A078898 -id:A078898 - cp's OEIS frontend

A250477 Number of times prime(n) (the n-th prime) occurs as the least prime factor among numbers 1 .. (prime(n)^2 * prime(n+1)): a(n) = A078898(A251720(n)).

6, 8, 12, 21, 33, 45, 63, 80, 116, 148, 182, 232, 265, 296, 356, 433, 490, 548, 625, 674, 740, 829, 919, 1055, 1187, 1252, 1313, 1376, 1446, 1657, 1897, 2029, 2134, 2301, 2484, 2605, 2785, 2946, 3110, 3301, 3439, 3654, 3869, 3978, 4086, 4349, 4811, 5147, 5273, 5395, 5604, 5787, 6049, 6403, 6684, 6954, 7153

Offset: 1

Antti Karttunen, Dec 14 2014

nonn

a(n) = Position of 6 on row n of array A249821. This is always larger than A250474(n), the position of 4 on row n, as 4 is guaranteed to be the first composite term on each row of A249821.
From Antti Karttunen, Mar 29 2015: (Start)
a(n) = 1 + number of positive integers <= (prime(n)*prime(n+1)) whose smallest prime factor is at least prime(n).
That a(n) >  A250474(n) can also be seen by realizing that prime(n) must occur at least as many times as the smallest prime factor for the numbers in range 1 .. (prime(n)^2 * prime(n+1)) than for numbers in (smaller) range 1 .. (prime(n)^3), and also by realizing that a(n) cannot be equal to A250474(n) because each row of A249822 is a permutation of natural numbers.
Or more simply, by considering the comment given in A256447 which follows from the new interpretation given above.
(End)

Antti Karttunen, Table of n, a(n) for n = 1..564
Antti Karttunen, Ratio a(n)/A250474(n+1), plotted up to n=65 with OEIS Plot2-utility

Column 6 of A249822. Cf. also A250474 (column 4), A250478 (column 8).
First differences: A256446. Cf. also A256447, A256448.
Cf. A002110, A006094, A020639, A078898, A249821, A251720, A281890.

f[n_] := Count[Range[Prime[n]^2*Prime[n + 1]], x_ /; Min[First /@ FactorInteger[x]] == Prime@ n]; Array[f, 20] (* Michael De Vlieger, Mar 30 2015 *)

allocatemem(234567890);
A002110(n) = prod(i=1, n, prime(i));
A250477(n) = { my(m); m = (prime(n) * prime(n+1)); sumdiv(A002110(n-1), d, (moebius(d)*(m\d))); };
for(n=1, 23, print1(A250477(n),", "));
\\ A more practical program:

allocatemem(234567890);
vecsize = (2^24)-4;
v020639 = vector(vecsize);
v020639[1] = 1; for(n=2,vecsize, v020639[n] = vecmin(factor(n)[, 1]));
A020639(n) = v020639[n];
A250477(n) = { my(p=prime(n),q=prime(n+1),u=p*q,k=1,s=1); while(k <= u, if(A020639(k) >= p, s++); k++); s; };
for(n=1, 564, write("b250477.txt", n, " ", A250477(n)));
\\ Antti Karttunen, Mar 29 2015

a(n) = A078898(A251720(n)).
a(1) = 1, a(n) = Sum_{d | A002110(n-1)} moebius(d) * floor(A006094(n) / d). [Follows when A251720, (p_n)^2 * p_{n+1} is substituted to the similar formula given for A078898. Here p_n is the n-th prime (A000040(n)), A006094(n) gives the product p_n * p{n+1} and A002110(n) gives the product of the first n primes. Because the latter is always squarefree, one could use here also Liouville's lambda (A008836) instead of Moebius mu (A008683)].
a(n) = A250474(n) + A256447(n).

A249813 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2 * a(A078898(n+1))) - 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 15, 32, 13, 10, 11, 64, 17, 128, 31, 14, 29, 256, 63, 12, 25, 18, 19, 512, 21, 1024, 127, 30, 33, 20, 255, 2048, 61, 26, 27, 4096, 57, 8192, 511, 22, 125, 16384, 23, 24, 49, 34, 35, 32768, 37, 28, 1023, 62, 41, 65536, 2047, 131072, 253, 58, 59, 36, 65, 262144, 39, 126, 509, 524288, 4095, 1048576, 121, 50, 51, 40, 53

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

This sequence is a "recursed variant" of A249812.

See also the comments at the inverse permutation A249814.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A249814.
Similar or related permutations: A246683, A249812, A250243.
Cf. A000079, A006093, A055396, A078898.
Differs from A246683 for the first time at n=20, where a(20) = 14, while A246683(20) = 18.

a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2 * a(A078898(n+1))) - 1).
As a composition of other permutations:
a(n) = A246683(A250243(n)).
Other identities. For all n >= 1, the following holds:
a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back.]
a(A006093(n)) = A000079(n-1).

A305798 Dirichlet convolution of A078898 with itself.

Original entry on oeis.org

1, 2, 2, 5, 2, 8, 2, 12, 5, 12, 2, 22, 2, 16, 8, 28, 2, 28, 2, 34, 10, 24, 2, 56, 5, 28, 14, 46, 2, 52, 2, 64, 14, 36, 8, 83, 2, 40, 16, 88, 2, 70, 2, 70, 26, 48, 2, 136, 5, 64, 20, 82, 2, 94, 10, 120, 22, 60, 2, 164, 2, 64, 34, 144, 12, 106, 2, 106, 26, 100, 2, 220, 2, 76, 36, 118, 8, 124, 2, 216, 42, 84, 2, 224, 14, 88, 32, 184, 2, 192, 10

Offset: 1

Antti Karttunen, Jun 13 2018

nonn

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

Cf. A078898, A305791, A305796, A305797, A305799.

up_to = 65537;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om,invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om,invec[i],(1+pt))); outvec; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A305798(n) = sumdiv(n,d,A078898(d)*A078898(n/d));

a(n) = Sum_{d|n} A078898(d)*A078898(n/d).

A249812 Permutation of natural numbers: a(n) = A000079(A055396(n+1)-1) * ((2*A078898(n+1))-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 13, 10, 15, 64, 17, 128, 19, 14, 21, 256, 23, 12, 25, 18, 27, 512, 29, 1024, 31, 22, 33, 20, 35, 2048, 37, 26, 39, 4096, 41, 8192, 43, 30, 45, 16384, 47, 24, 49, 34, 51, 32768, 53, 28, 55, 38, 57, 65536, 59, 131072, 61, 42, 63, 36, 65, 262144, 67, 46, 69, 524288, 71, 1048576, 73, 50, 75, 40, 77, 2097152, 79, 54, 81, 4194304, 83, 44

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

In the essence, a(n) tells which number in the array A135764 is at the same position where n is in the array A249741, the sieve of Eratosthenes minus 1. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e., a(2n+1) = 2n+1 for all n.

Equally: a(n) tells which number in array A054582 is at the same position where n is in the array A114881, as they are the transposes of above two arrays.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A249811.
Cf. A000079, A005408, A006093, A055396, A078898.
Similar or related permutations: A249813 ("deep variant"), A246675, A249816, A054582, A114881, A250252, A135764, A249741, A249742.
Differs from A246675 for the first time at n=20, where a(20)=14, while A246675(20)=18.

(define (A249812 n) (* (A000079 (- (A055396 (+ 1 n)) 1)) (+ -1 (* 2 (A078898 (+ 1 n))))))

a(n) = A000079(A055396(n+1)-1) * ((2*A078898(n+1))-1).
As a composition of related permutations:
a(n) = A054582(A250252(n)-1).
a(n) = A135764(A249742(n)).
a(n) = A246675(A249816(n)).
Other identities. For all n >= 1 the following holds:
a(A006093(n)) = A000079(n-1).

A250243 Permutation of natural numbers: a(n) = A246275(A055396(n+1), a(A078898(n+1))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 26, 21, 22, 23, 24, 25, 20, 27, 28, 29, 30, 31, 44, 33, 34, 35, 36, 37, 32, 51, 40, 41, 42, 43, 80, 45, 46, 47, 48, 49, 74, 39, 52, 53, 124, 55, 62, 57, 58, 59, 60, 61, 38, 87, 54, 65, 66, 67, 134, 69, 70, 71, 72, 73, 50, 63, 76, 101, 78, 79, 98, 81

Offset: 1

Antti Karttunen, Nov 16 2014

nonn

This is a "more recursed" variant of A249816. Preserves the parity of n.

Antti Karttunen, Table of n, a(n) for n = 1..5749
Index entries for sequences that are permutations of the natural numbers

Inverse: A250244.
Cf. A006093, A055396, A078898, A246275, A246278.
Similar or related permutations: A246684, A249813, A250246.
Differs from A249815 and A250244 for the first time at n=32, where a(32) = 44, while A249815(32) = A250244(32) = 38.
Differs from "shallow variant" A249816 for the first time at n=39, where a(39) = 51, while A249816(39) = 39.

a(n) = A246275(A055396(n+1), a(A078898(n+1))).
As a composition of other permutations:
a(n) = A246684(A249813(n)).
Other identities. For all n >= 1, the following holds:
a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back.]
a(A006093(n)) = A006093(n). [Primes minus one are among the fixed points].

A300247 Restricted growth sequence transform of A286457(n), filter combining A078898(n) and A246277(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 6, 2, 7, 2, 8, 4, 9, 2, 10, 2, 11, 12, 13, 2, 14, 3, 15, 16, 17, 2, 18, 2, 19, 20, 21, 4, 22, 2, 23, 24, 25, 2, 26, 2, 27, 28, 29, 2, 30, 3, 31, 32, 33, 2, 34, 12, 35, 36, 37, 2, 38, 2, 39, 40, 41, 42, 43, 2, 44, 45, 46, 2, 47, 2, 48, 49, 50, 4, 51, 2, 52, 53, 54, 2, 55, 56, 57, 58, 59, 2, 60, 12, 61, 62, 63, 64, 65, 2, 66

Offset: 1

Antti Karttunen, Mar 03 2018

For all i, j:
  a(i) = a(j) => A280492(i) = A280492(j).
  a(i) = a(j) => A300248(i) = A300248(j).
The latter follows because A046523(n) = A046523(2*A246277(n)).

			a(65) = a(119) (= 42) because A078898(65) = A078898(119) = 5 (both numbers occur in column 5 of A083221) and because A246277(65) = A246277(119) = 7 (both numbers occur in column 7 of A246278). Note that 65 = 5*13 = prime(3)*prime(6) and 119 = 7*17 = prime(4)*prime(7) = A003961(65). A246277(n) contains complete information about the (relative) differences between prime indices in the prime factorization of n.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A078898, A246277, A286457.

Cf. also A300248, A280492.

rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
write_to_bfile(start_offset,vec,bfilename) = { for(n=1, length(vec), write(bfilename, (n+start_offset)-1, " ", vec[n])); }
A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A078898(n) = { if(n<=1,n, my(spf=A020639(n),k=1,m=n/spf); while(m>1,if(A020639(m)>=spf,k++); m--); (k)); }; \\ Antti Karttunen, Mar 03 2018
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };
A286457(n) = if(1==n,0,(1/2)*(2 + ((A078898(n)+A246277(n))^2) - A078898(n) - 3*A246277(n)));
write_to_bfile(1,rgs_transform(vector(65537,n,A286457(n))),"b300247.txt");

A249810 a(1) = 0, a(n) = A078898(A003961(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 5, 2, 4, 1, 8, 1, 6, 3, 14, 1, 13, 1, 11, 4, 7, 1, 23, 2, 9, 9, 17, 1, 18, 1, 41, 5, 10, 3, 38, 1, 12, 6, 32, 1, 28, 1, 20, 12, 15, 1, 68, 2, 25, 7, 26, 1, 63, 4, 50, 8, 16, 1, 53, 1, 19, 19, 122, 5, 33, 1, 29, 10, 39, 1, 113, 1, 21, 17, 35, 3, 43, 1, 95, 42, 22, 1, 83, 6, 24, 11, 59, 1, 88, 4, 44, 13, 27, 7, 203

Offset: 1

Antti Karttunen, Dec 08 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..3071

Cf. A003961, A078898, A246277, A249817, A249818, A249820, A249821, A249822, A251721, A251722, A250469, A250470.

(define (A249810 n) (if (= 1 n) 0 (A078898 (A003961 n))))

a(1) = 0, a(n) = A078898(A003961(n)).

a(1) = 0, a(n) = A078898(n) + A249820(n).

A249816 Permutation of natural numbers: a(n) = A246275(A055396(n+1), A078898(n+1)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 26, 21, 22, 23, 24, 25, 20, 27, 28, 29, 30, 31, 44, 33, 34, 35, 36, 37, 32, 39, 40, 41, 42, 43, 80, 45, 46, 47, 48, 49, 74, 51, 52, 53, 124, 55, 62, 57, 58, 59, 60, 61, 38, 63, 54, 65, 66, 67, 134, 69, 70, 71, 72, 73, 50, 75, 76, 77, 78, 79, 98, 81, 82, 83

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

a(n) tells which number in square array A246275 is at the same position where n is in array A249741, the sieve of Eratosthenes minus 1. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e. a(2n+1) = 2n+1 for all n. Also, as the leftmost column in both arrays is primes minus one (A006093), they are also among the fixed points.

Equally: a(n) tells which number in array A246273 is at the same position where n is in the array A114881, as they are the transposes of above two arrays.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A249815.
Cf. A005408, A006093, A055396, A078898, A246278.
Similar or related permutations: A250243 ("deep variant"), A246676, A249812, A249818, A246273, A246275, A114881, A249741.
Differs from A249815 and A250244 for the first time at n=32, where a(32) = 44, while A249815(32) = A250244(32) = 38.
Differs from A250244 for the first time at n=39, where a(39) = 39, while A250243(39) = 51.

(define (A249816 n) (+ -1 (A246278bi (A055396 (+ 1 n)) (A078898 (+ 1 n)))))

a(n) = A246275(A055396(n+1), A078898(n+1)).
As a composition of other permutations:
a(n) = A246676(A249812(n)).
a(n) = A249818(n+1) - 1.
Other identities. For all n >= 1:
a(A005408(n-1)) = A005408(n-1) and a(A006093(n)) = A006093(n). [Fixes odd numbers and precedents of primes. Cf. comments above].

A249826 Permutation of natural numbers: a(n) = A078898(A003961(A003961(A003961(2*n)))).

Original entry on oeis.org

1, 2, 3, 14, 4, 21, 5, 92, 33, 25, 6, 144, 7, 32, 39, 641, 8, 226, 9, 170, 50, 36, 10, 1007, 46, 43, 355, 223, 11, 267, 12, 4482, 56, 55, 59, 1582, 13, 58, 68, 1190, 15, 350, 16, 249, 420, 70, 17, 7043, 78, 316, 86, 301, 18, 2485, 66, 1555, 91, 77, 19, 1869, 20, 81, 549, 31374, 80, 391, 22, 379, 109, 413, 23, 11068, 24, 88, 496, 406, 87, 473, 26, 8324, 3905, 99, 27

Offset: 1

Antti Karttunen, Dec 06 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..127
Index entries for sequences that are permutations of the natural numbers

Inverse: A249825.
Row 4 of A249822.
Cf. A003961, A251722, A048673, A078898, A246278, A249824, A249746, A250476.

(define (A249826 n) (A078898 (A003961 (A003961 (A003961 (* 2 n))))))

a(n) = A078898(A003961(A003961(A003961(2*n)))).
a(n) = A078898(A246278(4,n)).
As a composition of other permutations:
a(n) = A250476(A249824(n)).
a(n) = A250476(A249746(A048673(n))). [Composition of the first three rows of array A251722.]

A250248 Permutation of natural numbers: a(1) = 1, a(n) = A246278(a(A055396(n)),A078898(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 45, 34, 35, 36, 37, 38, 33, 40, 41, 42, 43, 44, 81, 46, 47, 48, 49, 50, 75, 52, 53, 54, 125, 56, 63, 58, 59, 60, 61, 62, 39, 64, 55, 66, 67, 68, 135, 70, 71, 72, 103, 74, 51, 76, 77, 78, 79, 80, 99, 82, 83

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..3432
Index entries for sequences that are permutations of the natural numbers

Inverse: A250247.
Similar permutations: A250250 for even more recursed variant of A249818.
Cf. A000040, A005843, A049076, A049084, A055396, A078442, A078898, A246278.
Differs from the "vanilla version" A249818 for the first time at n=73, where a(73) = 108, while A249818(73) = 73.

a(1) = 1, a(n) = A246278(a(A055396(n)), A078898(n)).
Other identities. For all n >= 1:
a(A005843(n)) = A005843(n). [Fixes even numbers].
a(p_n) = p_{a(n)}, or equally, a(n) = A049084(a(A000040(n))). [Restriction to primes induces the same sequence].
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves the "order of primeness of n"].

cp's OEIS Frontend

A250477 Number of times prime(n) (the n-th prime) occurs as the least prime factor among numbers 1 .. (prime(n)^2 * prime(n+1)): a(n) = A078898(A251720(n)).

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A249813 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2 * a(A078898(n+1))) - 1).

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A305798 Dirichlet convolution of A078898 with itself.

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A249812 Permutation of natural numbers: a(n) = A000079(A055396(n+1)-1) * ((2*A078898(n+1))-1).

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A250243 Permutation of natural numbers: a(n) = A246275(A055396(n+1), a(A078898(n+1))).

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A300247 Restricted growth sequence transform of A286457(n), filter combining A078898(n) and A246277(n).

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A249810 a(1) = 0, a(n) = A078898(A003961(n)).

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A249816 Permutation of natural numbers: a(n) = A246275(A055396(n+1), A078898(n+1)).

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A249826 Permutation of natural numbers: a(n) = A078898(A003961(A003961(A003961(2*n)))).

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