(q^2+q+1)/(3*n)))) \\ Martin Becker, Jun 11 2024

a(1) = 1, and a(n) = phi(v(n))/(3*e(n)), with phi = A000010 (Euler's totient), v(n) = A335865(n) = m(n)^2 + m(n) + 1, with m(n) = A000961(n), and e(n) = A025474(n), the exponent of the prime power dividing m(n), for n >= 2.

A291785 Iterate the map A291784: k -> (psi(k)+phi(k))/2, starting with n, until a power of a prime (A000961) is reached, or -1 if that never happens.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 7, 8, 9, 11, 11, 16, 13, 16, 16, 16, 17, 23, 19, 23, 23, 23, 23, 47, 25, 27, 27, 47, 29, 47, 31, 32, 83, 83, 83, 83, 37, 47, 47, 47, 41, 83, 43, 47, -1, 47, 47, -1, 49, -1, 83, 83, 53, 83, -1, -1, 59, 59, 59, -1, 61, 83, 83, 64, 83, 83, 67, -1, -1, -1, 71, -1, 73

Offset: 1

N. J. A. Sloane, Sep 02 2017

sign

Primes and prime powers are fixed points under the map f(k) = (psi(k)+phi(k))/2. (If n = p^k, then psi(n) = p^k(1+1/p), phi(n) = p^k(1-1/p), and their average is p^k, so n is a fixed point under the map.)
Since f(n)>n if n is not a prime power, there can be no nontrivial cycles.
Wall (1985) observes that the trajectories of 45 and 50 are unbounded, so a(45) = a(50) = -1.
Also 48 and many more terms seem to have unbounded trajectories. - Hugo Pfoertner, Sep 03 2017.
Obviously any number in the trajectory of a number with unbounded trajectory (in particular that of 45, A291787) again has this property. A291788 is the union of all these. - M. F. Hasler, Sep 03 2017

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004. See Section B41, p. 147.

C. R. Wall, Unbounded sequences of Euler-Dedekind means, Amer. Math. Monthly, 92 (1985), 587.

Cf. A000010, A001615, A291784, A291786, A291787, A291788.

A291785(n,L=n)={for(i=0,L,isprimepower(n=A291784(n))&&return(n));(-1)^(n>1)} \\ The search limit L=n is only experimental but appears quite conservative w.r.t. known data, cf. A291786. The algorithm assumes that there are no cycles except for the powers of primes. - M. F. Hasler, Sep 03 2017

More terms from Hugo Pfoertner, Sep 03 2017

A302040 Numbers k such that A078898(k) is a power of 2; an analog for A000961 based on factorization-kind of process involving the sieve of Eratosthenes (A083221).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 29, 31, 32, 37, 41, 43, 45, 47, 49, 53, 55, 59, 61, 64, 67, 71, 73, 79, 83, 89, 91, 93, 97, 101, 103, 107, 109, 113, 115, 121, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 189, 191, 193, 197, 199, 203, 211, 223, 227, 229, 233, 235, 239, 241, 247, 251, 256, 257

Offset: 1

Antti Karttunen, Apr 02 2018

nonn

Numbers k for which A302041(k) < 2, or equally, for which A302044(k) = 1.
Sequence A250245(A000961(k)) sorted into ascending order, or in other words, numbers k such that A250246(k) is a prime power (in A000961).
Numbers k such that all terms in iteration sequence k, A302042(k), A302042(A302042(k)), A302042(A302042(A302042(k))), ..., have an equal smallest prime factor (A020639) before the sequence settles to 1, in other words, that they all stay on the same row of A083221. This also forces the column position of each (A078898) to be a power of 2 (A000079).

			For k = 21 = 3*7, the smallest prime factor is 3. A302042(21) = 9, and A302042(9) = 3, both (9 and 3) which also have 3 as their smallest prime factor, and after that the sequence settles to 1, as A302042(3) = 1, thus 21 is included in this sequence.
For k = 27 = 3*3*3, the smallest prime factor is 3. However, A302042(27) = 7, thus 27 is not included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..6883
Index entries for sequences generated by sieves

Cf. A000961, A078898, A302036, A302041, A302044, A302053.

Cf. A000040, A000079, A001248 (subsequences).

for(n=1,257,if(2>A302041(n),print1(n,","))); \\ Other code as in A302041.

A335865 Moduli a(n) = v(n) for the simple difference sets of Singer type of order m(n) (v(n), m(n)+1, 1) in the additive group modulo v(n) = m(n)^2 + m(n) + 1, with m(n) = A000961(n).

Original entry on oeis.org

3, 7, 13, 21, 31, 57, 73, 91, 133, 183, 273, 307, 381, 553, 651, 757, 871, 993, 1057, 1407, 1723, 1893, 2257, 2451, 2863, 3541, 3783, 4161, 4557, 5113, 5403, 6321, 6643, 6973, 8011, 9507, 10303, 10713, 11557, 11991, 12883, 14763, 15751

Offset: 1

Wolfdieter Lang, Jul 26 2020

For details on these difference sets see A333852, with references, and a W. Lang link.
Because these simple difference sets of Singer type of order m = m(n) in the addive group (Z_{v(n)}, +) = RS(v(n)) = {0, 1, ..., v(n)-1} are also simple symmetric balanced incomplete block designs (BIBD), the number of blocks b(n) is also v(n) = a(n). This is the number of simple difference sets of each of the A335865(n) classes.
From Ed Pegg Jr, May 16 2019, edited by Hugo Pfoertner, May 13 2024: (Start)
(n^2+n+1,n+1) difference sets exist when n is a prime power.
(7,3), (1,2,4)
(13,4), (0,1,3,9)
(21,5), (3,6,7,12,14) (A095029)
(31,6), (1,5,11,24,25,27) (A095030)
(57,8), (0,1,6,15,22,26,45,55) (A095032)
(73,9), (0,1,12,20,26,30,33,35,57) (A095035)
(91,10), (0,2,6,7,18,21,31,54,63,71) (A095036)
(133,12), (1,10,11,13,27,31,68,75,83,110,115,121) (A095038)
(183,14), (1,13,20,21,23,44,61,72,77,86,90,116,122,169) (A095040) (End)
Is a(n) = A138077(n-1)? - R. J. Mathar, Sep 11 2020

			n = 2, m(2) = 2, a(2) = 2^2 + 2 + 1 = 7. The simple Singer difference set of order 2 is denoted by (7, 3, 1) (Fano plane). There are two classes (A335866(2) = 2) obtained from the representative difference sets {0, 1, 3} and {0, 1, 5} by element-wise addition of 1, 2, ..., 6 taken modulo 7. Each class consists of 7 simple difference sets.

Dan Gordon, Difference Sets

Cf. A000961, A333852, A335866, A138077.

a(n) = m(n)^2 + m(n) + 1 , with m(n) = A000961(n), for n >= 1.

Comments about difference sets moved from A138077 to here by Max Alekseyev, Apr 05 2022

A344975 Numbers k such that A011772(k) divides A344875(k), but k is not a power of prime (in A000961).

Original entry on oeis.org

6, 10, 18, 21, 24, 26, 28, 34, 36, 39, 40, 50, 55, 57, 58, 68, 74, 75, 78, 82, 93, 96, 98, 100, 106, 111, 120, 122, 129, 136, 146, 147, 150, 155, 162, 164, 171, 178, 183, 194, 196, 201, 202, 203, 205, 218, 219, 222, 224, 226, 237, 242, 250, 253, 274, 288, 291, 292, 294, 298, 300, 301, 305, 309, 314, 324, 327, 333

Offset: 1

Antti Karttunen, Jun 04 2021

nonn

This is not a subsequence of A344882. The first terms of this sequence that do not occur there are: 900, 1260, 1560, ... etc, see A344694. First terms of A344882 not present here are: 60, 66, 88, 92, 105, etc.

Cf. A000961, A011772, A344875, A344882, A344973.
Intersection of A024619 and A344974.
Cf. also A344595, A344694, A344978 (subsequences).

isA344975(n) = ((n>1)&&!isprimepower(n)&&(0==A344973(n)));

cp's OEIS Frontend

A068315 For numbers k such that A025474(k) > 1 and A025474(k+1) > 1, sequence gives A000961(k).

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A097621 In canonical prime factorization of n replace p^e with its index in A000961.

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A192015 Arithmetic derivative of prime powers: a(n) = A003415(A000961(n)).

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A024620 Positions of primes among the powers of primes (A000961).

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A024622 Position of 2^n among the powers of primes (A000961).

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A335866 Number of classes of simple difference sets of the Singer type (m^2 + m + 1, m + 1, 1) with m = m(n) = A000961(n), for n >= 1.

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