A109411 -id:A109411 - cp's OEIS frontend

A109825 Initial terms of groups in the partition of the sequence of natural numbers A109411.

1, 4, 5, 9, 10, 11, 16, 18, 21, 22, 23, 36, 39, 40, 43, 45, 47, 49, 50, 54, 60, 62, 63, 69, 70, 72, 74, 75, 89, 93, 94, 95, 96, 99, 104, 106, 107, 109, 111, 112, 115, 116, 126, 128, 135, 140, 144, 146, 147, 149, 151, 153, 159, 160, 162, 165, 170, 172, 175, 179, 184, 190

Offset: 1

Zak Seidov, Jul 03 2005

nonn

Cf. A109411, A109826.

s={1};a=1;Do[Do[If[Plus@@Last/@FactorInteger[(a+x)(x-a+1)/2]==2, a=x+1;AppendTo[s, a];Break[]], {x, a, 20000}], {k, 1, 100}];A109825=s

A109826 Final terms of groups in the partition of the sequence of natural numbers A109411.

Original entry on oeis.org

3, 4, 8, 9, 10, 15, 17, 20, 21, 22, 35, 38, 39, 42, 44, 46, 48, 49, 53, 59, 61, 62, 68, 69, 71, 73, 74, 88, 92, 93, 94, 95, 98, 103, 105, 106, 108, 110, 111, 114, 115, 125, 127, 134, 139, 143, 145, 146, 148, 150, 152, 158, 159, 161, 164, 169, 171, 174, 178, 183, 189

Offset: 1

Zak Seidov, Jul 03 2005

nonn

Cf. A109411, A109825.

s={3};a=4;Do[Do[If[Plus@@Last/@FactorInteger[(a+x)(x-a+1)/2]==2, a=x+1;AppendTo[s, x];Break[]], {x, a, 20000}], {k, 1, 100}];A109826=s

A109412 Semiprimes = one-term groups in the partition of the sequence of natural numbers A109411.

Original entry on oeis.org

4, 9, 10, 21, 22, 39, 49, 62, 69, 74, 93, 94, 95, 106, 111, 115, 146, 159, 203, 218, 219, 247, 254, 259, 295, 309, 319, 326, 327, 346, 355, 358, 394, 395, 398, 403, 427, 445, 446, 447, 451, 454, 551, 581, 586, 611, 622, 623, 671, 674, 679, 687, 694, 695, 698

Offset: 1

Zak Seidov, Jul 03 2005

nonn

Cf. A109411, A109825, A109826.

A109823 a(n) is the minimal b >= n such that sum of consecutive integers from n to b is a semiprime.

Original entry on oeis.org

3, 4, 7, 4, 8, 6, 8, 11, 9, 10, 15, 13, 16, 14, 15, 17, 18, 20, 20, 23, 21, 22, 35, 25, 25, 26, 28, 29, 32, 32, 36, 33, 33, 34, 35, 38, 42, 38, 39, 42, 45, 43, 44, 50, 46, 46, 48, 53, 49, 53, 51, 54, 56, 59, 55, 62, 57, 58, 60, 61, 62, 62, 68, 65, 65, 67, 70, 71, 69, 71, 72, 73

Offset: 1

Zak Seidov & Max Alekseyev, Jul 03 2005

nonn

If n is a semiprime, a(n)=n. It is not evident that for any n there is relative a(n), see A109411. For n <1000, the corresponding a(n) exists. Cf. A109824(n) = A109823(n) - n + 1.

Cf. A109411, A109824.

cis[n_]:=Module[{b=0},While[PrimeOmega[(n(n+b))/2]!=2,b++];n+b]; Array[ cis,80] (* Harvey P. Dale, Mar 30 2014 *)

{ a(n) = my(s, m); s = n; m = n; while(bigomega(s)!=2, m++; s += m); m }

A109824 a(n) is the number of consecutive integers starting with n summing up to a semiprime.

Original entry on oeis.org

3, 3, 5, 1, 4, 1, 2, 4, 1, 1, 5, 2, 4, 1, 1, 2, 2, 3, 2, 4, 1, 1, 13, 2, 1, 1, 2, 2, 4, 3, 6, 2, 1, 1, 1, 3, 6, 1, 1, 3, 5, 2, 2, 7, 2, 1, 2, 6, 1, 4, 1, 3, 4, 6, 1, 7, 1, 1, 2, 2, 2, 1, 6, 2, 1, 2, 4, 4, 1, 2, 2, 2, 4, 1, 14, 6, 1, 3, 2, 2, 5, 1, 13, 2, 1, 1, 1, 2, 4, 14, 1, 2, 1, 1, 1, 3, 4, 4, 5, 2

Offset: 1

Zak Seidov & Max Alekseyev, Jul 03 2005

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A109411, A109823.

Table[Position[Accumulate[Range[n,n+300]],?(PrimeOmega[#]==2&),1,1],{n,100}]//Flatten (* The parameter 300 is sufficient for n up to 100000 but might have to be increased for larger n. *) (* _Harvey P. Dale, Feb 14 2018 *)

{ a(n) = my(s, m); s=n; m=n; while(bigomega(s)!=2, m++; s+=m); m-n+1 }

A109824(n) = A109823(n) - n + 1.

A133837 Semiprimes from partition of sequence of positive integers.

Original entry on oeis.org

6, 4, 26, 9, 10, 65, 33, 57, 21, 22, 377, 111, 39, 123, 87, 91, 95, 49, 206, 339, 121, 62, 393, 69, 141, 145, 74, 1141, 362, 93, 94, 95, 291, 505, 209, 106, 215, 219, 111, 339, 115, 1205, 253, 917, 685, 566, 289, 146, 295, 299, 303, 933, 159, 321, 489, 835, 341

Offset: 1

Zak Seidov, Sep 26 2007

nonn

Partition the sequence of positive integers into groups of numbers that sum up to semiprimes: {1, 2, 3}, {4}, {5..8}, {9}, {10}, {11..15}, {16, 17}, {18..20}, {21}, {22}, {23..35}, {36..38}, {39}, {40..42}, {43, 44}, etc. Corresponding semiprimes are: 6, 4, 26, 9, 10, 65, 33, 57, 21, 22, 377, 111, 39, 123, 87, etc.

Is the sequence finite? See comment in A109411.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A109411.

s=Range[300];c=0;Label[1];i=1;p=s[[1]];While[i1,Goto[1]];Table[a[j],{j,c}]

cp's OEIS Frontend

A109825 Initial terms of groups in the partition of the sequence of natural numbers A109411.

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A109826 Final terms of groups in the partition of the sequence of natural numbers A109411.

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A109412 Semiprimes = one-term groups in the partition of the sequence of natural numbers A109411.

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A109823 a(n) is the minimal b >= n such that sum of consecutive integers from n to b is a semiprime.

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A109824 a(n) is the number of consecutive integers starting with n summing up to a semiprime.

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A133837 Semiprimes from partition of sequence of positive integers.

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