A025475 -id:A025475 - cp's OEIS frontend

A376310 Run-sums of the sequence of first differences of prime-powers.

3, 2, 2, 4, 3, 1, 2, 4, 8, 1, 5, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 4, 2, 1, 3, 6, 2, 10, 2, 12, 4, 2, 4, 6, 2, 10, 2, 4, 2, 24, 4, 2, 4, 6, 4, 8, 5, 1, 12, 2, 6, 4, 2, 6, 4, 14, 4, 2, 4, 14, 12, 4, 2, 4, 6, 2, 18, 4, 6, 8, 4, 8, 10, 2

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

			The sequence of prime-powers (A246655) is:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, ...
The sequence of first differences (A057820) of prime-powers is:
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, ...
with runs:
  (1,1,1),(2),(1,1),(2,2),(3),(1),(2),(4),(2,2,2,2),(1),(5),(4),(2),(4), ...
with sums A376310 (this sequence).

For primes instead of prime-powers we have A373822, halved A373823.
For squarefree numbers instead of prime-powers we have A376307.
For compression instead of run-sums we have A376308.
For run-lengths instead of run-sums we have A376309.
For positions of first appearances we have A376341, sorted A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A024619 and A361102 list non-prime-powers, first differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373948 encodes compression using compositions in standard order.
Cf. A001597, A006549, A007916, A025475, A037201, A053289, A054265, A110969, A120430, A174965, A251092, A375706.

Total/@Split[Differences[Select[Range[100],PrimePowerQ]]]

A056824 Numbers of the form p^(2k+1), p prime, k >= 1.

Original entry on oeis.org

8, 27, 32, 125, 128, 243, 343, 512, 1331, 2048, 2187, 2197, 3125, 4913, 6859, 8192, 12167, 16807, 19683, 24389, 29791, 32768, 50653, 68921, 78125, 79507, 103823, 131072, 148877, 161051, 177147, 205379, 226981, 300763, 357911, 371293, 389017

Offset: 1

Labos Elemer, Aug 29 2000

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A025475, A000961, A369632.

is(n)=my(e=isprimepower(n)); e>1 && e%2 \\ Charles R Greathouse IV, Jan 15 2015

list(lim)=my(v=List(apply(k->k^3, primes([2,sqrtnint(lim\1,3)]))),t); forstep(e=5,log(lim+.5)\log(2),2, forprime(p=2,,t=p^e; if(t>lim, break); listput(v,t))); Set(v) \\ Charles R Greathouse IV, Jan 15 2015

a(n) ~ n^3 log^3 n. - Charles R Greathouse IV, Jan 15 2015

Sum_{n>=1} 1/a(n) = Sum_{p prime} 1/(p*(p^2-1)) = 0.2214633713... (A369632). - Amiram Eldar, Dec 23 2020

Name edited by Altug Alkan, following a suggestion by Felix Fröhlich, May 17 2018

A071140 Numbers n such that sum of distinct primes dividing n is divisible by largest prime dividing n; n is neither a prime, nor a true power of prime.

Original entry on oeis.org

30, 60, 70, 90, 120, 140, 150, 180, 240, 270, 280, 286, 300, 350, 360, 450, 480, 490, 540, 560, 572, 600, 646, 700, 720, 750, 810, 900, 960, 980, 1080, 1120, 1144, 1200, 1292, 1350, 1400, 1440, 1500, 1620, 1750, 1798, 1800, 1920, 1960, 2160, 2240, 2250

Offset: 1

Labos Elemer, May 13 2002

nonn

a(n) are the numbers such that the difference between the largest and the smallest prime divisor equals the sum of the other distinct prime divisors. - Michel Lagneau, Nov 13 2011
The statement above is only true for 966 of the first 1000 terms. The first counterexample is a(140) = 15015. - Donovan Johnson, Apr 10 2013
Lagneau's definition can be simplified to the largest prime divisor equals the sum of the other distinct prime divisors. - Christian N. K. Anderson, Apr 15 2013

			n = 70 = 2*5*7 has a form of 2pq, where p and q are twin primes; n = 3135 = 3*5*11*19, sum = 3+5+11+19 = 38 = 2*19, divisible by 19.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

Subsequence of A024619.

a071140 n = a071140_list !! (n-1)
a071140_list = filter (\x -> a008472 x `mod` a006530 x == 0) a024619_list
-- Reinhard Zumkeller, Apr 18 2013

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Greater[s, 1], Print[{n, ba[n]}]], {n, 2, 1000000}]
(* Second program: *)
Select[Range@ 2250, And[Length@ # > 1, Divisible[Total@ #, Last@ #]] &[FactorInteger[#][[All, 1]] ] &] (* Michael De Vlieger, Jul 18 2017 *)

A008472(n)/A006530(n) is an integer and n has at least 3 distinct prime factors.

A008472(a(n)) mod A006530(a(n)) = 0 and A010055(a(n)) = 0. - Reinhard Zumkeller, Apr 18 2013

A071147 Smallest squarefree number k with exactly n prime factors such that the sum of the prime factors is divisible by the largest prime dividing k, or 0 if no such k exists.

Original entry on oeis.org

1, 2, 0, 30, 3135, 3570, 72930, 1231230, 14804790, 497668710, 14908423530, 278196808890, 12192694624110, 550939666387110, 21275256232500270, 1458502323630662310, 87988283090327810190, 3254611619240885033130, 261462818462495728868790, 9965666894849284108299810, 557940830126698960967415390, 90544636506979071680577724410

Offset: 0

Labos Elemer, May 13 2002

nonn

No solution exists for n=2, so a(2)=0.

			a(0) =       1 = 1;
a(1) =       2 = 2;
a(3) =      30 = 2 *  3 *  5;
a(4) =    3135 = 3 *  5 * 11 * 19;
a(5) =    3570 = 2 *  3 *  5 *  7 * 17;
a(6) =   72930 = 2 *  3 *  5 * 11 * 13 * 17;
a(7) = 1231230 = 2 *  3 *  5 *  7 * 11 * 13 * 41.

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071146.

A008472(k)/A006530(k) is an integer; k is squarefree and has exactly n prime factors.

Corrected and extended by Donovan Johnson, Apr 22 2008

Name corrected by Jon E. Schoenfield, Jul 08 2018

A225102 Numbers that can be represented as a sum of two distinct nontrivial prime powers (numbers of the form p^k where p is a prime number and k >= 2).

Original entry on oeis.org

12, 13, 17, 20, 24, 25, 29, 31, 33, 34, 35, 36, 40, 41, 43, 48, 52, 53, 57, 58, 59, 65, 68, 72, 73, 74, 76, 80, 81, 85, 89, 90, 91, 96, 97, 106, 108, 113, 125, 129, 130, 132, 133, 134, 136, 137, 141, 144, 145, 146, 148, 150, 152, 153, 155, 157, 160, 170, 173, 174, 177

Offset: 1

Alex Ratushnyak, Apr 28 2013

nonn

Indices of positive terms in A225099.

Nontrivial prime powers are A025475 except the first term A025475(1) = 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A025475, A225099.

#include 
#include 
#define TOP (1ULL<<17)
unsigned long long *powers, pwFlat[TOP], primes[TOP] = {2};
int main() {
  unsigned long long a, c, i, j, k, n, p, r, pp = 1, pfp = 0;
  powers = (unsigned long long*)malloc(TOP * TOP/8);
  memset(powers, 0, TOP * TOP/8);
  for (a = 3; a < TOP; a += 2) {
    for (p = 0; p < pp; ++p)  if (a % primes[p] == 0) break;
    if (p == pp)  primes[pp++] = a;
  }
  for (k = i = 0; i < pp; ++i)
    for (j = primes[i]*primes[i]; j < TOP*TOP; j *= primes[i])
      powers[j/64] |= 1ULL << (j & 63), ++k;
  if (k > TOP) exit(1);
  for (n = 0; n < TOP * TOP; ++n)
    if (powers[n/64] & (1ULL << (n & 63)))  pwFlat[pfp++] = n;
  for (n = 0; n < TOP * TOP; ++n) {
    for (c = i = 0; pwFlat[i] * 2 < n; ++i)
      r=n-pwFlat[i], c+= (powers[r/64] & (1ULL <<(r&63))) > 0;
    if (c)  printf("%llu, ", n);
  }
  return 0;
}

N:= 1000: # to get all terms <= N
P:= select(isprime, [2,seq(i,i=3..floor(sqrt(N)),2)]):
PP:= sort(map(p -> seq(p^t,t=2..floor(log[p](N))), P)):
sort(convert(select(`<=`,{seq(seq(PP[i]+PP[j],j=1..i-1),i=1..nops(PP))},N),list)); # Robert Israel, Feb 21 2017

nn = 177; p = Sort[Flatten[Table[Prime[n]^i, {n, PrimePi[Sqrt[nn]]}, {i, 2, Log[Prime[n], nn]}]]]; Select[Union[Flatten[Table[p[[i]] + p[[j]], {i, Length[p] - 1}, {j, i + 1, Length[p]}]]], # <= nn &] (* T. D. Noe, Apr 29 2013 *)

A376341 Position of first appearance of n in A057820, the sequence of first differences of prime-powers, or 0 if n does not appear.

Original entry on oeis.org

1, 5, 10, 13, 19, 25, 199, 35, 118, 48, 28195587, 61, 3745011205066703, 80, 6635, 312, 1079, 207, 3249254387600868788, 179, 43580, 216, 21151968922, 615, 762951923, 403, 1962, 466, 12371, 245, 1480223716, 783, 494890212533313, 1110, 2064590, 1235, 375744164943287809536

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

For odd n either a(n) or a(n)+1 is in A024622 (unless a(n) = 0), corresponding to cases where the smaller or the larger term in the pair of consecutive prime powers, respectively, is a power of 2. - Pontus von Brömssen, Sep 27 2024

			a(4) = 13, because the first occurrence of 4 in A057820 is at index 13. The corresponding first pair of consecutive prime powers with difference 4 is (19, 23), and a(4) = A025528(23) = 13.
a(61) = A024622(96), because the first pair of consecutive prime powers with difference 61 is (2^96, 2^96+61), and A025528(2^96+61) = A024622(96).

Pontus von Brömssen, Table of n, a(n) for n = 1..60

For compression instead of first appearances we have A376308.
For run-lengths instead of first appearances we have A376309.
For run-sums instead of first appearances we have A376310.
For squarefree numbers instead of prime-powers we have A376311.
The sorted version is A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A024619 and A361102 list non-prime-powers, first differences A375708.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A116861 counts partitions by compressed sum, by compressed length A116608.
Cf. A001597, A006549, A007916, A024622, A025475, A025528, A037201, A038664, A053289, A110969, A120430, A174965, A373400, A375706.

mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
q=Differences[Select[Range[100],#==1||PrimePowerQ[#]&]];
Table[Position[q,k][[1,1]],{k,mnrm[q]}]

A057820(a(n)) = n whenever a(n) > 0. - Pontus von Brömssen, Sep 24 2024

Definition modified by Pontus von Brömssen, Sep 26 2024

More terms from Pontus von Brömssen, Sep 27 2024

A376600 Inflection or undulation points in the sequence of non-prime-powers inclusive (A024619).

Original entry on oeis.org

2, 7, 9, 10, 11, 14, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 39, 41, 43, 44, 45, 47, 48, 50, 51, 52, 55, 56, 57, 58, 59, 62, 64, 66, 68, 70, 73, 74, 75, 76, 77, 80, 86, 87, 88, 90, 92, 93, 94, 95, 96, 97, 98, 100, 102, 103, 104, 107, 108, 109, 112, 114, 116

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A376599) are zero.

Inclusive means 1 is a prime-power but not a non-prime-power. For the exclusive version, add 1 to all terms.

			The non-prime-powers inclusive are (A024619):
  6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, ...
with first differences (A375735):
  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, ...
with first differences (A376599):
  -2, 0, -1, 2, -1, -1, 0, 1, 0, 0, 0, 1, -2, 0, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, ...
with zeros at (A376600):
  2, 7, 9, 10, 11, 14, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 39, 41, 43, 44, ...

Gus Wiseman, Inflection and undulation points in the non-prime-powers inclusive.

For first differences we had A375735, ones A375713(n)-1.
These are the zeros of A376599.
The complement is A376601.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A024619/A361102 list non-prime-powers inclusive.
A321346/A321378 count integer partitions into non-prime-powers, factorizations A322452.
For non-prime-powers: A375735/A375708 (first differences), A376599 (second differences), A376601 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power).
Cf. A025475, A053707, A057820, A064113, A093555, A174965, A251092, A333254, A376653, A376654.

Join@@Position[Differences[Select[Range[100], !(#==1||PrimePowerQ[#])&],2],0]

A068435 Consecutive prime powers without a prime between them.

Original entry on oeis.org

8, 9, 25, 27, 121, 125, 2187, 2197, 32761, 32768

Offset: 1

Jon Perry, Mar 09 2002

From David A. Corneth, Aug 24 2019: (Start)
Only 5 pairs are known up to 4*10^18. Legendre's conjecture states that there is a prime number between n^2 and (n + 1)^2 for every positive integer n. The conjecture has been verified up to n = 2*10^9. So to that bound we only have to check for two prime powers where at least one has an exponent of at least 3. That has been done to prime powers <= 10^22.
If there is another pair besides the first five listed with both numbers <= 10^22 then Legendre's conjecture is false.
Proof: If there is another such pair with both numbers <= 10^22 then it must be of the form [p^2, q^2] where p is a prime and q is the least prime larger than p. Then q - p >= 2 (as p != 2). So there is no prime between p^2 and q^2 and hence there is no prime between p^2 and (p+1)^2. This is a counterexample to Legendre's conjecture. (End)

			8 = 2^3, 9 = 3^2, there is no prime between 8 and 9.
25 = 5^2, 27 = 3^3, there is no prime between 25 and 27.

David A. Corneth, PARI program to search for pairs where not both numbers are squares of primes
Wikipedia, Legendre's conjecture

Cf. A014085, A025475, A060846, A067871, A068315, A246547, A366835.

Cf. A116086 and A116455 (for perfect powers, but not necessarily prime powers).

With[{upto=33000},Select[Partition[Select[Range[upto],PrimePowerQ],2,1],NoneTrue[#,PrimeQ]&]] (* Paolo Xausa, Oct 29 2023 *)

ispp(x) = !isprime(x) && isprimepower(x);
lista(nn=50000) = {my(prec = 0); for (i=1, nn, if (ispp(i), if (! prec, prec = i, if (primepi(i) == primepi(prec), print1(prec, ", ", i, ", ")); prec = i;);););} \\ Michel Marcus, Aug 24 2019

A071142 Numbers of the form 2pq where (p,q) is a twin prime pair.

Original entry on oeis.org

30, 70, 286, 646, 1798, 3526, 7198, 10366, 20806, 23326, 38086, 44998, 64798, 73726, 78406, 103966, 115198, 145798, 159046, 194686, 242206, 352798, 373246, 426886, 544966, 649798, 719998, 763846, 824326, 871198, 1312198, 1351366, 1371166, 1472326, 1555846

Offset: 1

Labos Elemer, May 13 2002

nonn

For each term k, A008472(k)/A006530(k) = (2+p+q)/q = (q+q)/q = 2.

			a(1) = 2 * (product of 1st twin prime pair) = 2*3*5 = 30.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A008472, A006530, A000961, A025475, A037074, A071139-A071147.

ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] sb[x_] := Apply[Plus, ba[x]] ma[x_] := Part[Reverse[Flatten[FactorInteger[x]]], 2] amo[x_] := Abs[MoebiusMu[x]] Do[s=sb[n]/ma[n]; If[IntegerQ[s]&&Equal[lf[n], 3]&& !Equal[amo[n], 0], Print[{n, ba[n]}]], {n, 2, 1000000}]

a(n) = 2*A037074(n).

Edited by Jon E. Schoenfield, Sep 30 2023

A085971 Union of primes and numbers that are not prime powers (A000040, A024619).

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Reinhard Zumkeller, Jul 06 2003

Complement of A025475;
A085972(n) = Max{k: a(k)<=n};
different from A007916 and A052485, as a(28)=36;
A085818(a(n)) = A000040(n).

With[{nn=100},Union[Join[Prime[Range[PrimePi[nn]]],DeleteCases[Range[2,80], ?(PrimePowerQ[#]&)]]]] (* _Harvey P. Dale, May 15 2019 *)

is(n)=isprimepower(n)<2 && n>1 \\ Charles R Greathouse IV, Oct 19 2015

from sympy import primepi, integer_nthroot
def A085971(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length())))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 20 2024

a(n) = n + o(sqrt n). - Charles R Greathouse IV, Oct 19 2015

cp's OEIS Frontend

A376310 Run-sums of the sequence of first differences of prime-powers.

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A056824 Numbers of the form p^(2k+1), p prime, k >= 1.

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A071140 Numbers n such that sum of distinct primes dividing n is divisible by largest prime dividing n; n is neither a prime, nor a true power of prime.

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A071147 Smallest squarefree number k with exactly n prime factors such that the sum of the prime factors is divisible by the largest prime dividing k, or 0 if no such k exists.

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A225102 Numbers that can be represented as a sum of two distinct nontrivial prime powers (numbers of the form p^k where p is a prime number and k >= 2).

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A376341 Position of first appearance of n in A057820, the sequence of first differences of prime-powers, or 0 if n does not appear.

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A376600 Inflection or undulation points in the sequence of non-prime-powers inclusive (A024619).

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A068435 Consecutive prime powers without a prime between them.

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A071142 Numbers of the form 2pq where (p,q) is a twin prime pair.