A025474 -id:A025474 - cp's OEIS frontend

A207193 Auxiliary function for computing the Carmichael lambda function (A002322).

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

Offset: 1

Reinhard Zumkeller, Feb 16 2012

nonn

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

Cf. A000961, A002322, A025473, A025474, A085730.

a207193 1 = 1
a207193 n | p == 2 && e > 2 = 2 ^ (e - 2)
          | otherwise       = (p - 1) * p ^ (e - 1)
          where p = a025473 n; e = a025474 n

f[p_, e_] := If[p == 2 && e > 2, 2^(e-2), (p-1)*p^(e-1)]; s[n_] := If[n == 1, 1, If[PrimePowerQ[n], f @@ (FactorInteger[n][[1]]), Nothing]]; Array[s, 200] (* Amiram Eldar, Apr 05 2025 *)

a(n) = f(A000961(n)), where f(1) = 1, and f(p^e) = 2^(e-2) if p = 2 and e > 2, and f(p^e) = (p-1)*p^(e-1) otherwise.

A257573 Exponents in A257278 = powers of primes p^m, p <= m.

Original entry on oeis.org

2, 3, 4, 3, 5, 6, 4, 7, 5, 8, 9, 6, 10, 11, 7, 5, 12, 8, 13, 6, 14, 9, 15, 10, 16, 7, 17, 11, 18, 8, 19, 12, 7, 20, 13, 9, 21, 22, 14, 8, 23, 10, 15, 24, 25, 9, 16, 11, 26, 17, 27, 12, 28, 10, 18, 29, 30, 19, 13, 11, 31, 20, 32, 14, 33, 21, 12, 34, 15, 22

Offset: 1

Reinhard Zumkeller, May 01 2015

nonn

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

Cf. A257278, A001222, A025474, A257572.

Haskell
```
a257573 = a001222 . a257278
```

seq[lim_] := Module[{s = {}, p = 2, r}, While[p^p <= lim, r = Range[p, Log[p, lim]]; AppendTo[s, Transpose[{r, p^r}]]; p = NextPrime[p]]; SortBy[Flatten[s, 1], Last][[;; , 1]]]; seq[10^13] (* Amiram Eldar, Apr 14 2025 *)

apply(bigomega,A257278) \\ (A257278 assumed to be defined as vector). - M. F. Hasler, May 02 2015

a(n) = A001222(A257278(n)).
A257278(n) = A257572(n) ^ a(n).
A257572(n) <= a(n) by definition of A257278.

Edited by M. F. Hasler, May 02 2015

A373946 Number of primitive polynomials of third degree over GF(m) with vanishing quadratic term with m = m(n) = A000961(n), for n >= 2.

Original entry on oeis.org

1, 1, 0, 4, 3, 18, 8, 16, 18, 48, 48, 27, 80, 48, 108, 108, 72, 300, 144, 224, 180, 308, 192, 336, 560, 240, 648, 420, 576, 540, 648, 768, 1080, 1200, 912, 1360, 1008, 1352, 1188, 1584, 960, 2340, 1620, 4410, 2112, 2432, 1980, 2952, 1560, 2592, 2025, 4592, 2448, 4872, 4576

Offset: 2

Martin Becker, Jun 23 2024

nonn

Apparently, a(n) = A373514(n) * A000010( 3 * A000961(n) - 3 ) * A025474(n) / 2, for n >= 2.

			For n=5, m=5, there are 20 primitive polynomials over GF(5) of the form x^3+a*x^2+b*x+c. Among these, 4 polynomials have a=0: x^3+3*x+2, x^3+3*x+3, x^3+4*x+2, and x^3+4*x+3. Thus, a(5) = 4.

Martin Becker, Table of n, a(n) for n = 2..400

Cf. A000961, A319213, A373514.

is_max_o = (x1, x0, m, e)-> {
  for(i = 1, #e, if(x1^e[i] == x0, return(0))); x1^m == x0;
}
count_them = (q)-> {
  z = ffprimroot(ffgen(q, 'c));
  m = q^3 - 1; f = factor(m); d = #f~;
  e = vector(d, i, m/f[d + 1 - i, 1]);
  co = vector(q - 1, i, z^(i - 1));
  r = 0;
  for(a = 1, q - 1,
    for(b = 1, q - 1,
      p = co[1]*x^3 + co[a]*x + co[b];
      x1 = Mod(x, p); x0 = x1^0;
      if(is_max_o(x1, x0, m, e) && polisirreducible(p), r += 1)
    )
  );
  r;
}
print1(count_them(2));
for(q = 3, 64, if(isprimepower(q), print1(", ", count_them(q))))

A379158 Numbers m such that the consecutive prime powers A246655(m) and A246655(m+1) are both prime.

Original entry on oeis.org

1, 4, 8, 11, 12, 16, 19, 20, 21, 24, 25, 28, 29, 30, 33, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 50, 51, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 71, 72, 73, 74, 75, 76, 79, 80, 81, 82, 83, 84, 87, 88, 89, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Gus Wiseman, Dec 23 2024

nonn

Also positions of 2 in A366835.

			The 4th and 5th prime powers are 5 and 7, which are both prime, so 4 is in the sequence.
The 12th and 13th prime powers are 19 and 23, which are both prime, so 12 is in the sequence.

Positions of adjacent primes in A246655 (prime powers).
Positions of 2 in A366835.
For just one prime we have A379155, positions of prime powers in A379157.
For no primes we have A379156, positions of prime powers in A068315.
The primes powers themselves are A379541.
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.
A366833 counts prime powers between primes, see A053607, A304521.
Cf. A025474, A067871, A080769, A178700, A274605, A345531, A377281, A377287, A378368.

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

A246655(a(n)) = A379541(n).

A379541 Prime numbers such that the next greatest prime power is also prime.

Original entry on oeis.org

2, 5, 11, 17, 19, 29, 37, 41, 43, 53, 59, 67, 71, 73, 83, 89, 97, 101, 103, 107, 109, 131, 137, 139, 149, 151, 157, 163, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 257, 263, 269, 271, 277, 281, 293, 307, 311, 313, 317, 331, 347, 349, 353

Offset: 1

Gus Wiseman, Dec 24 2024

nonn

			After 13 the next prime power is 16, which is not prime, so 13 is not in the sequence.
After 19 the next prime power is 23, which is prime, so 19 is in the sequence.

For no primes we have A068315, positions A379156.
Lesser of adjacent primes in A246655 (prime powers).
The indices of these primes are A377286.
For just one prime we have A379157, positions A379155.
Positions in the prime powers are A379158 = positions of 2 in A366835.
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.
A366833 counts prime powers between primes, see A053607, A304521.
Cf. A025474, A067871, A080769, A178700, A274605, A345531, A377281, A377287, A378368.

nextpripow[n_]:=NestWhile[#1+1&,n+1,!PrimePowerQ[#1]&];
Select[Range[100],PrimeQ[#]&&PrimeQ[nextpripow[#]]&]

a(n) = A246655(A379158(n)).

A085729 Sum of prime factors of prime powers.

Original entry on oeis.org

0, 2, 3, 4, 5, 7, 6, 6, 11, 13, 8, 17, 19, 23, 10, 9, 29, 31, 10, 37, 41, 43, 47, 14, 53, 59, 61, 12, 67, 71, 73, 79, 12, 83, 89, 97, 101, 103, 107, 109, 113, 22, 15, 127, 14, 131, 137, 139, 149, 151, 157, 163, 167, 26, 173, 179, 181, 191, 193, 197, 199, 211, 223

Offset: 1

Reinhard Zumkeller, Jul 20 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Power.
Eric Weisstein's World of Mathematics, Sum of Prime Factors.

Cf. A000961, A001414, A025473, A025474.

s[n_] := Module[{f = FactorInteger[n]}, If[Length[f] == 1, Times @@ f[[1]], Nothing]]; s[1] = 0; Array[s, 225] (* Amiram Eldar, May 14 2025 *)

a(n) = A001414(A000961(n)).

a(n) = e*p when n = p^e: a(n) = A025474(n)*A025473(n).

A086454 Number of divisors of prime powers: tau(p^e).

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 4, 3, 2, 2, 5, 2, 2, 2, 3, 4, 2, 2, 6, 2, 2, 2, 2, 3, 2, 2, 2, 7, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 3, 4, 2, 8, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 9, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Jul 20 2003

nonn

Eric Weisstein's World of Mathematics, Divisor Function.
Eric Weisstein's World of Mathematics, Prime Power.

Cf. A000005, A000961, A025474, A086455.

f[n_] := Module[{f = FactorInteger[n]}, If[Length[f] == 1, f[[1, 2]] + 1, Nothing]]; f[1] = 1; Array[f, 400] (* Amiram Eldar, Apr 09 2024 *)

a(n) = A000005(A000961(n)).
a(n) = e+1 for A000961(n) = p^e.
a(n) = A025474(n) + 1.

A088234 First differences of exponents of consecutive prime powers.

Original entry on oeis.org

1, 0, 1, -1, 0, 2, -1, -1, 0, 3, -3, 0, 0, 1, 1, -2, 0, 4, -4, 0, 0, 0, 1, -1, 0, 0, 5, -5, 0, 0, 0, 3, -3, 0, 0, 0, 0, 0, 0, 0, 1, 1, -2, 6, -6, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -4, 7, -7, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 1, -1, 0, 0

Offset: 1

Reinhard Zumkeller, Sep 25 2003

sign

a(n) = A025474(n+1) - A025474(n).

Cf. A088233, A000961, A057820.

Join[{1},Differences[FactorInteger[#][[1,2]]&/@Select[Range[ 400], PrimePowerQ]]] (* Harvey P. Dale, Jul 10 2020 *)

A247073 Triangle read by rows: T(n,k) is the number of k-th prime powers up to 2^n, for k = 1 to n.

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 6, 2, 1, 1, 11, 3, 2, 1, 1, 18, 4, 2, 1, 1, 1, 31, 5, 3, 2, 1, 1, 1, 54, 6, 3, 2, 2, 1, 1, 1, 97, 8, 4, 2, 2, 1, 1, 1, 1, 172, 11, 4, 3, 2, 2, 1, 1, 1, 1, 309, 14, 5, 3, 2, 2, 1, 1, 1, 1, 1, 564, 18, 6, 4, 3, 2, 2, 1, 1, 1, 1, 1, 1028, 24, 8, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1

Offset: 1

Michel Marcus, Nov 18 2014

			Up to 16, there are 6 primes (2, 3, 5, 7, 11, 13), 2 squared primes (4,9), 1 cube (8), and 1 fourth power (16), so 4th row is 6, 2, 1, 1.
Triangle starts:
1;
2, 1;
4, 1, 1;
6, 2, 1, 1;
11, 3, 2, 1, 1;
18, 4, 2, 1, 1, 1;
...

Reinhard Zumkeller, Rows n = 1..20 of triangle, flattened

Cf. A000961 (prime powers), A007053 (first column), A060967 (second column).

Cf. A025474.

import Data.List (sort, groupBy); import Data.Function (on)
a247073 n k = a247073_tabl !! (n-1) !! (k-1)
a247073_tabl = map a247073_row [1..]
a247073_row n = map length $ groupBy ((==) `on` fst) $ sort $
   takeWhile ((<= 2^n). snd) $ tail $ zip a025474_list a000961_list
-- Reinhard Zumkeller, Nov 18 2014

tabl(nn) = {for (n=1, nn, v = vector(2^n, i, i); vr = vector(n); for (k=1, #v, if (pp = isprimepower(v[k]), vr[pp] ++);); for (k=1, n, print1(vr[k], ", ");); print(););}

A264744 Exponent of the prime power A264734(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 1, 4, 1, 1, 1, 1

Offset: 1

Juri-Stepan Gerasimov, Nov 22 2015

Subsequence of A025474.

			a(9) = 4 is the exponent of 81 = A264734(9) = 3^4.

Cf. A025474, A264734 (prime power k such that both k - 2 and k + 2 is a prime power).

t = Prepend[Select[Range@ 100000, AllTrue[{# - 2, #, # + 2}, PrimePowerQ] &], 3]; Flatten@ Map[Last, FactorInteger@ # &@ t, {2}] (* Michael De Vlieger, Dec 03 2015, Version 10 *)

is(k) = isprimepower(k) || k==1;
for(k=1, 1e6, if(is(k) && is(k+2) && is(k-2), print1(bigomega(k), ", "))) \\ Altug Alkan, Nov 23 2015

cp's OEIS Frontend

A207193 Auxiliary function for computing the Carmichael lambda function (A002322).

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A257573 Exponents in A257278 = powers of primes p^m, p <= m.

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A373946 Number of primitive polynomials of third degree over GF(m) with vanishing quadratic term with m = m(n) = A000961(n), for n >= 2.

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A379158 Numbers m such that the consecutive prime powers A246655(m) and A246655(m+1) are both prime.

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A379541 Prime numbers such that the next greatest prime power is also prime.

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A085729 Sum of prime factors of prime powers.

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A086454 Number of divisors of prime powers: tau(p^e).

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Mathematica

Formula

A088234 First differences of exponents of consecutive prime powers.

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Mathematica

A247073 Triangle read by rows: T(n,k) is the number of k-th prime powers up to 2^n, for k = 1 to n.

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