Cody M. Haderlie - cp's OEIS frontend

A277495 a(1) = 1; a(n) = n if a(n-1) = 1; otherwise, a(n) = digit sum (n written in base a(n-1)).

1, 2, 2, 1, 5, 2, 3, 4, 3, 2, 3, 2, 3, 4, 6, 6, 7, 6, 4, 2, 3, 4, 5, 8, 4, 5, 3, 2, 4, 6, 6, 7, 9, 10, 8, 8, 9, 6, 4, 4, 5, 6, 3, 6, 5, 6, 7, 12, 5, 2, 4, 4, 5, 6, 5, 4, 6, 8, 10, 6, 6, 7, 3, 4, 2, 2, 3, 6, 9, 14, 6, 2, 3

Offset: 1

Cody M. Haderlie, Oct 17 2016

			a(15-1) = 4; 15 in base 4 is 33; 3+3 = 6, so a(15) = 6.

$RecursionLimit=100
a[1]=1;
a[n_]:=If[a[n-1]!=1,Total[IntegerDigits[n,a[n-1]]],n]
Array[a,100]

lista(nn) = {print1(a = 1, ", "); for (n=2, nn, if (a==1, a = n, a = vecsum(digits(n, a));); print1(a, ", "););} \\ Michel Marcus, Oct 21 2016

A272903 Least nonnegative integer x such that n^2+nx-2n-x is prime.

Original entry on oeis.org

2, 0, 1, 1, 1, 1, 5, 1, 1, 1, 1, 2, 1, 2, 1, 1, 5, 2, 1, 1, 1, 4, 5, 1, 17, 1, 3, 1, 13, 1, 1, 2, 3, 2, 1, 2, 5, 1, 1, 4, 1, 2, 5, 1, 1, 1, 13, 1, 11, 1, 3, 5, 1, 1, 1, 1, 5, 7, 1, 1, 11, 2, 3, 1, 1, 1, 7, 1, 3, 1, 11, 3, 5, 2, 3, 1, 17, 3, 7, 4, 3, 7, 1, 3, 1, 1, 3, 2, 1, 3, 11, 2, 1, 1, 7, 1, 17, 5, 9

Offset: 2

Cody M. Haderlie, May 09 2016

nonn

The offset is 2 because n^2+nx-2n-x = -1 for n = 1 and all x.

a(n) seems to be exclusively prime or a product of two previous terms, and less than n excluding n = 2. x is most frequently equal to 1.

Cody M. Haderlie, Table of n, a(n) for n = 2..10000

Table[Min[DeleteCases[Table[If[PrimeQ[n^2+n*x-2n-x],x],{x,0,n}],Null]],{n,2,100}] (*Assuming that xHarvey P. Dale, May 12 2019 *)

A272181 a(1) = 1; a(n) = n mod a(k), k being the largest value such that n mod a(k) > 1, or n if there is no such k.

Original entry on oeis.org

1, 2, 3, 2, 2, 6, 7, 2, 2, 3, 2, 5, 3, 2, 3, 2, 2, 3, 4, 2, 3, 2, 2, 4, 4, 2, 3, 3, 2, 2, 3, 2, 3, 2, 2, 36, 2, 2, 3, 4, 2, 2, 3, 2, 9, 2, 2, 3, 4, 2, 3, 7, 4, 2, 3, 2, 3, 2, 2, 4, 5, 2, 3, 4, 2, 2, 3, 2, 4, 2, 3, 2, 3, 2

Offset: 1

Cody M. Haderlie, Apr 21 2016

nonn

			a(5) = 2; k = 3 and 5 mod a(3) = 2.

Cody M. Haderlie, Table of n, a(n) for n = 1..100000

A271720 a(1)=1; for n>1, define a sequence {b(m), m >= 1} by b(1)=n b(2)=13, and b(m) = A020639(b(m-2)) + A006530(b(m-1)); then a(n) is the number of terms in that sequence before the first of the infinite string of 4s.

Original entry on oeis.org

8, 13, 15, 13, 4, 13, 12, 13, 15, 13, 4, 13, 16, 13, 15, 13, 12, 13, 15, 13, 15, 13, 4, 13, 4, 13, 15, 13, 8, 13, 12, 13, 15, 13, 4, 13, 12, 13, 15, 13, 4, 13, 8, 13, 15, 13, 12, 13, 12, 13, 15, 13, 12, 13, 4, 13, 15, 13, 4, 13, 8, 13, 15, 13, 4, 13, 12, 13, 15, 13, 8, 13, 11, 13, 15, 13, 12, 13

Offset: 1

Cody M. Haderlie, Apr 12 2016

nonn

Note that the majority of the terms (every other term, initially) are equal to b(2), which is 13. This happens with several other values of b(2) less than 20. Many other values for b(2) have been tested, and it seems that for all b(2) < 100000000, a(n) < 20.

Records 8, 13, 15, 16, 19, 20, 24, ... occur at 1, 2, 3, 13, 349, 3919, 55633, ...

			n = 6; the sequence is:
6, 13, 15, 18, 6, 5, 7, 12, 10, 7, 9, 10, 8, 4, 4, 4, ...
There are 13 terms before the first of the infinite 4s; a(6) = 13.
For n = 55633 the sequence is: 55633, 13, 55646, 27836, 6961, 6963, 7172, 166, 85, 19, 24, 22, 13, 15, 18, 6, 5, 7, 12, 10, 7, 9, 10, 8, 4, 4, 4, ... . As the first 4 comes as the 25th term, a(55633) = 24. - _Antti Karttunen_, Oct 01 2018

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

Table[
Clear[h];
h[1]=x;
h[2]=13;
h[n_]:=FactorInteger[h[n-1]][[-1,1]]+FactorInteger[h[n-2]][[1,1]];
Position[Array[h,100],4][[1,1]]-1,
{x,1,100}] (*This only works for x≠4*)

A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1); \\ From A006530
A020639(n) = if(1==n, n, factor(n)[1, 1]);
A271720(n) = { my(up=1001, bvec = vector(up), m=1); bvec[1] = n; bvec[2] = 13; for(n=3,oo,bvec[n] = A020639(bvec[n-2])+A006530(bvec[n-1]); if(4==bvec[n], return(n-1))); }; \\ Antti Karttunen, Oct 01 2018

A269026 a(1)=1; for n>1, define a sequence {b(m), m >= 1} by b(1)=a(n-1), b(2)=n, and b(m) = A020639(b(m-2)) + A006530(b(m-1)); then a(n) is the number of terms in that sequence before the first of the infinite string of 4s.

Original entry on oeis.org

1, 9, 12, 1, 4, 10, 5, 6, 8, 6, 5, 3, 15, 7, 12, 2, 17, 7, 4, 6, 13, 11, 8, 10, 9, 3, 12, 9, 11, 3, 12, 2, 16, 6, 12, 10, 5, 11, 12, 6, 9, 7, 12, 14, 13, 11, 16, 10, 5, 7, 12, 14, 8, 10, 5, 11, 4, 10, 17, 15, 15, 7, 8, 2, 5, 3, 15, 7, 4, 9, 12, 10, 5, 10, 13, 3, 11, 11, 11

Offset: 1

Cody M. Haderlie, Apr 11 2016

nonn

			n = 3:
a(n-1) = a(2) = 9;
b(1) = 9, b(2) = 3;
the sequence generated is: 9, 3, 6, 6, 5, 7, 12, 10, 7, 9, 10, 8, 4, 4, 4, ...
There are 12 terms before the first of the infinite 4s, so a(3) = 12.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A006530, A020639, A271621.

spf(n) = if (n==1, 1, vecmin(factor(n)[,1]));
gpf(n) = if (n==1, 1, vecmax(factor(n)[,1]));
nbt(a, n) = {x = a; y = n; nb = 0; while (!((x==4) && (y==4)), z = spf(x) + gpf(y); x = y; y = z; nb++;); nb;}
lista(nn) = { print1(a=1, ", "); for (n=2, nn, na = nbt(a, n); print1(na, ", "); a = na;);} \\ Michel Marcus, Apr 12 2016

A271621 a(1) = 2, a(2) = 3, a(n) = A020639(a(n-2)) + A006530(a(n-1)).

Original entry on oeis.org

2, 3, 5, 8, 7, 9, 10, 8, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Cody M. Haderlie, Apr 10 2016

nonn

Any sequence generated by this formula and any values for a(1) and a(2) will have a finite number of terms not equal to 4; i.e., all such sequences will eventually end up at 4 (and all following terms will be 4; 4 is the only term that can appear more than twice in a row in a sequence because it is the only number equal to the sum of its least and greatest prime factors). Example: a(1) = 77713; a(2) = 16; the sequence is: 77713, 16, 77715, 159, 56, 10, 7, 9, 10, 8, 4, 4, 4, ...

			a(1) = 13; a(2) = 46.
lpf(13) = 13; gpf(46) = 23.
a(3) = 13 + 23 = 36.

Cf. A020639, A006530, A074320.

a[1] = 2; a[2] = 3; a[n_] := a[n] = FactorInteger[a[n - 2]][[1, 1]] +
FactorInteger[a[n - 1]][[-1, 1]]; Array[a, {120}] (* Michael De Vlieger, Apr 12 2016 *)

spf(n) = if (n==1, 1, vecmin(factor(n)[,1]));
gpf(n) = if (n==1, 1, vecmax(factor(n)[,1]));
lista(nn) = {print1(x=2, ", "); print1(y=3, ", "); for (n=1, nn, ny = spf(x) + gpf(y); print1(ny, ", "); x = y; y = ny;);} \\ Michel Marcus, Apr 15 2016

a(n) = lpf(a(n-2)) + gpf(a(n-1)), where lpf(n) is the least prime dividing n and gpf(n) is the greatest prime dividing n.

A267822 First differences of A271063.

Original entry on oeis.org

1, 2, 2, 3, 7, 5, 3, 25, 22, 17, 72, 161, 89, 72, 50, 25, 22, 17, 10, 7, 5, 3, 2, 5, 624, 622, 619, 614, 607, 597, 580, 558, 533, 483, 411, 6466, 5983, 5450, 4892, 4312, 3715, 3108, 2494, 43297, 40803, 37695, 33980

Offset: 1

Cody M. Haderlie, Apr 09 2016

nonn

Cody M. Haderlie, Table of n, a(n) for n = 1..1981

Cf. A271063.

a(n) = A271063(n + 1) - A271063(n).

A271441 a(1) = 2; if gpf(a(n-1)) <= n-1 then a(n) = a(n-1) + a(gpf(a(n-1))), else a(n) = a(n-1) + 1, where gpf(m) is the greatest prime factor of m.

Original entry on oeis.org

2, 3, 4, 7, 8, 11, 12, 16, 19, 20, 28, 40, 48, 52, 100, 108, 112, 124, 125, 133, 258, 259, 260, 308, 336, 348, 349, 350, 362, 363, 391, 651, 1042, 1043, 1044, 1406, 1407, 1408, 1436, 1437, 1438, 1439, 1440, 1448, 1449, 1709, 1710, 1835, 1836, 1948

Offset: 1

Cody M. Haderlie, Apr 07 2016

Choosing the initial value a(1) = 2 seems to produce the most irregular sequence.

			Since a(11) = 28, gpf(28) = 7 and a(7) = 12, then a(12) = 28 + 12 = 40.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A006530 (gpf).

Nest[Append[#, (x = #[[-1]]) + If[(p = FactorInteger[x][[-1, 1]]) <= Length@#, #[[p]], 1]] &, {2}, 49] (* Ivan Neretin, Jan 27 2017 *)

gpf(n) = if (n==1, 1, vecmax(factor(n)[,1]));
lista(nn) = {va = vector(nn); print1(va[1] = 2, ", "); for (n=2, nn, if (gpf(va[n-1]) <= n-1, va[n] = va[n-1] + va[gpf(va[n-1])], va[n] = va[n-1]+1); print1(va[n], ", "););} \\ Michel Marcus, Apr 09 2016

A271418 Define a sequence by b(1) = n, b(m) = b(m-1) + b(m-1)/A052126(m) + 1; then a(n) is the least m such that b(m) belongs to A270807.

Original entry on oeis.org

1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 4, 1, 2, 3, 4, 1, 3, 1, 4, 2, 4, 1, 4, 1, 4, 4, 4, 3, 2, 2, 3, 2, 2, 1, 4, 2, 2, 2, 2, 1, 4, 1, 3, 3, 2, 3, 2, 2, 3, 2, 3, 1, 4, 1

Offset: 1

Cody M. Haderlie, Apr 06 2016

			the sequence where b(1) = 2 is 2, 4, 7, 9, 13, 15, ...
7 is shared with A270807
there are 3 terms in 2, 4, 7; a(2) = 3

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A052126, A269304, A270807.

f[n_] := n + n/FactorInteger[n][[-1, 1]] + 1; s = NestList[f, 1, 10^2]; Table[Length@ NestWhileList[f, n, ! MemberQ[s, #] &], {n, 120}] (* Michael De Vlieger, Apr 08 2016 *)

a(75) corrected by Chai Wah Wu, Apr 06 2016

A271063 a(1) = 2, a(2) = 3; thereafter a(n) = a(n-1) + a(|n-a(T)|), where a(T) is the largest term in the sequence before a(n) such that 0 < |n-a(T)| < n.

Original entry on oeis.org

2, 3, 5, 7, 10, 17, 22, 25, 50, 72, 89, 161, 322, 411, 483, 533, 558, 580, 597, 607, 614, 619, 622, 624, 629, 1253, 1875, 2494, 3108, 3715, 4312, 4892, 5450, 5983, 6466, 6877, 13343, 19326, 24776, 29668, 33980, 37695, 40803, 43297, 86594, 127397, 165092, 199072, 228740, 253516

Offset: 1

Cody M. Haderlie, Apr 05 2016

nonn

When a(1) = 1, the formula generates the natural numbers.

			a(11) = 89
a(T) = a(7) = 22
|12-22| = 10
a(10) = 72
89 + 72 = 161
a(12) = 161

Cody M. Haderlie, Table of n, a(n) for n = 1..1982

a = {2, 3}; Do[AppendTo[a, a[[n - 1]] + Part[a, #] &@ SelectFirst[ Reverse@ Abs[a - n], 0 < # < n &]], {n, 3, 50}]; a (* Michael De Vlieger, Apr 08 2016 *)

lista(nn) = {va = vector(nn); print1(va[1] = 2, ", "); print1(va[2] = 3, ", "); for (n=3, nn, T = 0; forstep(k = n-1, 1, -1, vabs = abs(n - va[k]); if ((vabs < n) && (vabs > 0), T = k; break);); va[n] = va[n-1] + va[abs(n-va[T])]; print1(va[n], ", "););} \\ Michel Marcus, Apr 08 2016

cp's OEIS Frontend

User: Cody M. Haderlie

A277495 a(1) = 1; a(n) = n if a(n-1) = 1; otherwise, a(n) = digit sum (n written in base a(n-1)).

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A272903 Least nonnegative integer x such that n^2+nx-2n-x is prime.

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A272181 a(1) = 1; a(n) = n mod a(k), k being the largest value such that n mod a(k) > 1, or n if there is no such k.

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A271720 a(1)=1; for n>1, define a sequence {b(m), m >= 1} by b(1)=n b(2)=13, and b(m) = A020639(b(m-2)) + A006530(b(m-1)); then a(n) is the number of terms in that sequence before the first of the infinite string of 4s.

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A269026 a(1)=1; for n>1, define a sequence {b(m), m >= 1} by b(1)=a(n-1), b(2)=n, and b(m) = A020639(b(m-2)) + A006530(b(m-1)); then a(n) is the number of terms in that sequence before the first of the infinite string of 4s.

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A271621 a(1) = 2, a(2) = 3, a(n) = A020639(a(n-2)) + A006530(a(n-1)).

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Formula

A267822 First differences of A271063.

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A271441 a(1) = 2; if gpf(a(n-1)) <= n-1 then a(n) = a(n-1) + a(gpf(a(n-1))), else a(n) = a(n-1) + 1, where gpf(m) is the greatest prime factor of m.

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A271418 Define a sequence by b(1) = n, b(m) = b(m-1) + b(m-1)/A052126(m) + 1; then a(n) is the least m such that b(m) belongs to A270807.

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