A060355 -id:A060355 - cp's OEIS frontend

A199801 Primitive elements of A060355: n such that n and n+1 are powerful but n is not of the form 4m(m+1) where m and m+1 are powerful.

8, 675, 9800, 12167, 235224, 465124, 11309768, 4931691075, 5425069447, 13051463048, 8192480787000, 11968683934831, 15061377048200, 28821995554247

Offset: 1

Charles R Greathouse IV, Nov 10 2011

nonn

This sequence is infinite. See the comment dated Nov 19 2012 in A060355. - Franklin T. Adams-Watters, Nov 09 2013

			8 and 9 are powerful and although 8 = 4*1*(1+1), 1+1 is not powerful so 8 is in this sequence.
288 and 289 are powerful but 288 = 4*8*(8+1) and both 8 and 8+1 are powerful so 288 is not in this sequence.
675 and 676 are powerful and 675 is not of the form 4m(m+1) so 675 is in this sequence.

Donovan Johnson, Table of n, a(n) for n = 1..24 (terms < 10^22)

Subsequence of A060355 and hence of A001694.

A056169 Number of unitary prime divisors of n.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 1, 0, 2, 0, 1, 1, 3, 1, 0, 2, 2, 2, 0, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 0, 2, 3, 1, 1, 2, 3, 1, 0, 1, 2, 1, 1, 2, 3, 1, 1, 0, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 1, 0, 1, 3, 1, 1, 3

Offset: 1

Labos Elemer, Jul 27 2000

The zeros of this sequences are the powerful numbers (A001694). There are no arbitrarily long subsequences with a given upper bound; for example, every sequence of 4 values includes one divisible by 2 but not 4, so there are no more than 3 consecutive zeros. Similarly, there can be no more than 23 consecutive values with none divisible by both 2 and 3 but neither 4 nor 9 (so a(n) >= 2), etc. In general, this gives an upper bound that is a (relatively) small multiple of the k-th primorial number (prime(k)#). One suspects that the actual upper bounds for such subsequences are quite a bit lower; e.g., Erdős conjectured that there are no three consecutive powerful numbers. - Franklin T. Adams-Watters, Aug 08 2006
In particular, for every A048670(k)*A002110(k) consecutive terms, at least one is greater than or equal to k. - Charlie Neder, Jan 03 2019
Following Catalan's conjecture (which became Mihăilescu's theorem in 2002), the first case of two consecutive zeros in this sequence is for a(8) and a(9), because 8 = 2^3 and 9 = 3^2, and there are no other consecutive zeros for consecutive powers. However, there are other pairs of consecutive zeros at powerful numbers (A001694, A060355). The next example is a(288) = a(289) = 0, because 288 = 2^5 * 3^2 and 289 = 17^2, then also a(675) and a(676). - Bernard Schott, Jan 06 2019
a(2k-1) is the number of primes p such that p || x + y and p^2 || x^(2k-1) + y^(2k-1) for some positive integers x and y. For any positive integers x, y and k > 1, there is no prime p such that p || x + y and p^2 || x^(2k) + y^(2k). - Jinyuan Wang, Apr 08 2020

			9 = 3^2 so a(9) = 0; 10 = 2 * 5 so a(10) = 2; 11 = 11^1 so a(11) = 1.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Catalan's Conjecture.
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A001694, A002110, A034444, A056170, A055231, A076445, A162642, A275812, A295659, A295662, A295664, A001694.
See also A060355.
Cf. A077761, A085548.

a056169 = length . filter (== 1) . a124010_row
-- Reinhard Zumkeller, Sep 10 2013

a:= n-> nops(select(i-> i[2]=1, ifactors(n)[2])):
seq(a(n), n=1..120);  # Alois P. Heinz, Mar 27 2017

Join[{0},Table[Count[Transpose[FactorInteger[n]][[2]],1],{n,2,110}]] (* Harvey P. Dale, Mar 15 2012 *)
Table[DivisorSum[n, 1 &, And[PrimeQ@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Nov 28 2017 *)

a(n)=my(f=factor(n)[,2]); sum(i=1,#f,f[i]==1) \\ Charles R Greathouse IV, Apr 29 2015

from sympy import factorint
def a(n):
    f=factorint(n)
    return 0 if n==1 else sum(1 for i in f if f[i]==1)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 19 2017

;; With memoization-macro definec.
(definec (A056169 n) (if (= 1 n) 0 (+ (if (= 1 (A067029 n)) 1 0) (A056169 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

A prime factor of n is unitary iff its exponent is 1 in prime factorization of n. In general, gcd(p, n/p) = 1 or = p.
Additive with a(p^e) = 1 if e = 1, 0 otherwise.
a(n) = #{k: A124010(n,k) = 1, k = 1..A001221}. - Reinhard Zumkeller, Sep 10 2013
From Antti Karttunen, Nov 28 2017: (Start)
a(1) = 0; for n > 1, a(n) = A063524(A067029(n)) + a(A028234(n)).
a(n) = A001221(A055231(n)) = A001222(A055231(n)).
a(n) = A001221(n) - A056170(n) = A001221(n) - A001221(A000188(n)).
a(n) = A001222(n) - A275812(n).
a(n) = A162642(n) - A295662(n).
a(n) <= A162642(n) <= a(n) + A295659(n).
a(n) <= A295664(n).
(End)
Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B - C), where B is Mertens's constant (A077761) and C = Sum_{p prime} (1/p^2) = 0.452247... (A085548). - Amiram Eldar, Sep 28 2023

A062739 Odd powerful numbers.

Original entry on oeis.org

1, 9, 25, 27, 49, 81, 121, 125, 169, 225, 243, 289, 343, 361, 441, 529, 625, 675, 729, 841, 961, 1089, 1125, 1225, 1323, 1331, 1369, 1521, 1681, 1849, 2025, 2187, 2197, 2209, 2401, 2601, 2809, 3025, 3087, 3125, 3249, 3267, 3375, 3481, 3721, 3969, 4225

Offset: 1

Labos Elemer, Jul 12 2001

nonn

Smallest term of this sequence not also in A075109 is 675, followed by 1125. - Alonso del Arte, Nov 22 2011

			Consecutive-odd examples from Sentance: {25,27},{70225,70227},{189750625,189750627}

R. K. Guy, Unsolved Problems in Number Theory, B16

Zak Seidov, Table of n, a(n) for n = 1..10000
W. A. Sentance, Occurrences of consecutive odd powerful numbers, Amer. Math. Monthly, 88 (1981), 272-274.

Cf. A001694, A060355, A060859, A060860, A082695.

Cf. A076445 (consecutive odd powerful numbers).

Powerful[n_Integer] := (n ==1) || Min[Transpose[FactorInteger[n]][[2]]]>=2; Select[Range[5000],OddQ[ # ]&&Powerful[ # ]&] (* T. D. Noe, May 04 2006 *)
Join[{1},Select[Range[3,4301,2],Min[FactorInteger[#][[All,2]]]>1&]] (* Harvey P. Dale, Jan 08 2021 *)

It is not true that a(n) = A001694(2n-1).

Sum_{n>=1} 1/a(n) = (2/3) * Sum_{n>=1} 1/A001694(n) = 2*zeta(2)*zeta(3)/(3*zeta(6)) = (2/3) * A082695 = 1.2957309... - Amiram Eldar, Jun 23 2020

Checked by T. D. Noe, May 04 2006

A076445 The smaller of a pair of powerful numbers (A001694) that differ by 2.

Original entry on oeis.org

25, 70225, 130576327, 189750625, 512706121225, 13837575261123, 99612037019889, 1385331749802025, 3743165875258953025, 10114032809617941274225, 8905398244301708746029223, 27328112908421802064005625, 73840550964522899559001927225

Offset: 1

Jud McCranie, Oct 15 2002

nonn

Erdos conjectured that there aren't three consecutive powerful numbers and no examples are known. There are an infinite number of powerful numbers differing by 1 (cf. A060355). A requirement for three consecutive powerful numbers is a pair that differ by 2 (necessarily odd). These pairs are much more rare.

Sentance gives a method for constructing families of these numbers from the solutions of Pell equations x^2-my^2=1 for certain m whose square root has a particularly simple form as a continued fraction. Sentance's result can be generalized to any m such that A002350(m) is even. These m, which generate all consecutive odd powerful numbers, are in A118894. - T. D. Noe, May 04 2006

			25=5^2 and 27=3^3 are powerful numbers differing by 2, so 25 is in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, B16

Max Alekseyev, Conjectured table of n, a(n) for n = 1..33 [These terms certainly belong to the sequence, but they are not known to be consecutive.]
R. A. Mollin and P. G. Walsh, On powerful numbers, IJMMS 9:4 (1986), 801-806.
W. A. Sentance, Occurrences of consecutive odd powerful numbers, Amer. Math. Monthly, 88 (1981), 272-274.
Eric Weisstein's World of Mathematics, Powerful numbers

Cf. A001694, A060355.

a(8)-a(10) from Geoffrey Reynolds (geoff(AT)hisplace.co.nz), Feb 15 2005

More terms from T. D. Noe, May 04 2006

A354558 Numbers k such that k and k+1 are both divisible by the square of their largest prime factor.

Original entry on oeis.org

8, 49, 242, 288, 675, 1444, 1681, 2400, 2645, 6727, 6859, 9408, 9800, 10647, 12167, 13689, 18490, 23762, 24299, 26010, 36517, 47915, 48734, 57121, 58080, 59535, 75809, 85697, 101250, 103246, 113568, 118579, 131043, 142884, 158949, 182182, 201019, 212194, 235224

Offset: 1

Amiram Eldar, May 30 2022

nonn

Numbers k such that P(k)^2 | k and P(k+1)^2 | (k+1), where P(k) = A006530(k).

			8 = 2^3 is a term since P(8) = 2 and 2^2 | 8, 9 = 3^2, P(9) = 3 and 3^2 | 9.
675 = 3^3 * 5^2 is a term since P(675) = 5, 5^2 | 675, 676 = 2^2 * 13^2, P(676) = 13 and 13^2 | 676.

Michael S. Branicky, Table of n, a(n) for n = 1..12072 (all terms <= 10^11)
Jean-Marie De Koninck, Nicolas Doyon, and Florian Luca, Consecutive integers divisible by the square of their largest prime factors, Journal of Combinatorics and Number Theory, Vol. 5, No. 2 (2013), pp. 81-93; Researchgate link.
Jean-Marie De Koninck and Matthieu Moineau, Consecutive Integers Divisible by a Power of their Largest Prime Factor, J. Integer Seq., Vol. 21 (2018), Article 18.9.3.
Régis de la Bretèche and Sary Drappeau, Niveau de répartition des polynômes quadratiques et crible majorant pour les entiers friables, Journal of the European Mathematical Society, Vol. 22, No. 5 (2020), pp. 1577-1624; arXiv preprint, arXiv:1703.03197 [math.NT], 2017-2019.
Daniel Suteu, Table of n, a(n) for n = 1..112589 (all terms <= 10^13)

Subsequence of A070003.
Subsequences: A060355, A354560, A354562, A354563, A354564, A354565, A354566.
Cf. A006530, A071178, A354559.

q[n_] := FactorInteger[n][[-1, 2]] > 1; Select[Range[250000], q[#] && q[# + 1] &]

from sympy import factorint
def c(n): f = factorint(n); return f[max(f)] >= 2
def ok(n): return n > 1 and c(n) and c(n+1)
print(list(filter(ok, range(235225)))) # Michael S. Branicky, May 30 2022

x^(1/4)/log(x) << N(x) << x*exp(-c*sqrt(2*log(x)*log(log(x)))), where N(x) is the number of terms <= x, c = 25/24 (De Koninck et al., 2013), or 4/sqrt(5) (de la Bretèche and Drappeau, 2020).

A349062 Powerful numbers (A001694) with a record gap to the next powerful number.

Original entry on oeis.org

1, 4, 9, 16, 36, 49, 81, 144, 169, 256, 289, 441, 529, 576, 676, 729, 900, 1024, 1156, 1225, 1372, 1444, 1600, 1849, 2209, 2401, 2916, 3600, 3721, 4096, 4356, 4624, 4761, 5041, 5625, 6400, 6561, 7225, 7396, 8281, 9025, 9409, 9801, 11025, 11236, 11664, 12544, 14400

Offset: 1

Amiram Eldar, Nov 07 2021

nonn

This sequence is infinite since the asymptotic density of the powerful numbers is 0.
The corresponding record gaps are 3, 4, 7, 9, 13, 15, 19, 25, 27, 32, 35, 43, ...
Apparently, most of the terms are squares. The nonsquare terms are 1372, 465125, 4879688, ... (A371191).

			The sequence of powerful numbers begins with 1, 4, 8, 9, 16, 25, 27, 32, 36 and 49. The differences between these terms are 3, 4, 1, 7, 9, 2, 5, 4 and 13. The record values, 3, 4, 7, 9 and 13 occur after the powerful numbers 1, 4, 9, 16 and 36, the first 5 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Cf. A001694, A060355, A076445, A371191.

powQ[n_] := Min[FactorInteger[n][[;; , 2]]] > 1; seq[nmax_] := Module[{s = {}, n1 = 1, gapmax = 0, gap}, Do[If[powQ[n], gap = n - n1; If[gap > gapmax, gapmax = gap; AppendTo[s, n1]]; n1 = n], {n, 2, nmax}]; s]; seq[10^5]

A060859 Powerful numbers of the form k^2 - 1.

Original entry on oeis.org

8, 288, 675, 9800, 235224, 332928, 1825200, 11309768, 384199200, 592192224, 4931691075, 13051463048, 221322261600, 443365544448, 865363202000, 8192480787000, 13325427460800, 15061377048200, 511643454094368

Offset: 1

Labos Elemer, May 04 2001

nonn

If k^2-1 is a term, then k-1 is a term of A335851. - Amiram Eldar, Feb 23 2024

			From _Jon E. Schoenfield_, Sep 06 2017: (Start)
n     k        a(n) =  k^2 - 1          a(n) + 1 = k^2
=   ===   =========================   ==================
1     3        8 = 2^3                  3^2 = 3^2
2    17      288 = 2^5 * 3^2           17^2 = 17^2
3    26      675 = 5^2 * 3^3           26^2 = 2^2 * 13^2
4    99     9800 = 2^3 * 5^2 * 7^2     99^2 = 3^4 * 11^2
5   485   235224 = 2^3 * 3^5 * 11^2   485^2 = 5^2 * 97^2
6   577   332928 = 2^7 * 3^2 * 17^2   577^2 = 577^2
(End)

Amiram Eldar, Table of n, a(n) for n = 1..55 (terms below 10^36; terms 1..30 from Donovan Johnson)
Index entries for sequences related to powerful numbers.

Proper subset of A060355.

Cf. A001694, A060860, A335851.

Select[Range[10^6]^2 - 1, Min[FactorInteger[#][[All, -1]]] > 1 &] (* Michael De Vlieger, Sep 05 2017 *)
seq[max_] := Module[{p = Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]], q, i}, q = Union[p, 2*Select[p, # <= max && OddQ[#] &]]; i = Position[Differences[q], 2] // Flatten; q[[i]]*(q[[i]] + 2)]; seq[10^10] (* Amiram Eldar, Feb 23 2024 *)

isok(n) = issquare(n+1) && ispowerful(n); \\ Michel Marcus, Sep 05 2017

a(n) = k^2 - 1 and a(n) + 1 = k^2 are consecutive powerful numbers.

a(n) = A060860(n)^2 - 1. - Amiram Eldar, Feb 23 2024

Corrected and extended by Jud McCranie, Jul 08 2001
Offset corrected by Donovan Johnson, Nov 15 2011
Name simplified by Jon E. Schoenfield, Nov 30 2023

A060860 Numbers k such that k^2-1 and k^2 are consecutive powerful numbers.

Original entry on oeis.org

3, 17, 26, 99, 485, 577, 1351, 3363, 19601, 24335, 70226, 114243, 470449, 665857, 930249, 2862251, 3650401, 3880899, 22619537, 39480499, 130576328, 131836323, 189750626, 456335045, 768398401, 1184384449, 4478554083, 9863382151, 10850138895, 26102926097

Offset: 1

Labos Elemer, May 04 2001

nonn

a(31) > 10^11. - Donovan Johnson, Nov 15 2011

a(n) - 1 is a term of A335851. - Amiram Eldar, Feb 23 2024

			592192224 = 2^5*3^2*13^2*23^3 = 24334*24336, 592192225 = 5^2*31^2*157^2 = 24335^2.

Amiram Eldar, Table of n, a(n) for n = 1..55
Index entries for sequences related to powerful numbers.

Cf. A060355, A060859, A001694, A335851.

seq[max_] := Module[{p = Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]], q, i}, q = Union[p, 2*Select[p, # <= max && OddQ[#] &]]; i = Position[Differences[q], 2] // Flatten; Sqrt[q[[i]]*(q[[i]] + 2) + 1]]; seq[10^10] (* Amiram Eldar, Feb 23 2024 *)

a(n) = sqrt(A060859(n) + 1). - Amiram Eldar, Feb 23 2024

Corrected and extended by Jud McCranie, Jul 08 2001
a(21)-a(24) from Donovan Johnson, Apr 27 2008
a(25)-a(26) from Donovan Johnson, Dec 07 2008
a(27)-a(28) from Donovan Johnson, Jun 17 2011
a(29)-a(30) from Donovan Johnson, Nov 15 2011

A348119 Numbers k such that k and k+1 are both numbers whose powerful part is larger than their powerfree part (A328014).

Original entry on oeis.org

8, 24, 48, 49, 63, 80, 175, 224, 242, 288, 324, 350, 351, 360, 512, 539, 575, 675, 735, 832, 960, 1088, 1215, 1224, 1368, 1444, 1681, 1700, 1862, 2057, 2106, 2299, 2303, 2375, 2400, 2600, 2624, 2645, 2808, 3024, 3249, 3750, 3887, 3968, 4224, 4374, 4624, 4900, 5040

Offset: 1

Amiram Eldar, Oct 01 2021

nonn

			8 is a term since both 8 and 9 are in A328014.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A328014.

A060355 is a subsequence.

f[p_, e_] := If[e==1, p, 1]; s[n_] := Times @@ (f @@@ FactorInteger[n]); Select[Range[5040], s[#]^2 < # && s[#+1]^2 < #+1 &]

A078326 Numbers n such that n-1 and n are a pair of consecutive powerful numbers.

Original entry on oeis.org

9, 289, 676, 9801, 12168, 235225, 332929, 465125, 1825201, 11309769, 384199201, 592192225, 4931691076, 5425069448, 13051463049, 221322261601, 443365544449, 865363202001, 8192480787001, 11968683934832, 13325427460801, 15061377048201, 28821995554248

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

a(n) = u*rad(u) = v*rad(v)+1 for appropriate u, v, where rad(n) = A007947(n) is the squarefree kernel.

Also numbers n such that n(n-1) is a powerful number. - Charles R Greathouse IV, Aug 08 2013

Donovan Johnson, Table of n, a(n) for n = 1..39 (terms < 10^22)
Jérôme Germoni, Nombres puissants au bac S, Images des Mathématiques, CNRS, 2021 (in French).
Eric Weisstein's World of Mathematics, Powerful Number

Cf. A078310, A064549, A078315, A078325, A060355, A078326.

is(n)=ispowerful(n-1)&&ispowerful(n) \\ Charles R Greathouse IV, Aug 08 2013

a(n) = A060355(n)+1.

a(22)-a(23) from Donovan Johnson, Jul 29 2011

cp's OEIS Frontend

A199801 Primitive elements of A060355: n such that n and n+1 are powerful but n is not of the form 4m(m+1) where m and m+1 are powerful.

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A056169 Number of unitary prime divisors of n.

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A062739 Odd powerful numbers.

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A076445 The smaller of a pair of powerful numbers (A001694) that differ by 2.

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A354558 Numbers k such that k and k+1 are both divisible by the square of their largest prime factor.

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A349062 Powerful numbers (A001694) with a record gap to the next powerful number.

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A060859 Powerful numbers of the form k^2 - 1.

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