A001694 -id:A001694 - cp's OEIS frontend

A116064 Powerful numbers (A001694) made of nontrivial runs of identical digits.

7744, 665500, 774400, 5556600, 6655000, 7744000, 55566000, 66550000, 77440000, 447722888, 448855888, 555660000, 665500000, 774400000, 1155775544, 5500002244, 5511005500, 5556600000, 6655000000, 7744000000, 11000004488, 11122233444, 22244466888

Offset: 1

Giovanni Resta, Feb 13 2006

A run of length 1 is trivial.

			448855888 = 2^4*13^3*113^2.

Amiram Eldar, Table of n, a(n) for n = 1..201 (terms below 10^18)
Index entries for sequences related to powerful numbers.

Intersection of A001694 and A033023.

a(16)-a(23) from Donovan Johnson, Jun 29 2011

A227297 Suppose that (m, m+1) is a pair of consecutive powerful numbers as defined by A001694. This sequence gives the values of m for which neither m nor m+1 are perfect squares.

Original entry on oeis.org

12167, 5425069447, 11968683934831, 28821995554247, 48689748233307, 161461422688535037152, 3887785221910670811499

Offset: 1

Ant King, Jul 07 2013

a(1) to a(5) were found by Jaroslaw Wroblewski, who also proved that this sequence is infinite (see link to Problem 53 below). However, there are no more terms less than 500^6 = 1.5625*10^16.

A subsequence of A060355 and of A001694.

			12167 is a term because (12167, 12168) are a pair of consecutive powerful numbers, neither of which are perfect squares.
235224 is not a term because although (235224, 235225) are a pair of consecutive powerful numbers, the larger member of the pair is a square number (= 485^2).

Richard K. Guy, Unsolved Problems in Number Theory, 2nd ed., New York, Springer-Verlag, (1994), pp. 70-74. (See Powerful numbers, section B16.)

Solomon W. Golomb, Powerful numbers, Amer. Math. Monthly, Vol. 77, No. 8 (October 1970), 848-852.
Carlos Rivera, Problem 53: Powerful numbers revisited, The Prime Puzzles & Problems Connection.
David T. Walker, Consecutive integer pairs of powerful numbers and related Diophantine equations, Fibonacci Quart., Vol. 14, No. 2 (1976), pp. 111-116.
Wikipedia, Powerful number.
Index entries for sequences related to powerful numbers.

Cf. A060355, A001694.

a(6)-a(7) from the b-file at A060355 added by Amiram Eldar, Mar 22 2025

A342551 a(n) is the smallest m such that A008477(m) is the n-th powerful number (A001694).

Original entry on oeis.org

1, 4, 9, 8, 16, 32, 27, 25, 64, 128, 81, 72, 512, 1024, 108, 2048, 243, 49, 4096, 8192, 16384, 288, 729, 32768, 125, 225, 200, 131072, 262144, 2187, 524288, 1152, 1048576, 432, 2097152, 4194304, 972, 196, 8388608, 648, 33554432, 4608, 864, 67108864, 19683, 268435456

Offset: 1

Bernard Schott, Mar 27 2021

nonn

As A008477 is not injective and terms A008477(n) are precisely the powerful numbers, this sequence lists the smallest preimage of each powerful number.
There are these three possibilities (see corresponding examples):
-> If A008477(q) = q is a fixed point in A008478 and if q = A001694(u) then a(u) = q.
-> If k and m are in A062307 and satisfy A008477(k) = m  and A008477(m) = k, if m = A001694(s) and k = A001694(t), then a(t) = m and a(s) = k;
-> If A008477(j) = v where v is a powerful number not in {A008478 U A062307} and j is the smallest preimage of v with v = A001694(z) then a(z) = j.

			-> A008477(16) = 16 is a fixed point and 16 is the 5th powerful number, so a(5) = 16.
-> 25 and 32 are in A062307 and satisfy A008477(25) = 32 and A008477(32) = 25, as 25 = A001694(6) and 32 = A001694(8), so a(6) = 32 and a(8) = 25.
-> A008477(81) = A008477(256) = 64 that is the 11th powerful number, since 81 is the smallest preimage of 64, so a(11) = 81.

Cf. A001694, A008477, A008478, A062307.

pwf(n) = my(k=1, nb=1); while (nb != n, k++; if (ispowerful(k), nb++)); k; \\ A001694
f(n) = factorback(factor(n)*[0, 1; 1, 0]); \\ A008477
a(n) = my(k=1, p=pwf(n)); while (f(k) != p, k++); k; \\ Michel Marcus, Mar 28 2021

More terms from Amiram Eldar, Mar 27 2021

A348018 a(n) is the index of A064549(n) = n * Product_{p prime|n} p in the sequence of powerful numbers (A001694).

Original entry on oeis.org

1, 2, 4, 3, 6, 9, 10, 5, 7, 14, 16, 12, 20, 21, 24, 8, 28, 15, 31, 22, 35, 36, 39, 19, 17, 44, 13, 32, 48, 50, 51, 11, 56, 59, 60, 23, 65, 67, 68, 33, 71, 73, 75, 52, 43, 81, 84, 27, 30, 37, 90, 64, 94, 29, 97, 46, 102, 104, 107, 74, 110, 111, 62, 18, 117, 119

Offset: 1

Amiram Eldar, Sep 24 2021

nonn

A permutation of the positive integers.

The inverse permutation of A306458.

			The sequence of powerful numbers (A001694) begins with 1, 4, 8, 9, ...
The position of A064549(1) = 1 in A001694 is 1, so a(1) = 1.
The position of A064549(3) = 9 in A001694 is 4, so a(3) = 4.

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

Cf. A001694, A064549, A306458.

powQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; pows = Join[{1}, Select[Range[10^4], powQ]]; TakeWhile[Table[FirstPosition[pows, n * Times @@ (First /@ FactorInteger[n])][[1]], {n, 1, 100}], NumericQ]

A001694(a(n)) = A064549(n).
A306458(a(n)) = a(A306458(n)) = n.
The fixed points of this permutation are 1, 2, 12, 1208, 1256, 1288 and no more below 3*10^5.

A360722 a(n) is the sum of infinitary divisors of n that are powerful (A001694).

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 13, 10, 1, 1, 5, 1, 1, 1, 17, 1, 10, 1, 5, 1, 1, 1, 13, 26, 1, 37, 5, 1, 1, 1, 49, 1, 1, 1, 50, 1, 1, 1, 13, 1, 1, 1, 5, 10, 1, 1, 17, 50, 26, 1, 5, 1, 37, 1, 13, 1, 1, 1, 5, 1, 1, 10, 85, 1, 1, 1, 5, 1, 1, 1, 130, 1, 1, 26, 5, 1, 1, 1, 17

Offset: 1

Amiram Eldar, Feb 18 2023

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

Cf. A001694, A049417, A077609, A077610, A360723.

Similar sequences: A183097, A360721.

f[p_, e_] := Times @@ (p^(2^(-1 + Flatten @ Position[Reverse@IntegerDigits[e, 2], ?(# == 1 &)])) + 1) - If[OddQ[e], p, 0]; a[1] = 1; a[n] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k))+1, 1)) - if(f[i, 2]%2, f[i, 1], 0));}

Multiplicative with a(p^e) = f(p, e) if e is even, and f(p, e) - p is e is odd, where f(p, e) = Product{k>=1, e_k=1} (p^(2^k) + 1), where e = Sum_{k} e_k * 2^k is the binary representation of e, i.e., e_k is bit k of e.
a(n) <= A049417(n), with equality if and only if n is a square.
a(n) <= A183097(n), with equality if and only if n is not in A360723.

A362984 Decimal expansion of the asymptotic mean of the abundancy index of the powerful numbers (A001694).

Original entry on oeis.org

2, 1, 4, 9, 6, 8, 6, 9, 0, 3, 0, 1, 5, 2, 6, 7, 6, 5, 1, 2, 8, 2, 1, 9, 0, 4, 2, 1, 0, 5, 1, 0, 9, 4, 1, 6, 1, 4, 5, 9, 8, 7, 6, 5, 3, 2, 7, 5, 1, 0, 0, 9, 9, 9, 8, 7, 3, 2, 7, 3, 3, 4, 3, 7, 8, 9, 7, 6, 2, 7, 1, 7, 9, 4, 0, 3, 6, 4, 2, 3, 6, 5, 7, 4, 2, 7, 4, 2, 3, 7, 7, 1, 7, 0, 2, 4, 2, 2, 8, 9, 7, 3, 8, 6, 2

Offset: 1

Amiram Eldar, May 12 2023

The abundancy index of a positive integer k is A000203(k)/k = A017665(k)/A017666(k).

The asymptotic mean of the abundancy index over all the positive integers is lim_{m->oo} (1/m) * Sum_{k=1..m} A000203(k)/k = Pi^2/6 = zeta(2) = 1.644934... (A013661).

			2.14968690301526765128219042105109416145987653275100999873...

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (6).

Cf. A000203, A001694, A017665, A017666, A180114.

Similar constants (the asymptotic mean of the abundancy index of other sequences): A013661 (all positive integers), A082020 (cubefree), A111003 (odd), A157292 (5-free), A157294 (7-free), A157296 (9-free), A240976 (squares), A245058 (even), A306633 (squarefree), A362985 (cubefull).

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{2, -3, 4, -6, 7, -7, 7, -6, 5, -3, 2, -1}, {0, 0, 0, 4, 5, 6, 0, -12, -9, -5, 0, 22}, m]; RealDigits[(2^4 + 2^2 + 2^(3/2) - 1)/(2^4 - 2)*(3^4 + 3^2 + 3^(3/2) - 1)/(3^4 - 3) * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n/2] - 1/2^(n/2) - 1/3^(n/2))/n, {n, 4, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 120][[1]]

prodeulerrat((p^8 + p^4 + p^3 - 1)/(p^8 - p^2), 1/2)

Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A180114(k)/A001694(k).

Equals Product_{p prime} (p^4 + p^2 + p^(3/2) - 1)/(p^4 - p) = Product_{p prime} (1 + (p^2 + p^(3/2) + p - 1)/(p^4 - p)) (Jakimczuk and Lalín, 2022).

A364991 Primitive coreful 3-abundant numbers: coreful 3-abundant numbers (A340109) that are powerful numbers (A001694).

Original entry on oeis.org

5400, 7200, 10800, 14400, 16200, 18000, 21168, 21600, 27000, 28800, 32400, 36000, 42336, 43200, 48600, 54000, 56448, 57600, 63504, 64800, 72000, 81000, 84672, 86400, 88200, 90000, 97200, 98784, 104544, 108000, 112896, 115200, 127008, 129600, 135000, 144000, 145800

Offset: 1

Amiram Eldar, Aug 15 2023

nonn

Powerful numbers k such that csigma(k) > 3*k, where csigma(k) = A057723(k) is the sum of the coreful divisors of k.

If m is a term and k is a squarefree number coprime to m, then csigma(k*m) = csigma(k) * csigma(m) = k * csigma(m) > 3*k*m, so k*m is a coreful 3-abundant number. Therefore, the sequence of coreful 3-abundant numbers (A340109) can be generated from this sequence by multiplying with coprime squarefree numbers. The asymptotic density of the coreful 3-abundant numbers can be calculated from this sequence (see comment in A340109).

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

Intersection of A001694 and A340109.

Subsequence of A356871.

f[p_, e_] := (p^(e+1)-1)/(p-1)-1; g[1] = 1; g[n_] := If[AllTrue[(fct = FactorInteger[n])[[;; , 2]], #>1 &], Times @@ f @@@ fct, 0]; seq[kmax_] := Module[{s = {}}, Do[If[g[k] > 3*k, AppendTo[s, k]], {k, 1, kmax}]; s]; seq[500000]

s(f) = prod(i = 1, #f~, sigma(f[i, 1]^f[i, 2]) - 1);
lista(kmax) = {my(f); for(k=2, kmax, f=factor(k); if(vecmin(f[,2]) > 1 && s(f) > 3*k, print1(k, ", ")));}

A379545 Triangle read by rows where row n lists powerful divisors d | n (i.e., d in A001694).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 1, 1, 4, 8, 1, 9, 1, 1, 1, 4, 1, 1, 1, 1, 4, 8, 16, 1, 1, 9, 1, 1, 4, 1, 1, 1, 1, 4, 8, 1, 25, 1, 1, 9, 27, 1, 4, 1, 1, 1, 1, 4, 8, 16, 32, 1, 1, 1, 1, 4, 9, 36, 1, 1, 1, 1, 4, 8, 1, 1, 1, 1, 4, 1, 9, 1, 1, 1, 4, 8, 16, 1, 49, 1, 25, 1, 1, 4

Offset: 1

Michael De Vlieger, Feb 13 2025

Intersection of row n of A027750 and A001694.

			D(1) = {1} = row 1 of this sequence since 1 | 1 is powerful.
D(2) = {1, 2}; of these, only 1 is powerful.
D(4) = {1, 2, 4}; of these, only 2 is not powerful, so row 4 = {1, 4}.
D(6) = {1, 2, 3, 6}; of these, only 1 is powerful.
D(8) = {1, 2, 4, 8}; of these, only 2 is not powerful, so row 4 = {1, 4, 8}.
D(12) = {1, 2, 3, 4, 6, 12}; of these, only {1, 4} are powerful.
D(36) = {1, 2, 3, 4, 6, 9, 12, 18, 36}; of these, only {1, 4, 9, 36} are powerful, etc.
Table begins:
   n:  row n
  ----------------
   1:  1;
   2:  1;
   3:  1;
   4:  1, 4;
   5:  1;
   6:  1;
   7:  1;
   8:  1, 4, 8;
   9:  1, 9;
  10:  1;
  11:  1;
  12:  1, 4;
  13:  1;
  14:  1;
  15:  1;
  16:  1, 4, 8, 16;
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11445 (rows n = 1..6000, flattened).

Cf. A000005, A001694, A005361, A007947, A027750, A332785, A380819.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[Select[Divisors[n], Divisible[#, rad[#]^2] &], {n, 2, 60}] // Flatten

row(n) = select(x -> ispowerful(x), divisors(n)); \\ Amiram Eldar, May 02 2025

First term in row n is 1.
Row n does not contain squarefree factors of n, and also does not contain factors in A332785.
Length of row n = A005361(n) = tau(n/rad(n)), where tau = A000005 and rad = A007947.
For squarefree n, row n = {1}.
Let D(n) = row n of A027750. For prime p and m > 0, row p^m = D(p^m) \ {p}, since d = 1 and p = p^j, j > 1 are powerful, but primes are squarefree (and not powerful).

A113938 Numbers which are powerful(1) (A001694) and triangular at the same time.

Original entry on oeis.org

1, 36, 1225, 29403, 41616, 1413721, 48024900, 74024028, 1631432881, 27665282700, 55420693056, 1024060098375, 1665678432600, 1882672131025, 63955431761796, 194838725161125, 2172602007770041, 26030209161894003, 73804512832419600

Offset: 1

Giovanni Resta, Jan 31 2006

nonn

			29403=T(242) and 29403=3^5*11^2.

Donovan Johnson, Table of n, a(n) for n = 1..47 (terms < 10^37)

Intersection of A001694 and A000217.

a(12)-a(19) from Donovan Johnson, Dec 07 2008

A115685 Prime numbers whose digit reversal is a powerful(1) number (A001694).

Original entry on oeis.org

23, 61, 163, 293, 487, 521, 691, 821, 1297, 1861, 2531, 2731, 4201, 4441, 4483, 5209, 5227, 8429, 8867, 9049, 9631, 12391, 14437, 16141, 16987, 25153, 25703, 29741, 52163, 61483, 63211, 65579, 65707, 65899, 67057, 67901, 69481, 80687

Offset: 1

Giovanni Resta, Jan 31 2006

			8867 is prime and 7688 = 2^3 * 31^2.

Chai Wah Wu, Table of n, a(n) for n = 1..8201 (terms 1..458 from Robert Israel)

Cf. A001694.

N:= 10000; # to get all entries <= N
F:= proc(p)
     local L,i,q,f;
     if not isprime(p) then return false end if;
     L:= convert(p,base,10);
     q:= add(10^(i-1)*L[-i],i=1..nops(L));
     f:= ifactors(q)[2];
     not has(map2(op,2,f),1);
    end proc;
select(F,[2*i+1 $ i=1..floor((N-1)/2)]);
# Robert Israel, Feb 11 2013

fQ[n_] := Min[ Last@# & /@ FactorInteger@ FromDigits@ Reverse@ IntegerDigits@ n] > 1; Select[ Prime@ Range@ 8000, fQ] (* Robert G. Wilson v, Feb 11 2013 *)

cp's OEIS Frontend

A116064 Powerful numbers (A001694) made of nontrivial runs of identical digits.

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A227297 Suppose that (m, m+1) is a pair of consecutive powerful numbers as defined by A001694. This sequence gives the values of m for which neither m nor m+1 are perfect squares.

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A342551 a(n) is the smallest m such that A008477(m) is the n-th powerful number (A001694).

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A348018 a(n) is the index of A064549(n) = n * Product_{p prime|n} p in the sequence of powerful numbers (A001694).

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A360722 a(n) is the sum of infinitary divisors of n that are powerful (A001694).

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A362984 Decimal expansion of the asymptotic mean of the abundancy index of the powerful numbers (A001694).

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A364991 Primitive coreful 3-abundant numbers: coreful 3-abundant numbers (A340109) that are powerful numbers (A001694).

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A379545 Triangle read by rows where row n lists powerful divisors d | n (i.e., d in A001694).

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