A269861 -id:A269861 - cp's OEIS frontend

A269862 Least monotonic left inverse of A269861.

0, 0, 0, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 7, 8, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13, 13, 13, 13, 14, 14, 15, 15, 16, 16, 17, 18, 19, 20, 21, 22, 22, 22, 23, 23, 23, 24, 25, 26, 26, 27, 28, 29, 30, 30, 30, 31, 31, 32, 33, 34, 34, 35, 35, 35, 35, 35, 35, 36, 36, 36, 36, 37, 37, 38, 38, 38, 39, 39, 39, 39, 40, 41

Offset: 1

Antti Karttunen, Mar 16 2016

nonn

a(n) = number of terms of A269861 <= n.

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

Cf. A048673, A269861.

f[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; s = Select[Range@ 150, Xor[EvenQ@ f@ #, EvenQ@ #] &]; Table[Count[s, k_ /; k <= n], {n, 87}] (* Michael De Vlieger, Mar 17 2016 *)

a(1) = 0, for n > 1, a(n) = (A048673(n)-n reduced mod 2) + a(n-1).
Other identities. For all n >= 1:
a(A269861(n)) = n.

A270431 Numbers n such that A048673(n) and A064216(n) are of opposite parity.

Original entry on oeis.org

6, 7, 11, 14, 15, 18, 19, 21, 22, 23, 24, 28, 35, 38, 43, 44, 45, 46, 47, 51, 54, 55, 56, 57, 59, 60, 61, 63, 66, 67, 70, 71, 72, 73, 76, 78, 79, 83, 84, 86, 87, 88, 89, 91, 92, 94, 95, 96, 103, 107, 110, 112, 115, 118, 119, 122, 123, 127, 129, 131, 134, 135, 138, 140, 142, 143, 146, 150, 152, 153, 157, 158, 159, 162

Offset: 1

Antti Karttunen, Mar 17 2016

nonn

See comments in A270434.

Antti Karttunen, Table of n, a(n) for n = 1..17000

Complement: A270430.
Left inverse: A270433.
Cf. A000035, A048673, A064216, A270434.
Cf. also A269861.

f[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; g[n_] := Times @@ Power[If[# == 1, 1, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger[2 n - 1]; Select[Range@ 162, Xor[EvenQ@ f@ #, EvenQ@ g@ #] &] (* Michael De Vlieger, Mar 17 2016 *)

Other identities. For all n >= 1:

A270433(a(n)) = n.

A269860 Numbers n such that n and A048673(n) are of the same parity.

Original entry on oeis.org

1, 2, 3, 6, 8, 9, 11, 13, 18, 20, 22, 23, 24, 25, 26, 27, 28, 31, 32, 33, 35, 37, 39, 46, 47, 49, 50, 54, 59, 60, 62, 66, 68, 69, 70, 71, 72, 74, 75, 76, 78, 80, 81, 83, 84, 85, 88, 89, 93, 94, 95, 96, 97, 98, 99, 104, 105, 107, 109, 111, 112, 116, 117, 118, 119, 121, 128, 131, 133, 138, 139, 141, 142, 143, 145, 147, 150

Offset: 1

Antti Karttunen, Mar 16 2016

nonn

Union of odd terms of A246261 and even terms of A246263.

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

Complement: A269861.
Cf. A048674 (a subsequence).
Cf. A000035, A048673, A246261, A246263.
Cf. also A270430.

f[n_] := (Times @@ Power[If[# == 1, 1, NextPrime@ #] & /@ First@ #, Last@ #] + 1)/2 &@ Transpose@ FactorInteger@ n; Select[Range@ 150, Xor[EvenQ@ f@ #, OddQ@ #] &] (* Michael De Vlieger, Mar 17 2016 *)

A349573 a(n) = A048673(n) - n, where A048673(n) = (A003961(n)+1) / 2, and A003961(n) shifts the prime factorization of n one step towards larger primes.

Original entry on oeis.org

0, 0, 0, 1, -1, 2, -1, 6, 4, 1, -4, 11, -4, 3, 3, 25, -7, 20, -7, 12, 7, -2, -8, 44, 0, 0, 36, 22, -13, 23, -12, 90, 0, -5, 4, 77, -16, -3, 4, 55, -19, 41, -19, 15, 43, -2, -20, 155, 12, 24, -3, 25, -23, 134, -9, 93, 1, -11, -28, 98, -27, -6, 75, 301, -5, 32, -31, 18, 4, 46, -34, 266, -33, -12, 48, 28, -5, 50, -37

Offset: 1

Antti Karttunen, Nov 23 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization

Cf. A003961, A048673, A064216, A349571.

Cf. A048674 (positions of zeros), A246351 (negative terms), A246281 (nonpositive terms), A246352 (nonnegative terms), A246282 (positive terms), A269860 (even terms), A269861 (odd terms).

f[p_, e_] := NextPrime[p]^e; a[1] = 0; a[n_] := (1 + Times @@ f @@@ FactorInteger[n])/2 - n; Array[a, 100] (* Amiram Eldar, Nov 23 2021 *)

A048673(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(1+factorback(f)); };
A349573(n) = (A048673(n)-n);

a(n) = A048673(n) - n.

a(n) = Sum_{d|n, dA349571(n/d).

cp's OEIS Frontend

A269862 Least monotonic left inverse of A269861.

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A270431 Numbers n such that A048673(n) and A064216(n) are of opposite parity.

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A269860 Numbers n such that n and A048673(n) are of the same parity.

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A349573 a(n) = A048673(n) - n, where A048673(n) = (A003961(n)+1) / 2, and A003961(n) shifts the prime factorization of n one step towards larger primes.

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