A097248 -id:A097248 - cp's OEIS frontend

A332461 a(n) = Product_{d|n, d>1} A000040(A297113(d)), where A000040(n) gives the n-th prime, and A297113(n) = the excess of n plus the index of the largest dividing prime (A046660 + A061395).

1, 2, 3, 6, 5, 18, 7, 30, 15, 50, 11, 270, 13, 98, 75, 210, 17, 450, 19, 1050, 147, 242, 23, 9450, 35, 338, 105, 3234, 29, 11250, 31, 2310, 363, 578, 245, 47250, 37, 722, 507, 57750, 41, 43218, 43, 9438, 2625, 1058, 47, 727650, 77, 2450, 867, 17238, 53, 22050, 605, 210210, 1083, 1682, 59, 8268750, 61, 1922, 8085, 30030

Offset: 1

Antti Karttunen, Feb 22 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Cf. A000040, A002110, A046660, A046523, A048675, A061395, A097248, A156552, A297113, A324202, A332462.

A297113(n) = if(1==n, 0, (primepi(vecmax(factor(n)[, 1])) + (bigomega(n)-omega(n))));
A332461(n) = if(1==n,1, my(m=1); fordiv(n,d,if(d>1, m *= prime(A297113(d)))); (m));

a(n) = Product_{d|n, d>1} A000040(A297113(d)).
a(p) = p for all primes p.
For all n >= 0, a(2^n) = A002110(n).
For all n >= 1:
A046523(a(n)) = A324202(n).
A048675(a(n)) = A156552(n).
A097248(a(n)) = A332462(n).

A332462 a(n) = A019565(A156552(n)).

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 7, 30, 15, 14, 11, 42, 13, 22, 21, 210, 17, 70, 19, 66, 33, 26, 23, 330, 35, 34, 105, 78, 29, 110, 31, 2310, 39, 38, 55, 462, 37, 46, 51, 390, 41, 130, 43, 102, 165, 58, 47, 2730, 77, 154, 57, 114, 53, 770, 65, 510, 69, 62, 59, 546, 61, 74, 195, 30030, 85, 170, 67, 138, 87, 182, 71, 4290, 73, 82, 231, 174, 91

Offset: 1

Antti Karttunen, Feb 22 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from indices in prime factorization

Permutation of squarefree numbers, A005117.

Cf. A002110, A005940, A019565, A048675, A097248, A156552, A332461.

A332462(n) = A097248(A332461(n)); \\ Needs also code from A097248 and A332461.

a(n) = A019565(A156552(n)) = A097248(A332461(n)).
a(p) = p for all primes p.
For all n >= 1, A048675(a(n)) = A156552(n).
For all n >= 0, a(A005940(1+n)) = A019565(n).
For all n >= 0, a(2^n) = A002110(n).

A331751 Numbers k such that A048675(sigma(k)) is equal to A048675(2*k).

Original entry on oeis.org

2, 6, 27, 28, 84, 270, 496, 1053, 1120, 1488, 1625, 1638, 3360, 3780, 4875, 8128, 10530, 24384, 66960, 147420, 167400, 406224, 611226, 775000, 872960, 943250, 1097280, 1245699, 1255338, 1303533, 1464320, 1686400, 1740024, 1922375, 1952500, 2011625, 2193408, 2325000, 2611440, 2618880, 2829750, 2941029, 4392960

Offset: 1

Antti Karttunen, Feb 05 2020

nonn

Numbers k such that A097248(sigma(k)) is equal to A097248(2*k).
Numbers k such that A331750(k) is equal to 1+A048675(k), which in turn is equal to A048675(A225546(2*k)) = A048675(2*A225546(k)).
Among the first 60 terms, 15 are odd: 27, 1053, 1625, 4875, 1245699, 1303533, 1922375, 2011625, 2941029, 5767125, 6034875, 12733875, 17137575, 26316675, 29362905, and only 1053 = 3^4 * 13 is in A228058.
Note that the condition A090880(sigma(k)) == A090880(2*k) appears to be much more constrained.

			For n = 1053 = 3^4 * 13^1, A331750(1053) = A331750(81) + A331750(13) = 32+9 = 41, while A048675(2*1053) = A048675(2)+A048675(81)+A048675(13) = 1+8+32 = 41 also, thus 1053 is included in this sequence.
For n = 3360 = 2^5 * 3^1 * 5^1 * 7^1, A331750(3360) = A331750(32)+A331750(3)+A331750(5)+A331750(7) = 12+2+3+3 = 20, while A048675(2*3360) = A048675(2)+A048675(32)+A048675(3)+A048675(5)+A048675(7) = 1+5+2+4+8 = 20 also, thus 3360 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..60 (up to the term 33550336).
Index entries for sequences where any odd perfect numbers must occur

Cf. A000203, A048675, A090880, A097246, A097248, A331750, A331752, A332208.

Cf. A000396 (a subsequence).

A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i, 1]+1)^(f[i, 2]\2))*((f[i, 1])^(f[i, 2]%2))); };
A097248(n) = { my(k=A097246(n)); while(k<>n, n = k; k = A097246(k)); k; };
isA331751(n) = (A097248(2*n)==A097248(sigma(n)));

A097249 a(n) is the number of times we must iterate A097246, starting at n, before the result is squarefree.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Aug 03 2004

nonn

a(n) = Min{k: r(n,k)=r(n,k+1)}, where r(n,k)=A097246(r(n,k-1)), r(n,0)=n;

a(A005117(n))=0; a(A097250(n))=n and a(m)A097250(n).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A005117, A097246, A097248, A097250, A277899.

f[n_] := Product[{p, e} = pe; NextPrime[p]^Quotient[e, 2] p^Mod[e, 2], {pe, FactorInteger[n]}];
a[n_] := (NestWhileList[f, n, !SquareFreeQ[#]&] // Length) - 1;
Array[a, 105] (* Jean-François Alcover, Nov 18 2021 *)

A097246(n) = { my(f=factor(n)); prod(i=1, #f~, (nextprime(f[i,1]+1)^(f[i,2]\2))*((f[i,1])^(f[i,2]%2))); };
A097249(n) = if(issquarefree(n),0,1+A097249(A097246(n))); \\ Antti Karttunen, Jul 29 2018

If A008966(n) = 1 [when n is in A005117], a(n) = 0, otherwise a(n) = 1 + a(A097246(n)). - Antti Karttunen, Jul 29 2018

Edited by Sam Alexander, Jan 05 2005

A334110 The squares of squarefree numbers (A062503), ordered lexicographically according to their prime factors. a(n) = Product_{k in I} prime(k+1)^2, where I are the indices of nonzero binary digits in n = Sum_{k in I} 2^k.

Original entry on oeis.org

1, 4, 9, 36, 25, 100, 225, 900, 49, 196, 441, 1764, 1225, 4900, 11025, 44100, 121, 484, 1089, 4356, 3025, 12100, 27225, 108900, 5929, 23716, 53361, 213444, 148225, 592900, 1334025, 5336100, 169, 676, 1521, 6084, 4225, 16900, 38025, 152100, 8281, 33124, 74529, 298116, 207025, 828100, 1863225, 7452900, 20449, 81796, 184041

Offset: 0

Antti Karttunen and Peter Munn, May 01 2020

nonn

For the lexicographic ordering, the prime factors must be written in nonincreasing order. If we write the factors in nondecreasing order, we get a lexicographically ordered set with an order type that is greater than a natural number index - the resulting sequence does not include all qualifying numbers. (Note also that the symbols used for the lexicographic order are the prime numbers, not their digits.)
a(n) is the n-th power of 4 in the monoid defined in A331590.
Conjecture: a(n) is the position of the first occurrence of n in A334109.

			The initial terms are shown below, equated with the product of their prime factors to exhibit the lexicographic ordering. The list starts with 1, since 1 is factored as the empty product and the empty list is first in lexicographic order.
    1 = .
    4 = 2*2.
    9 = 3*3.
   36 = 3*3*2*2.
   25 = 5*5.
  100 = 5*5*2*2.
  225 = 5*5*3*3.
  900 = 5*5*3*3*2*2.
   49 = 7*7.
  196 = 7*7*2*2.
  441 = 7*7*3*3.

Cf. A000079, A019565 (square roots), A048675, A097248, A225546, A267116, A332382, A334109 (a left inverse).
Column 2 of A329332. Permutation of A062503.
After 1, the right children of the leftmost edge of A334860, or respectively, the left children of the rightmost edge of A334866.
Subsequences: A001248, A061742, A166329.
Subsequence of A052330.
A003961, A003987, A059897, A331590 are used to express relationship between terms of this sequence.

Array[If[# == 0, 1, Times @@ Flatten@ Map[Function[{p, e}, Map[Prime[Log2@ # + 1]^(2^(PrimePi@ p - 1)) &, DeleteCases[NumberExpand[e, 2], 0]]] @@ # &, FactorInteger[3^#]]] &, 51, 0] (* Michael De Vlieger, May 26 2020 *)

A334110(n) = { my(p=2,m=1); while(n, if(n%2, m *= p^2); n >>= 1; p = nextprime(1+p)); (m); };

a(n) = A019565(n)^2.
For n >= 1, a(A000079(n-1)) = A001248(n).
For all n >= 0, A334109(a(n)) = n.
a(n+k) = A331590(a(n), a(k)).
a(n XOR k) = A059897(a(n), a(k)), where XOR denotes bitwise exclusive-or, A003987.
a(n) = A225546(3^n).
a(2n) = A003961(a(n)).
a(2n+1) = 4 * a(2n).
a(2^k-1) = A061742(k).
A267116(a(n)) = 2.
A048675(a(n)) = 2n.
A097248(a(n)) = A332382(n) = A019565(2n).

A322869 a(n) = A000120(A048675(n)).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Dec 31 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences computed from indices in prime factorization

Cf. A000120, A001221, A001222, A048675, A097248, A322821, A322862, A322868.

Cf. A002110 (position of the first occurrence of n).

Array[If[# == 1, 0, DigitCount[Total@ Map[#2*2^(PrimePi@ #1 - 1) & @@ # &, FactorInteger[#]], 2, 1]] &, 105]  (* Michael De Vlieger, Dec 31 2018 *)

A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; }; \\ From A048675
A322869(n) = hammingweight(A048675(n));

a(n) = A000120(A048675(n)) = A000120(A322821(n)).
a(n) = A001221(A097248(n)) = A001222(A097248(n)).
If n is squarefree, then a(n) = A001221(n) = A322862(n).
a(A002110(n)) = n.

A332824 a(n) = Product_{d|n} A019565(phi(d)), where phi is Euler totient function A000010.

Original entry on oeis.org

2, 4, 6, 12, 10, 36, 30, 60, 90, 100, 42, 540, 70, 900, 210, 420, 22, 8100, 66, 2100, 3150, 1764, 330, 18900, 550, 4900, 2970, 94500, 770, 44100, 2310, 4620, 6930, 484, 11550, 4252500, 130, 4356, 16170, 115500, 182, 9922500, 546, 291060, 242550, 108900, 2730, 1455300, 8190, 302500, 858, 1131900, 1430, 8820900, 19110

Offset: 1

Antti Karttunen, Feb 25 2020

nonn

Antti Karttunen, Table of n, a(n) for n = 1..8192

Cf. A000010, A019565, A097248, A318834, A332825, A332826, A332827.

Cf. A048675 (a left inverse).

A019565(n) = {my(j,v); factorback(Mat(vector(if(n, #n=vecextract(binary(n), "-1..1")), j, [prime(j), n[j]])~))}; \\ From A019565
A332824(n) = { my(m=1); fordiv(n,d,m *= A019565(eulerphi(d))); (m); };

a(n) = Product_{d|n} A332825(d).
a(n) = A318834(n) * A332825(n).
A048675(a(n)) = n.
A097248(a(n)) = A019565(n).

A376406 a(0) = 1, and for n > 0, a(n) = A019565(Sum_{i=0..n-1} a(i)), where A019565 is the base-2 exp-function.

Original entry on oeis.org

1, 2, 6, 14, 330, 10166, 12075690, 1174153011328084322, 73582975079922326904310062621361286633125176265747127754

Offset: 0

Antti Karttunen, Nov 04 2024

nonn

a(9) has 272 digits and a(10) has 1523 digits.
The lexicographically earliest infinite sequence x for which A048675(x(n)) gives the partial sums of x (shifted right once). This follows because the "least k" condition in the alternative formula also ensures that each k is squarefree, as we have A097248(n) = A019565(A048675(n)) <= n for all n, with equivalence only when n is squarefree.
Compare also to A376408.

			Starting with a(0) = 1, we take partial sums of previous terms, and apply A019565 to get the next term, and in the rightmost column, we "unbox" that term by applying A048675 to get A376407(n), which thus gives the partial sums of a(0)..a(n-1):
a(0)                               = 1          -> 0
a(1) = A019565(1)                  = 2,         -> 1     = 1
a(2) = A019565(1+2)                = 6,         -> 3     = 1+2
a(3) = A019565(1+2+6)              = 14,        -> 9     = 1+2+6
a(4) = A019565(1+2+6+14)           = 330,       -> 23    = 1+2+6+14
a(5) = A019565(1+2+6+14+330)       = 10166,     -> 353   = 1+2+6+14+330
a(6) = A019565(1+2+6+14+330+10166) = 12075690,  -> 10519 = 1+2+6+14+330+10166
etc.

Antti Karttunen, Table of n, a(n) for n = 0..9
Index entries for sequences related to binary expansion of n

Cf. A019565, A048675, A097248, A376408.
Cf. A376407 (= A048675(a(n)), also gives the partial sums from its second term onward).
Subsequence of A005117.
Cf. also analogous sequences A002110 (for A276085), A093502 (for A056239), A376399 (for A276075).

up_to = 12;
A019565(n) = { my(m=1, p=1); while(n>0, p = nextprime(1+p); if(n%2, m *= p); n >>= 1); (m); };
A376406list(up_to) = { my(v=vector(up_to), s=1); v[1]=1; for(n=2,up_to,v[n] = A019565(s); s += v[n]); (v); };
v376406 = A376406list(1+up_to);
A376406(n) = v376406[1+n];

a(n) = A019565(A376407(n)) = A019565(Sum_{i=0..n-1} a(i)).
a(0) = 1, and for n > 0, a(n) is the least k such that A048675(k) = a(n-1) + A048675(a(n-1)), where A048675 is the base-2 log-function.
For n > 0, a(n) <= a(n-1) * A019565(a(n-1)).

A277886 If n is squarefree, a(n) = n, else a(n) = A000040(1+A277885(n)) * (n/(A249739(n)^2)).

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 6, 5, 10, 11, 9, 13, 14, 15, 12, 17, 10, 19, 15, 21, 22, 23, 18, 7, 26, 15, 21, 29, 30, 31, 24, 33, 34, 35, 27, 37, 38, 39, 30, 41, 42, 43, 33, 25, 46, 47, 36, 11, 14, 51, 39, 53, 30, 55, 42, 57, 58, 59, 45, 61, 62, 35, 48, 65, 66, 67, 51, 69, 70, 71, 54, 73, 74, 21, 57, 77, 78, 79, 60, 45

Offset: 1

Antti Karttunen, Nov 08 2016

nonn

If n has non-unitary prime divisors, then divide it by the square of the smallest of them and multiply by a single instance of the next larger prime.

This differs from related A097246 for the first time at n=16. For both sequences A097248 gives the eventual stable points reached when starting iterating from n.

			For n = 12 = 2*2*3, the smallest non-unitary prime divisor (and in this case the only one) is 2, thus we divide with 2^2 and multiply with the next larger prime 3, to get ((2^2 * 3)/(2^2))*3 = 3*3, thus a(12) = 9.
For n = 16 = 2^4, we divide two instances of 2 out and multiply by a single instance of 3 to get 2*2*3 = 12.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000040, A048675, A097246, A097248, A249739, A277885, A277887, A277888, A277896.

Table[If[SquareFreeQ@ n, n, Prime[1 + PrimePi@ Min[Select[FactorInteger[n][[All, 1]], ! CoprimeQ[#, n/#] &] /. {} -> 0]] (n/If[SquareFreeQ@ n, 1, p = 2; While[! Divisible[n, p^2], p = NextPrime@ p]; p]^2)], {n, 81}] (* Michael De Vlieger, Nov 15 2016 *)

(define (A277886 n) (if (zero? (A277885 n)) n (* (A000040 (+ 1 (A277885 n))) (/ n (expt (A249739 n) 2)))))

If A277885(n) = 0 [when n is squarefree], then a(n) = n, otherwise a(n) = A000040(1+A277885(n)) * (n/(A249739(n)^2)).
Other identities. For all n >= 1:
A048675(a(n)) = A048675(n).

A097247 A097246(A097246(n)).

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 6, 5, 10, 11, 5, 13, 14, 15, 5, 17, 10, 19, 15, 21, 22, 23, 10, 7, 26, 15, 21, 29, 30, 31, 10, 33, 34, 35, 15, 37, 38, 39, 30, 41, 42, 43, 33, 7, 46, 47, 15, 11, 14, 51, 39, 53, 30, 55, 42, 57, 58, 59, 25, 61, 62, 35, 15, 65, 66, 67, 51, 69, 70, 71, 30, 73, 74

Offset: 1

Reinhard Zumkeller, Aug 03 2004

nonn

This sequence is not multiplicative, see example.

			m=20 and n=63 are coprime:
a(20) = A097246(A097246(2*2*5)) = A097246(3*5)=15,
a(63) = A097246(A097246(3*3*7)) = A097246(5*7)=35 and
a(20*63) = A097246(A097246(2*2*3*3*5*7)) = A097246(3*5*5*7) =
3*7*7 = 147 <> a(20)*a(63) = 15*35 = 525: the sequence is not multiplicative.

Cf. A097248.

cp's OEIS Frontend

A332461 a(n) = Product_{d|n, d>1} A000040(A297113(d)), where A000040(n) gives the n-th prime, and A297113(n) = the excess of n plus the index of the largest dividing prime (A046660 + A061395).

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A332462 a(n) = A019565(A156552(n)).

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A331751 Numbers k such that A048675(sigma(k)) is equal to A048675(2*k).

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A097249 a(n) is the number of times we must iterate A097246, starting at n, before the result is squarefree.

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A334110 The squares of squarefree numbers (A062503), ordered lexicographically according to their prime factors. a(n) = Product_{k in I} prime(k+1)^2, where I are the indices of nonzero binary digits in n = Sum_{k in I} 2^k.

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A322869 a(n) = A000120(A048675(n)).

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A332824 a(n) = Product_{d|n} A019565(phi(d)), where phi is Euler totient function A000010.

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A376406 a(0) = 1, and for n > 0, a(n) = A019565(Sum_{i=0..n-1} a(i)), where A019565 is the base-2 exp-function.

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