A007947 -id:A007947 - cp's OEIS frontend

A147800 Minimal value of A007947(m*(5^n-m)) with m coprime to 5.

2, 6, 22, 42, 222, 366, 2046, 13962, 10626, 79926, 293262

Offset: 1

M. F. Hasler, Nov 13 2008

The minima are reached for m values given in A147803.
This is related to the abc conjecture.
All terms of this sequence are even, so one could also consider A147800/2 = 1, 3, 11, 21, 111, 183, 1023, 6981, 5313, 39963, 146631, ...

Cf. A007947, A147803 (m values), A143702 (analog for 2^n), A147801 (analog for 3^n), A147298 (general case).

A147800(n, p=5) = {my(m=n=p^n); for(a=1, (n-1)\2, a%p || next; A007947(n-a)*A007947(a)A007947((n-a)*a)); m; }

A147802 Least m coprime to 3 minimizing A007947(m*(3^n-m)).

Original entry on oeis.org

1, 1, 2, 1, 1, 25, 139, 289, 31, 1, 16096, 49, 7424, 588665, 619115, 83521, 8000000, 1515625, 505620842, 17643776, 244140625, 5443635008

Offset: 1

M. F. Hasler, Nov 13 2008

Related to the abc conjecture: Since m is coprime to 3, it is also coprime to 3^n and thus to 3^n-m. Thus A007947(m*(3^n-m)*3^n) = 3*A007947(m(3^n-m)).

Cf. A007947, A143700 (analog for 2^n), A147300 (general case).

rad[n_] := Times @@ FactorInteger[n][[All, 1]];
a[n_] := MinimalBy[Select[Range[3^n - 2], CoprimeQ[#, 3] &], rad[# (3^n - #)] &][[1]];
Reap[Do[Print[n, " ", an = a[n]]; Sow[an], {n, 1, 16}]][[2, 1]] (* Jean-François Alcover, Mar 27 2020 *)

A147802(n, p=3) = {local(b, m=n=p^n); for(a=1, (n-1)\2, a%p || next; A007947(n-a)*A007947(a)A007947((n-a)*b=a)); b; }

a(17) from Jean-François Alcover, Mar 28 2020

a(18)-a(22) from Giovanni Resta, Mar 29 2020

A255423 The least number k > A255334(n) for which A000203(k) = A000203(A255334(n)) and A007947(k) = A007947(A255334(n)), where A000203 gives the sum of divisors, and A007947 gives the squarefree kernel of n.

Original entry on oeis.org

2058, 10290, 22638, 26754, 34986, 39102, 47334, 51450, 59682, 52728, 63798, 76146, 84378, 88494, 96726, 109074, 113190, 121422, 125538, 133770, 137886, 146118, 150234, 162582, 170814, 174930, 183162, 195510, 199626, 207858, 211974, 220206, 224322, 232554, 236670, 249018, 257250, 261366, 269598, 281946, 286062, 294294

Offset: 1

Antti Karttunen, Apr 06 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..6589

Cf. A000203, A007947, A255334.

Cf. also A255335 (same sequence sorted into ascending order), A255424 (squarefree kernel of a(n)), A255426 (same terms with but with their squarefree kernel divided out of them).

A007947(n) = factorback(factorint(n)[, 1]);
nextone(n) = { if(!n,return(0)); my(r=A007947(n), s=sigma(n), k=n+r); while(kA007947(k) == r), return(k), k = k+r)); return(0); };
i=0; for(n=1, 2^25, k = nextone(n); if(k, i++; write("b255423.txt", i, " ", k))); \\ Andrew Lelechenko, May 09 2014

a(n) = A255424(n) * A255426(n).

A255424 Squarefree kernel of A255334: a(n) = A007947(A255334(n)).

Original entry on oeis.org

42, 210, 462, 546, 714, 798, 966, 210, 1218, 78, 1302, 1554, 1722, 1806, 1974, 2226, 2310, 2478, 2562, 2730, 2814, 2982, 3066, 3318, 3486, 3570, 3738, 3990, 4074, 4242, 4326, 4494, 4578, 4746, 4830, 462, 210, 5334, 5502, 5754, 5838, 6006, 6090, 390, 6258, 6342, 6510, 6594, 6846, 7014, 546, 7266, 7518, 7602, 7770, 7854, 8022, 8106, 8274, 8358, 8610

Offset: 1

Antti Karttunen, Apr 06 2015

nonn

Sequence gives value of A007947(n) for numbers n for which there exists k > n such that A000203(k) = A000203(n) and A007947(k) = A007947(n), where A000203 gives the sum of divisors, and A007947 gives the squarefree kernel of n. The sequence is ordered according to the magnitude of n, and contains duplicates, because there are cases of multiple such pairs having same squarefree kernel.

The first duplicate occurs as a(2) = a(8) = 210.

Antti Karttunen, Table of n, a(n) for n = 1..6589

Cf. A007947, A255334, A255412, A255423, A255425, A255426.

(define (A255424 n) (A007947 (A255334 n)))

a(n) = A007947(A255334(n)).

a(n) = A007947(A255423(n)). [Equally, squarefree kernel of A255423(n).]

A326142 Sum of all other divisors of n except its largest squarefree divisor: a(n) = sigma(n) - A007947(n).

Original entry on oeis.org

0, 1, 1, 5, 1, 6, 1, 13, 10, 8, 1, 22, 1, 10, 9, 29, 1, 33, 1, 32, 11, 14, 1, 54, 26, 16, 37, 42, 1, 42, 1, 61, 15, 20, 13, 85, 1, 22, 17, 80, 1, 54, 1, 62, 63, 26, 1, 118, 50, 83, 21, 72, 1, 114, 17, 106, 23, 32, 1, 138, 1, 34, 83, 125, 19, 78, 1, 92, 27, 74, 1, 189, 1, 40, 109, 102, 19, 90, 1, 176, 118, 44, 1, 182, 23, 46, 33

Offset: 1

Antti Karttunen, Jun 09 2019

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000203, A007947, A066503, A326143.
Cf. also A326053, A326061, A326126.
Cf. A013661, A065463.

rad[n_] := Times @@ FactorInteger[n][[;; , 1]]; a[n_] := DivisorSigma[1, n] - rad[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2023 *)

A007947(n) = factorback(factorint(n)[, 1]);
A326142(n) = (sigma(n)-A007947(n));

a(n) = A000203(n) - A007947(n).
a(n) = n + A326143(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 - A065463 = 0.940491... . - Amiram Eldar, Dec 05 2023

A364702 Numbers k in A361098 that are not divisible by A007947(k)^2.

Original entry on oeis.org

48, 50, 54, 75, 80, 96, 98, 112, 135, 147, 160, 162, 189, 192, 224, 240, 242, 245, 250, 252, 270, 294, 300, 320, 336, 338, 350, 352, 360, 363, 375, 378, 384, 396, 405, 416, 448, 450, 468, 480, 486, 490, 504, 507, 525, 528, 540, 550, 560, 567, 578, 588, 594, 600

Offset: 1

Michael De Vlieger, Aug 03 2023

nonn

Subset of A126706, the set of numbers k neither prime powers nor squarefree, i.e., k such that A001222(k) > A001221(k) > 1.
Let p = A119288(k) be the second smallest prime factor of k. Let q = A053669(k) be the smallest prime that does not divide k. Let r = rad(k) = A007947(k) be the squarefree kernel of k. Define sequence S = A361098 = {k : Omega(k) > omega(k) > 1, q*r < k, p*r <= k} = A361098.
Sequence T = A286708 represents numbers in A001694 that are not prime powers. Numbers k in T are such that k = m*r^2, m >= 1, by definition. Since we may rewrite q*r < k instead as q*r < m*r^2, it is clear since omega(r) > 1, that q < r. Further, we may rewrite p*r <= k instead as p*r <= m*r^2, and since p | r, p < r as omega(r) > 1, we see that S contains T.
This sequence gives k that are in S but not in T.

			Let B = A126706.
B(1) = 12 is not in the sequence since 3*6 > 12.
B(2) = 18 is not in the sequence, since, though 3*6 = 18, 5*6 > 18.
B(6) = S(1) = 36 is not in the sequence since, though 3*6 < 36 and 5*6 < 36, rad(36)^2 = 6^2 | 36, hence B(6) = T(1).
B(10) = S(2) = a(1) = 48 is in the sequence since rad(48) = 6, and 6^2 does not divide 48.
B(11) = S(3) = a(2) = 50 is in the sequence since rad(50) = 10, and 10^2 does not divide 50, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A001694, A007947, A053669, A119288, A126706, A286708, A361098.

nn = 2^10; a053669[n_] := If[OddQ[n], 2, p = 2; While[Divisible[n, p], p = NextPrime[p]]; p]; s = Select[Range[nn], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]; Reap[Do[n = s[[j]]; If[And[#1*a053669[n] < n, #1*#2 <= n, ! Divisible[n, #1^2]] & @@ {Times @@ #, #[[2]]} &@ FactorInteger[n][[All, 1]], Sow[n]], {j, Length[s]}] ][[-1, -1]]

This sequence is A361098 \ A286708.

A370896 Partial alternating sums of the squarefree kernel function (A007947).

Original entry on oeis.org

1, -1, 2, 0, 5, -1, 6, 4, 7, -3, 8, 2, 15, 1, 16, 14, 31, 25, 44, 34, 55, 33, 56, 50, 55, 29, 32, 18, 47, 17, 48, 46, 79, 45, 80, 74, 111, 73, 112, 102, 143, 101, 144, 122, 137, 91, 138, 132, 139, 129, 180, 154, 207, 201, 256, 242, 299, 241, 300, 270, 331, 269

Offset: 1

Amiram Eldar, Mar 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A007947, A065463, A073355.

Similar sequences: A068762, A068773, A307704, A357817, A362028.

rad[n_] := Times @@ (First[#]& /@ FactorInteger[n]); Accumulate[Array[(-1)^(#+1) * rad[#] &, 100]]

rad(n) = vecprod(factor(n)[, 1]);
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += (-1)^(k+1) * rad(k); print1(s, ", "))};

a(n) = Sum_{k=1..n} (-1)^(k+1) * A007947(k).

a(n) = c * n^2 + O(R(n)), where c = A065463 / 10 = 0.07044422..., R(n) = x^(3/2)*exp(-c_1*log(n)^(3/5)/log(log(n))^(1/5)) unconditionally, or x^(7/5)*exp(c_2*log(n)/log(log(n))) assuming the Riemann hypothesis, and c_1 and c_2 are positive constants (Tóth, 2017).

A376846 Number of m <= n such that rad(m) | n and Omega(m) > Omega(n), where rad = A007947 and Omega = A001222.

Original entry on oeis.org

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

Offset: 1

Michael De Vlieger, Oct 06 2024

nonn

Number of m not exceeding n such that the squarefree kernel of m divides n, and m has more prime factors with repetition than does n.

Number of m in row n of A162306 such that Omega(m) > Omega(n).

			Table of select n such that a(n) > 0:
   n  a(n)  List of m such that Omega(m) > Omega(n).
  -------------------------------------------------
  10   1    {8}
  14   1    {8}
  18   1    {16}
  20   1    {16}
  22   2    {8, 16}
  26   2    {8, 16}
  28   1    {16}
  30   2    {16, 24}
  33   1    {27}
  34   3    {8, 16, 32}
  36   1    {32}
  38   3    {8, 16, 32}
  39   1    {27}
  40   1    {32}
  42   4    {16, 24, 32, 36}

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Hasse diagrams of m in select rows n of A162306 indicating in red those m such that Omega(m) > Omega(n).
Michael De Vlieger, Numbers k for which floor(log k / log lpf(k)) <= bigomega(k), 2024, about zeros in this sequence.

Cf. A000961, A001222, A007947, A010846, A162306.

rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
{0}~Join~Table[With[{k = PrimeOmega[n]}, Count[Range[n], _?(And[Divisible[n, rad[#]], PrimeOmega[#] > k] &)]], {n, 2, 120}]

a(n) = card({m <= n : rad(m) | n, Omega(m) > Omega(n) }).
a(n) = 0 for prime power n (in A000961).
a(n) < A010846(n).

A381800 a(n) = number of distinct residues r mod n of numbers k such that rad(k) | n, where rad = A007947.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 4, 3, 7, 2, 8, 2, 6, 8, 5, 2, 12, 2, 9, 9, 13, 2, 11, 3, 15, 4, 9, 2, 19, 2, 6, 9, 11, 12, 16, 2, 21, 6, 12, 2, 24, 2, 16, 15, 14, 2, 16, 3, 28, 20, 17, 2, 31, 8, 12, 21, 31, 2, 28, 2, 8, 13, 7, 10, 32, 2, 13, 15, 35, 2, 20, 2, 39, 29, 24

Offset: 1

Michael De Vlieger, Mar 07 2025

nonn

			 n  a(n)  row n of A381801
----------------------------------------------
 1    1   {0}
 2    2   {0,1}
 3    2   {0,1}
 4    3   {0,1,2}
 6    5   {0,1,2,3,4}
 8    4   {0,1,2,4}
10    7   {0,1,2,4,5,6,8}
12    8   {0,1,2,3,4,6,8,9}
14    6   {0,1,2,4,7,8}
15    8   {0,1,3,5,6,9,10,12}
18   12   {0,1,2,3,4,6,8,9,10,12,14,16}
20    9   {0,1,2,4,5,8,10,12,16}
21    9   {0,1,3,6,7,9,12,15,18}
22   13   {0,1,2,4,6,8,10,11,12,14,16,18,20}
24   11   {0,1,2,3,4,6,8,9,12,16,18}
26   15   {0,1,2,4,6,8,10,12,13,14,16,18,20,22,24}
28    9   {0,1,2,4,7,8,14,16,21}
30   19   {0,1,2,3,4,5,6,8,9,10,12,15,16,18,20,21,24,25,27}
36   16   {0,1,2,3,4,6,8,9,12,16,18,20,24,27,28,32}

Michael De Vlieger, Table of n, a(n) for n = 1..5000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..5000, showing a(n) for prime n in red, a(n) for proper prime power n in gold, a(n) such that n is squarefree and composite in green, and a(n) such that n is neither squarefree nor prime power in blue and magenta, where the latter color also signifies n is powerful but not a prime power.
Michael De Vlieger, Faster code for A381800 and A381801, 2025.

Cf. A000005, A010846, A024619, A051953, A381798, A381801.

Table[CountDistinct@ Flatten@ Mod[TensorProduct @@ Map[(p = #; NestWhileList[Mod[p*#, n] &, 1, UnsameQ, All]) &, FactorInteger[n][[All, 1]]], n], {n, 120}]

a(n) = length of row n of A381801.
a(1) = 1 since 1 is the empty product.
A010846(n) <= a(n) <= A051953(n).
a(n) >= 2 for n > 1.
For prime p, a(p) = A010846(p^m) = A000005(p^m) = A381798(p) = 2.
For prime power p^m, m > 0, a(p^m) = A010846(p^m) = A000005(p^m) = A381798(p^m) = m+1.
For n in A024619, a(n) > A381798(n).

A078318 Sum of divisors of n*rad(n)+1, where rad = A007947 (squarefree kernel).

Original entry on oeis.org

3, 6, 18, 13, 42, 38, 93, 18, 56, 102, 186, 74, 324, 198, 342, 48, 540, 110, 546, 272, 756, 588, 972, 180, 312, 678, 126, 528, 1266, 972, 1596, 84, 1980, 1260, 1842, 256, 2484, 1842, 2286, 402, 2613, 2124, 3534, 1440, 1281, 2220, 4536, 307, 660, 672

Offset: 1

Reinhard Zumkeller, Nov 23 2002

nonn

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

Cf. A000203, A007947, A078310, A078317, A078319, A078320.

a078318 = a000203 . a078310  -- Reinhard Zumkeller, Jul 23 2013

a[n_] := DivisorSigma[1, 1 + n * Times @@ FactorInteger[n][[;;, 1]]]; Array[a, 100] (* Amiram Eldar, Apr 10 2025 *)

rad(n)=vecprod(factor(n)[,1])
a(n)=sigma(n*rad(n)+1) \\ Charles R Greathouse IV, Jul 09 2013

a(n) = A000203(A078310(n)).

cp's OEIS Frontend

A147800 Minimal value of A007947(m*(5^n-m)) with m coprime to 5.

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A147802 Least m coprime to 3 minimizing A007947(m*(3^n-m)).

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A255423 The least number k > A255334(n) for which A000203(k) = A000203(A255334(n)) and A007947(k) = A007947(A255334(n)), where A000203 gives the sum of divisors, and A007947 gives the squarefree kernel of n.

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A255424 Squarefree kernel of A255334: a(n) = A007947(A255334(n)).

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A326142 Sum of all other divisors of n except its largest squarefree divisor: a(n) = sigma(n) - A007947(n).

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A364702 Numbers k in A361098 that are not divisible by A007947(k)^2.

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A370896 Partial alternating sums of the squarefree kernel function (A007947).

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A376846 Number of m <= n such that rad(m) | n and Omega(m) > Omega(n), where rad = A007947 and Omega = A001222.

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