A277324 -id:A277324 - cp's OEIS frontend

A277333 Left inverse of A260443, giving 0 as a result when n is outside of the range of A260443.

0, 1, 2, 0, 4, 3, 8, 0, 0, 0, 16, 0, 32, 0, 6, 0, 64, 5, 128, 0, 0, 0, 256, 0, 0, 0, 0, 0, 512, 7, 1024, 0, 0, 0, 12, 0, 2048, 0, 0, 0, 4096, 0, 8192, 0, 0, 0, 16384, 0, 0, 0, 0, 0, 32768, 0, 0, 0, 0, 0, 65536, 0, 131072, 0, 0, 0, 0, 0, 262144, 0, 0, 0, 524288, 0, 1048576, 0, 10, 0, 24, 0, 2097152, 0, 0, 0, 4194304, 0, 0, 0, 0, 0, 8388608, 9

Offset: 1

Antti Karttunen, Oct 10 2016

nonn

			a(1) = 0 because A260443(0) = 1. For n > 1, a(n) = 0 only if n does not occur in the range of A260443.
a(6) = 3 because A260443(3) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..5000

Cf. A005811, A007949, A048675, A064989, A260443.

Cf. A277316, A260442 (from 2 onward, the positions of nonzeros), A277317 (positions of primes).

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ Michel Marcus, Oct 10 2016
A260443(n) = if(n<2, n+1, if(!(n%2), A003961(A260443(n/2)), A260443((n-1)/2) * A260443((n+1)/2)));
A277333(n) = { my(k=A048675(n)); if(A260443(k) == n, k, 0); } ;
for(n=1, 5000, write("b277333.txt", n, " ", A277333(n)));

(define (A277333 n) (let ((k (A048675 n))) (if (= (A260443 k) n) k 0)))

If A260443(A048675(n)) = n, then a(n) = A048675(n), otherwise a(n) = 0.
Other identities. For all n >= 0:
a(A260443(n)) = n.
a(2n+1) = 2*a(A064989(2n+1)).
If a(2n) > 0 [by necessity an odd number in that case], then A005811((a(2n)-1)/2) = A007949(2n). [See comment in A277324.]

A284576 a(n) = A059896(A260443(n), A260443(1+n)).

Original entry on oeis.org

2, 6, 6, 30, 90, 270, 30, 210, 630, 6750, 6750, 1890, 15750, 47250, 210, 2310, 6930, 47250, 47250, 330750, 992250, 425250, 47250, 103950, 173250, 2315250, 2315250, 519750, 8489250, 25467750, 2310, 30030, 90090, 519750, 25467750, 3638250, 1910081250, 13023281250, 1447031250, 1400726250, 4202178750, 104186250, 2604656250

Offset: 0

Antti Karttunen, Apr 11 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A059896, A260443, A277324, A284577, A284578, A285106.

(define (A284576 n) (A059896bi (A260443 n) (A260443 (+ 1 n)))) ;; For A059896bi, see A059896.

a(n) = A059896(A260443(n), A260443(1+n)).
a(n) = A284577(n) * A284578(n).
a(n) = A277324(n) / A284578(n).

A284577 a(n) = A059897(A260443(n), A260443(1+n)).

Original entry on oeis.org

2, 6, 2, 30, 90, 270, 2, 210, 630, 6750, 2250, 378, 15750, 47250, 2, 2310, 6930, 6750, 630, 66150, 198450, 3402, 90, 14850, 24750, 92610, 30870, 14850, 8489250, 25467750, 2, 30030, 90090, 6750, 339570, 14850, 382016250, 372093750, 9843750, 1400726250, 4202178750, 3402, 198450, 20465156250, 7796250, 83531250, 90, 859950, 1433250, 1890

Offset: 0

Antti Karttunen, Apr 11 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A059897, A260443, A277324, A284576, A284578, A285107.

(define (A284577 n) (A059897bi (A260443 n) (A260443 (+ 1 n)))) ;; For A059897bi, see A059897.

a(n) = A059897(A260443(n), A260443(1+n)).
a(n) = A277324(n) / A000290(A284578(n)).
A001222(a(n)) = A285107(n).

A284578 a(n) = A059895(A260443(n), A260443(1+n)).

Original entry on oeis.org

1, 1, 3, 1, 1, 1, 15, 1, 1, 1, 3, 5, 1, 1, 105, 1, 1, 7, 75, 5, 5, 125, 525, 7, 7, 25, 75, 35, 1, 1, 1155, 1, 1, 77, 75, 245, 5, 35, 147, 1, 1, 30625, 13125, 7, 245, 245, 40425, 11, 11, 13475, 1029, 245, 245, 1715, 1617, 11, 77, 1225, 3675, 385, 1, 1, 15015, 1, 1, 1001, 75, 245, 2695, 1715, 3, 1, 7, 48125, 7203, 1, 35, 300125, 363, 1, 1, 75625

Offset: 0

Antti Karttunen, Apr 11 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A000196, A059895, A260443, A277324, A284576, A284577, A285108.

(define (A284578 n) (A059895bi (A260443 n) (A260443 (+ 1 n)))) ;; For A059895bi, see A059895.

a(n) = A059895(A260443(n), A260443(1+n)).
a(n) = A277324(n) / A284576(n) = A000196(A277324(n)/A284577(n)).
a(n) = A284576(n) / A284577(n).
A001222(a(n)) = A285108(n).

A277189 Odd bisection of A277020: a(n) = A277020(2n+1).

Original entry on oeis.org

1, 5, 13, 21, 45, 93, 109, 85, 173, 477, 957, 733, 749, 1501, 877, 341, 685, 3549, 12221, 7133, 14269, 49021, 28605, 5853, 5869, 30685, 61373, 23517, 12013, 24029, 7021, 1365, 2733, 28125, 192445, 97245, 384957, 2031485, 1032125, 113629, 227261, 4128637, 16252669, 3112829, 1564605, 6225789, 913341, 46813, 46829, 915421

Offset: 0

Antti Karttunen, Oct 08 2016

Antti Karttunen, Table of n, a(n) for n = 0..8191

Cf. A156552, A277020, A277324.

Terms form a proper subset of A247648.

(define (A277189 n) (A277020 (+ n n 1)))

a(n) = A277020(2*n + 1).
a(n) = A156552(A277324(n)).
Other identities. For all n >= 0:
a(n) = 1 mod 4.

Offset changed from 1 to 0 by Antti Karttunen, Oct 11 2016

A277200 Even terms in A260442 (in A260443).

Original entry on oeis.org

2, 6, 18, 30, 90, 210, 270, 450, 630, 2310, 6750, 6930, 9450, 15750, 20250, 22050, 30030, 47250, 90090, 330750, 510510, 727650, 1212750, 1531530, 1653750, 2668050, 3543750, 4961250, 8489250, 9699690, 18191250, 24806250, 25467750, 29099070, 40020750, 53156250, 57881250, 104053950, 173423250, 173643750

Offset: 1

Antti Karttunen, Oct 14 2016

nonn

All odd terms larger > 1 in A260442 can be obtained from these terms by shifting their prime factorization some number of steps towards larger primes with A003961.

Cf. A003961, A260442, A260443.
Sequence A277324 sorted into ascending order.
Subsequence of A055932.
Cf. A002110, A277317 (subsequences, apart from their initial terms).
Also all terms of A277318 apart from initial 3 are included in this sequence.

allocatemem(2^30);
A061395(n) =  if(1==n, 0, primepi(vecmax(factor(n)[, 1]))); \\ After M. F. Hasler's code for A006530.
A048675(n) = my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; \\ Michel Marcus, Oct 10 2016
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2))));
isA277200(n) = (!(n%2) && (omega(n) == A061395(n)) && (A260443(A048675(n)) == n));
i=1; n=0; while(i < 100, n++; if(isA277200(n), write("b277200.txt", i, " ", n); i++));

A277323 Even bisection of A260443 (the odd terms): a(n) = A260443(2*n).

Original entry on oeis.org

1, 3, 5, 15, 7, 75, 35, 105, 11, 525, 245, 2625, 77, 3675, 385, 1155, 13, 5775, 2695, 128625, 847, 643125, 18865, 202125, 143, 282975, 29645, 1414875, 1001, 444675, 5005, 15015, 17, 75075, 35035, 15563625, 11011, 346644375, 2282665, 108945375, 1859, 544726875, 15978655, 12132553125, 121121, 3813088125, 2697695

Offset: 0

Antti Karttunen, Oct 10 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A000035, A005811, A007949, A101979, A112765, A260443, A277324.

a[n_] := a[n] = Which[n < 2, n + 1, EvenQ@ n, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1] &@ a[n/2], True, a[#] a[# + 1] &[(n - 1)/2]]; Table[a[2 n], {n, 0, 46}] (* Michael De Vlieger, Apr 05 2017 *)

a(n) = A260443(2*n).
a(0) = 1; for n >= 1, a(n) = A003961(A260443(n)).
Other identities. For all n >= 0:
A007949(a(n)) = A000035(n).
A112765(a(n)) is the interleaving of A000035 and A005811, probably A101979.

A283975 Odd bisection of A264977.

Original entry on oeis.org

1, 3, 1, 7, 5, 7, 1, 15, 13, 7, 5, 11, 13, 15, 1, 31, 29, 7, 13, 3, 1, 11, 5, 19, 21, 15, 13, 19, 29, 31, 1, 63, 61, 7, 29, 19, 25, 3, 13, 11, 9, 11, 1, 23, 25, 19, 5, 35, 37, 15, 21, 11, 9, 19, 13, 43, 37, 31, 29, 35, 61, 63, 1, 127, 125, 7, 61, 51, 41, 19, 29, 59, 49, 3, 25, 31, 17, 11, 13, 27, 25, 11, 9, 31, 21, 23, 1, 47, 33, 19, 25, 63, 41, 35, 5, 67, 69

Offset: 0

Antti Karttunen, Mar 25 2017

Antti Karttunen, Table of n, a(n) for n = 0..8190
Index entries for sequences related to binary expansion of n

Cf. A000120, A248663, A264977, A277324.

Cf. also A284266.

A264977[n_]:= If[n<2, n, If[EvenQ[n], 2 A264977[n/2], BitXor[A264977[(n - 1)/2], A264977[(n + 1)/2]]]]; Table[A264977[2n + 1], {n, 0, 150}] (* Indranil Ghosh, Mar 28 2017 *)

a(n) = if(n<2, n, if(n%2, bitxor(a((n - 1)/2), a((n + 1)/2)), 2*a(n/2))); for(n=0, 150, print1(a(2*n + 1),", ")) \\ Indranil Ghosh, Mar 28 2017

(define (A283975 n) (A264977 (+ n n 1)))

a(n) = A264977((2*n)+1).
a(n) = A248663(A277324(n)).
A000120(a(n)) = A284266(n).

A284008 a(n) = lcm(A260443(n), A260443(1+n)).

Original entry on oeis.org

2, 6, 6, 30, 90, 90, 30, 210, 630, 450, 1350, 1890, 3150, 3150, 210, 2310, 6930, 3150, 47250, 330750, 992250, 141750, 47250, 103950, 173250, 110250, 330750, 519750, 242550, 242550, 2310, 30030, 90090, 34650, 3638250, 3638250, 272868750, 173643750, 11576250, 200103750, 600311250, 34728750, 2604656250, 28651218750, 1910081250, 272868750, 3638250, 9459450, 15765750

Offset: 0

Antti Karttunen, Mar 22 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..4096
Index entries for sequences related to lcm's

Cf. A260443, A277198, A277324, A277330, A284009.

a[p_?PrimeQ] := a[p] = Prime[ PrimePi[p] + 1]; a[1] = 1; a[n_] := a[n] = Times @@ ( a[First[#]] ^ Last[#] & ) /@ FactorInteger[n]; (* after Jean-François Alcover, in A003961 *) A[n_]:= If[n<2, n + 1, If[EvenQ[n], a[A[n/2]], A[(n - 1)/2] A[(n + 1)/2]]] ;Table[A[n], {n, 0, 51}] (* sequence A260443 *) Table[LCM[A[n], A[n + 1]], {n, 0, 101}] (* Indranil Ghosh, Mar 22 2017 *)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From Michel Marcus
A260443(n) = if(n<2, n+1, if(n%2, A260443(n\2)*A260443(n\2+1), A003961(A260443(n\2))));
A284008(n) = lcm(A260443(n),A260443(1+n));

(define (A284008 n) (lcm (A260443 n) (A260443 (+ 1 n))))
(define (A284008 n) (/ (A277324 n) (A277198 n)))

a(n) = lcm(A260443(n), A260443(1+n)).
a(n) = A277324(n) / A277198(n).
Other identities. For all n >= 0:
A001222(a(n)) = A284009(n).

cp's OEIS Frontend

A277333 Left inverse of A260443, giving 0 as a result when n is outside of the range of A260443.

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A284576 a(n) = A059896(A260443(n), A260443(1+n)).

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A284577 a(n) = A059897(A260443(n), A260443(1+n)).

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A284578 a(n) = A059895(A260443(n), A260443(1+n)).

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A277189 Odd bisection of A277020: a(n) = A277020(2n+1).

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A277200 Even terms in A260442 (in A260443).

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A277323 Even bisection of A260443 (the odd terms): a(n) = A260443(2*n).

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A283975 Odd bisection of A264977.

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A284008 a(n) = lcm(A260443(n), A260443(1+n)).

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