A025474 -id:A025474 - cp's OEIS frontend

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

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)

A137502 Reverse sequence of powers in prime decomposition of n.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 10, 2, 18, 2, 14, 6, 16, 2, 12, 2, 50, 10, 22, 2, 54, 4, 26, 8, 98, 2, 30, 2, 32, 14, 34, 6, 36, 2, 38, 22, 250, 2, 70, 2, 242, 18, 46, 2, 162, 4, 20, 26, 338, 2, 24, 10, 686, 34, 58, 2, 150, 2, 62, 50, 64, 14, 154, 2, 578, 38, 42, 2, 108, 2, 74, 12, 722, 6, 286

Offset: 1

Philippe Lallouet (philip.lallouet(AT)orange.fr), Apr 22 2008

The term a(1) = 1 added on the grounds that as 1 has an empty prime factorization, it stays same when reversed. - Antti Karttunen, May 20 2014
In the prime decomposition of n we use all the primes up to the highest prime divisor, exponents of zero being allowed except for the largest prime.
If n = (p(1)^e1)*(p(2)^e2)*.......*(p(k)^ek) (ek>0, other ei>=0 and p(n) = n-th prime) then we reverse the sequence e1, e2 , ..., ek to build a(n): a(n) = (p(1)^ek)*(p(2)^e(k-1))* . . . . *(p(k)^e1)
As p(1)=2 and ek is never zero for n>1, a(n) is always even for n>1.
If n is prime then a(n) = 2 and if n is a power of prime, a(n) is the same power of 2.
(That is, a(A000961(n)) = A000079(A025474(n)) for all n.) - Antti Karttunen, May 20 2014.
If the sequence e1, e2, ..., ek is palindromic, a(n)=n. (A242418 gives such n).
For any given even number Q, we can by reversing the sequence of its powers define not only one but an infinity (by adding as many zeros as we want on the left end) of n such that a(n) = Q. Hence the sequence is a permutation of even integers where each even integer is infinitely repeated.
For example as Q = 1224 = (2^3)*(3^2)*(5^0)*(7^0)*(11^0)*(13^0)*(17^1),
Q = a((2^1)*(3^0)*(5^0)*(7^0)*(11^0)*(13^2)*(17^3)) = a(1660594) but also of an infinity of other ones, the first one being a((2^0)*(3^1)*(5^0)*(7^0)*(11^0)*(13^0)*(17^2)*(19^1)) = a(5946753).
Please see A241916 for a variant which results an ordinary permutation of all natural numbers. - Antti Karttunen, May 20 2014

			As 9 = (2^0)*(3^2), hence a(9) = (2^2)*(3^0) = 4.
As 50 = (2^1)*(3^0)*(5^2), hence a(50) = (2^2)*(3^0)*(5^1) = 2*2*5 = 20.
As 57 = (2^0)*(3^1)*(5^0)*(7^0)*(11^0)*(13^0)*(17^0)*(19^1), hence a(57) = (2^1)*(3^0)*(5^0)*(7^0)*(11^0)*(13^0)*(17^1)*(19^0) = 2*17 = 34.

Antti Karttunen, Table of n, a(n) for n = 1..10000

A242418 gives the fixed points.

Cf. A241916, A006530, A000961, A025474.

f[n_] := If[n == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ n]; g[w_List] := Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, w]; Table[g@ Reverse@ f@ n, {n, 120}] (* Michael De Vlieger, Aug 27 2016 *)

(define (A137502 n) (if (< n 2) n (/ (* 2 (A241916 n)) (A006530 n)))) ;; Antti Karttunen, May 20 2014

a(1) = 1, and for n>1, a(n) = 2*A241916(n) / A006530(n). - Antti Karttunen, May 20 2014

Edited by N. J. A. Sloane, Jan 16 2009.

Term a(1)=1 prepended, and erroneous terms (first at n=50) corrected, Antti Karttunen, May 20 2014

A192134 Difference between n-th prime power and its arithmetic derivative.

Original entry on oeis.org

1, 1, 2, 0, 4, 6, -4, 3, 10, 12, -16, 16, 18, 22, 15, 0, 28, 30, -48, 36, 40, 42, 46, 35, 52, 58, 60, -128, 66, 70, 72, 78, -27, 82, 88, 96, 100, 102, 106, 108, 112, 99, 50, 126, -320, 130, 136, 138, 148, 150, 156, 162, 166, 143, 172, 178, 180, 190, 192, 196

Offset: 1

Reinhard Zumkeller, Jun 26 2011

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

Cf. A000040, A000961, A001248, A003415, A005722, A006093, A025473, A025474, A095874, A192015, A192133.

a192134 n = a000961 n - a192015 n  -- Reinhard Zumkeller, Apr 16 2014

f[n_] := If[n == 1, 1, If[PrimePowerQ[n], {p, e} = FactorInteger[n][[1]]; n - e*p^(e-1), Nothing]]; Array[f, 300] (* Amiram Eldar, Apr 11 2025 *)

a(n) = A000961(n)-A192015(n) = A000961(n)-A003415(A000961(n)) = A192133(n)*A025473(n)^(A025474(n)-1) = A192133(n)*A000961(n)/A025473(n).
a(A095874(A000040(n))) = A006093(n).
a(A095874(A001248(n))) = A005722(n) + 1.

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.

A379156 Positions in A246655 (prime powers) of terms q such that there is no prime between q and the next prime power.

Original entry on oeis.org

6, 14, 41, 359, 3589

Offset: 1

Gus Wiseman, Dec 22 2024

The powers of primes themselves are 8, 25, 121, 2187, 32761, ... (A068315).

The prime powers themselves are A068315, for just one prime A379157.
For perfect powers instead of prime powers we have A274605.
Positions of 0 in A366835.
For just one prime we have A379155, for perfect powers A378368.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A031218 gives the greatest prime power <= n.
A065514 gives the greatest prime power < prime(n), difference A377289.
A131605 finds perfect powers that are not prime powers.
A246655 lists the prime powers.
A366833 counts prime powers between primes, see A053607, A304521.
Cf. A025474, A046933, A067871, A080101, A080769, A175106, A178700, A345531, A377281, A377287, A377432, A377434.

v=Select[Range[100],PrimePowerQ];
Select[Range[Length[v]-1],FreeQ[Range[v[[#]],v[[#+1]]],_?PrimeQ]&]

A246655(a(n)) = A068315(n).

A086455 Sum of divisors of prime powers: sigma(p^e).

Original entry on oeis.org

1, 3, 4, 7, 6, 8, 15, 13, 12, 14, 31, 18, 20, 24, 31, 40, 30, 32, 63, 38, 42, 44, 48, 57, 54, 60, 62, 127, 68, 72, 74, 80, 121, 84, 90, 98, 102, 104, 108, 110, 114, 133, 156, 128, 255, 132, 138, 140, 150, 152, 158, 164, 168, 183, 174, 180, 182, 192, 194, 198

Offset: 1

Reinhard Zumkeller, Jul 20 2003

nonn

R. J. Mathar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Function.
Eric Weisstein's World of Mathematics, Prime Power.

Cf. A000203, A000961, A025473, A025474, A086454.

A086455 := proc(n)
    numtheory[sigma](A000961(n)) ;
end proc: # R. J. Mathar, Jun 04 2016

DivisorSigma[1, #]& /@ Join[{1}, Select[Range[2, 200], PrimePowerQ]] (* Jean-François Alcover, Feb 10 2018 *)

list(lim) = apply(sigma, select(x -> x == 1 || isprimepower(x), vector(lim, i, i))); \\ Amiram Eldar, May 07 2025

a(n) = A000203(A000961(n)).
a(n) = (p^(e+1)-1)/(p-1), where p^e = A000961(n).
a(n) = (A025473(n)^(A025474(n)+1)-1)/(A025473(n)-1).

A117331 Lexicographically earliest permutation of prime powers such that the exponents of succeeding terms increase at most by 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 8, 11, 13, 17, 19, 23, 25, 27, 16, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 81, 32, 64, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227

Offset: 1

Reinhard Zumkeller, Mar 08 2006

nonn

A025474(A095874(a(n+1))) - A025474(A095874(a(n))) <= 1;
A117332(n) = A095874(a(n));
a(A117333(n)) = A000961(n).

			a(13)..a(16): 23,5^2,3^3,2^4;
a(38)..a(43): 113,11^2,5^3,3^4,2^5,2^6;
a(239)..a(248): 1367,37^2,11^3,5^4,3^5,3^6,2^7,2^8,2^9,2^10.

Eric Weisstein's World of Mathematics, Prime Power

A192133 Difference of base and exponent of prime powers (cf. A000961).

Original entry on oeis.org

1, 1, 2, 0, 4, 6, -1, 1, 10, 12, -2, 16, 18, 22, 3, 0, 28, 30, -3, 36, 40, 42, 46, 5, 52, 58, 60, -4, 66, 70, 72, 78, -1, 82, 88, 96, 100, 102, 106, 108, 112, 9, 2, 126, -5, 130, 136, 138, 148, 150, 156, 162, 166, 11, 172, 178, 180, 190, 192, 196, 198, 210

Offset: 1

Reinhard Zumkeller, Jun 26 2011

sign

a(1) = 1 by convention, in accordance with A025473(1) = 1 and A025474(1) = 0.

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

A006093 and A090076 are subsequences.

Cf. A000961, A025473, A025474, A192134.

s[n_] := Module[{f = FactorInteger[n]}, If[Length[f] == 1, f[[1, 1]] - f[[1, 2]], Nothing]]; s[1] = 1; Array[s, 250] (* Amiram Eldar, May 16 2025 *)

a(n) = A025473(n)-A025474(n) = A192134(n)*A025473(n)/A000961(n).

A085730 Euler's totient function applied to the sequence of prime powers.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 4, 6, 10, 12, 8, 16, 18, 22, 20, 18, 28, 30, 16, 36, 40, 42, 46, 42, 52, 58, 60, 32, 66, 70, 72, 78, 54, 82, 88, 96, 100, 102, 106, 108, 112, 110, 100, 126, 64, 130, 136, 138, 148, 150, 156, 162, 166, 156, 172, 178, 180, 190, 192, 196, 198, 210

Offset: 1

Reinhard Zumkeller, Jul 20 2003

nonn

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

Cf. A000010, A000961, A025473, A025474, A207193.

a085730 1 = 1
a085730 n = (p - 1) * p ^ (e - 1)
   where p =  a025473 n; e =  a025474 n
-- Reinhard Zumkeller, Feb 16 2012

f[p_, e_] := (p-1)*p^(e-1); s[n_] := If[n == 1, 1, If[PrimePowerQ[n], f @@ (FactorInteger[n][[1]]), Nothing]]; Array[s, 220] (* Amiram Eldar, Apr 05 2025 *)

list(lim)=my(v=List(primes(primepi(lim)))); listput(v,1); for(e=2, log(lim+.5)\log(2),forprime(p=2,(lim+.5)^(1/e),listput(v, p^e))); apply(n->eulerphi(n),vecsort(Vec(v))) \\ Charles R Greathouse IV, Apr 30 2012

a(n) = A000010(A000961(n)).
a(p^e) = (p-1)*p^(e-1).
a(n) = (A025473(n)-1)*A025473(n)^(A025474(n)-1).

A192083 Arithmetic derivative of squares of prime powers: a(n) = A003415(A056798(n)).

Original entry on oeis.org

0, 4, 6, 32, 10, 14, 192, 108, 22, 26, 1024, 34, 38, 46, 500, 1458, 58, 62, 5120, 74, 82, 86, 94, 1372, 106, 118, 122, 24576, 134, 142, 146, 158, 17496, 166, 178, 194, 202, 206, 214, 218, 226, 5324, 18750, 254, 114688, 262, 274, 278, 298, 302, 314, 326, 334

Offset: 1

Reinhard Zumkeller, Jun 26 2011

nonn

A001787 and A024622 give record values and where they occur.

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

Cf. A003415, A024622, A025473, A025474, A056798, A192084.

Subsequences: A001787, A027471, A051674, A079705, A100484.

s[n_] := If[PrimePowerQ[n], f = FactorInteger[n][[1]]; 2*f[[2]]*n^(2 - 1/f[[2]]), Nothing]; s[1] = 0; Array[s, 200] (* Amiram Eldar, Apr 06 2025 *)

a(n) = 2 * A025474(n) * A025473(n)^(2*A025474(n) - 1).

A192084(n) = A003415(a(n)).

cp's OEIS Frontend

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

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A137502 Reverse sequence of powers in prime decomposition of n.

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A192134 Difference between n-th prime power and its arithmetic derivative.

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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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A379156 Positions in A246655 (prime powers) of terms q such that there is no prime between q and the next prime power.

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A086455 Sum of divisors of prime powers: sigma(p^e).

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A117331 Lexicographically earliest permutation of prime powers such that the exponents of succeeding terms increase at most by 1.

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A192133 Difference of base and exponent of prime powers (cf. A000961).

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