A117563 -id:A117563 - cp's OEIS frontend

A130882 a(n) = smallest k such that A002808(n+1) = A002808(n) + (A002808(n) mod k), or 0 if no such k exists.

0, 4, 7, 2, 4, 5, 13, 2, 7, 4, 19, 2, 4, 23, 2, 5, 2, 13, 4, 31, 2, 3, 2, 17, 37, 2, 19, 4, 43, 2, 4, 47, 2, 7, 2, 5, 53, 2, 5, 2, 4, 29, 61, 2, 3, 2, 4, 67, 2, 4, 5, 73, 2, 3, 2, 4, 79, 2, 4, 83, 2, 5, 2, 43, 89, 2, 7, 2, 3, 2, 47, 97

Offset: 1

Rémi Eismann, Aug 21 2007 - Jan 09 2011

nonn

a(n) is the "weight" of composite numbers.

The decomposition of composite numbers into weight * level + gap is A002808(n) = a(n) * A179621(n) + A073783(n) if a(n) > 0.

			For n = 1 we have A002808(n) = 4, A002808(n+1) = 6; there is no k such that 6 - 4 = 2 = (4 mod k), hence a(1) = 0.
For n = 3 we have A002808(n) = 8, A002808(n+1) = 9; 7 is the smallest k such that 9 - 8 = 1 = (8 mod k), hence a(3) = 7.
For n = 24 we have A002808(n) = 36, A002808(n+1) = 38; 17 is the smallest k such that 38 - 36 = 2 = (36 mod k), hence a(24) = 17.

Remi Eismann, Table of n, a(n) for n=1..9999

Cf. A020639, A117078, A117563, A001223, A118534, A090369, A090368, A130533, A130650, A130703.

A130889 a(n) = smallest k such that A000959(n+1) = A000959(n) + (A000959(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 5, 5, 11, 9, 17, 19, 29, 29, 31, 37, 47, 13, 59, 5, 5, 71, 71, 71, 9, 29, 31, 9, 107, 103, 5, 5, 131, 43, 131, 11, 5, 157, 167, 51, 5, 191, 7, 197, 199, 29, 5, 43, 227, 233, 233, 223, 257, 15, 9, 263, 281, 281, 281, 97, 13, 59, 317, 7, 17, 17, 47, 11, 353, 71, 349, 379, 389

Offset: 1

Rémi Eismann, Aug 21 2007 - Jan 23 2011

nonn

a(n) is the "weight" of lucky numbers.

The decomposition of lucky numbers into weight * level + gap is A000959(n) = a(n) * A184828(n) + A031883(n) if a(n) > 0.

			For n = 1 we have A000959(n) = 1, A000959(n+1) = 3; there is no k such that 3 - 1 = 2 = (1 mod k), hence a(1) = 0.
For n = 3 we have A000959(n) = 7, A000959(n+1) = 9; 5 is the smallest k such that 9 - 7 = 2 = (7 mod k), hence a(3) = 5.
For n = 24 we have A000959(n) = 105, A000959(n+1) = 111; 9 is the smallest k such that 111 - 105 = 6 = (105 mod k), hence a(24) = 9.

Remi Eismann, Table of n, a(n) for n=1..9999

Cf. A000959, A031883, A184828, A184827, A117078, A117563, A001223, A118534.

A125565 Primes p=prime(i) of level (1,12), i.e., such that A118534(i)=prime(i-12).

Original entry on oeis.org

15014557, 27001043, 29602093, 50234633, 87028433, 91814759, 94529221, 103336843, 112840309, 113774329, 113961299, 114887657, 115528969, 118974901, 129235273, 144352123, 146127721, 160370491, 163559197, 169274999, 188168059, 188895919, 191829409, 198823447

Offset: 1

Rémi Eismann, Jan 27 2007

nonn

This subsequence of A125830 and of A162174 gives primes of level (1,12): If prime(i) has level 1 in A117563 and 2*prime(i) - prime(i+1) = prime(i-k), then we say that prime(i) has level (1,k).

			prime(5316164) - prime(5316163) = 91814831 - 91814759 = 91814759 - 91814687 = prime(5316163) - prime(5316163-12) and prime(5316163) has level 1 in A117563, so prime(5316163)=91814759 has level (1,12).

Fabien Sibenaler, Table of n, a(n) for n = 1..1800

Cf. A006562 (primes of level (1,1)), A117078, A117563, A006562, A117876, A118464, A118467, A119402, A119403, A119404.

Definition and comment reworded following suggestions from the author. - M. F. Hasler, Nov 30 2009

A125572 Primes p=prime(i) of level (1,13), i.e., such that A118534(i)=prime(i-13).

Original entry on oeis.org

35630467, 118877047, 123823081, 140061577, 155032793, 175204303, 184606997, 188871349, 189489733, 232093339, 244004749, 278518081, 309055367, 310542257, 313596551, 315659909, 329918227, 340761691, 389220347, 398329523, 411405833, 422745641, 480428801, 485608819

Offset: 1

Rémi Eismann and Fabien Sibenaler, Jan 27 2007

nonn

This subsequence of A125830 and of A162174 gives primes of level (1,13): If prime(i) has level 1 in A117563 and 2*prime(i) - prime(i+1) = prime(i-k), then we say that prime(i) has level (1,k).

			prime(10272256) - prime(10272255) = 184607153 - 184606997 = 184606997 - 184606841 = prime(10272255) - prime(10272255-13) and prime(10272255) has level 1 in A117563, so prime(10272255)=184606997 has level (1,13).

Fabien Sibenaler, Table of n, a(n) for n = 1..700

Cf. A006562 (primes of level (1,1)), A117078, A117563, A117876, A118464, A118467, A119402, A119403, A119404.

Definition and comment reworded following suggestions from the authors. - M. F. Hasler, Nov 30 2009

A125574 Primes p=prime(i) of level (1,14), i.e., such that A118534(i)=prime(i-14).

Original entry on oeis.org

31515413, 69730637, 132102911, 132375259, 215483129, 284491367, 325689253, 388190689, 548369603, 620829113, 633418787, 638213603, 670216277, 793852487, 797759539, 960200149, 1038197399, 1050359137, 1092920249, 1331713301, 1342954871, 1349496367, 1365964199

Offset: 1

Rémi Eismann and Fabien Sibenaler, Jan 27 2007

nonn

This subsequence of A125830 and of A162174 gives primes of level (1,14): If prime(i) has level 1 in A117563 and 2*prime(i) - prime(i+1) = prime(i-k), then we say that prime(i) has level (1,k).

			prime(15456800) - prime(15456799) = 284491601 - 284491367 = 284491367 - 284491133 = prime(15456799) - prime(15456799-14) and prime(15456799) has level 1 in A117563, so prime(15456799) = 284491367 has level (1,14).

Fabien Sibenaler, Table of n, a(n) for n = 1..250

Cf. A117078, A117563, A006562 (primes of level (1,1)), A117876, A118464, A118467, A119402, A119403, A119404.

lista(nn) = my(c=15, v=primes(15)); forprime(p=53, nn, if(2*v[c]-p==v[c=c%15+1], print1(precprime(p-1), ", ")); v[c]=p); \\ Jinyuan Wang, Jun 18 2021

Definition and comment reworded following suggestions from the authors. - M. F. Hasler, Nov 30 2009

A118123 a(n) = number of k's such that prime(n+1) = prime(n) + (prime(n) mod k).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 3, 2, 1, 3, 1, 2, 3, 2, 1, 1, 3, 2, 4, 3, 1, 4, 3, 3, 2, 5, 4, 7, 6, 2, 2, 2, 7, 2, 5, 2, 1, 2, 3, 1, 3, 3, 7, 6, 7, 2, 1, 2, 8, 7, 1, 3, 5, 4, 1, 1, 3, 2, 6, 5, 5, 3, 2, 3, 2, 2, 4, 2, 7, 6, 1, 6, 2, 1, 6, 3, 2, 2, 2, 5, 3, 2, 7, 3, 6, 3, 6, 2, 7, 6, 5, 2, 6, 5, 10, 3, 2, 3, 2, 2, 2, 3, 1, 9, 2

Offset: 1

Rémi Eismann and Robert G. Wilson v, May 12 2006

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A117078, A117563.

f[n_] := If[n == 1, 0, Block[{p = Prime@n, np = Prime[n + 1]}, Length@Select[Divisors[2p - np], # >= np - p &]]]; Array[f, 105]
nk[n_]:=Count[Mod[n,Range[n-1]],?(#==NextPrime[n]-n&)]; nk/@Prime[ Range[ 110]] (* _Harvey P. Dale, May 27 2016 *)

A118123(n)={my(d=prime(n+1)-n=prime(n)); sumdiv(n-d,k,k>d)}

a(n) = # { k>0 | prime(n+1) - prime(n) = prime(n) % k }, where p % k is the remainder of p divided by k.

Edited by M. F. Hasler, Nov 07 2009

A119595 Primes for which the weight as defined in A117078 is 15 and the gap as defined in A001223 is 8.

Original entry on oeis.org

743, 1193, 1523, 1733, 2003, 2243, 2273, 3623, 4583, 4943, 5573, 5693, 6143, 6203, 6473, 7673, 8573, 8933, 9803, 10103, 11243, 11813, 12413, 12503, 13163, 14423, 14843, 15053, 15233, 15383, 16103

Offset: 1

Rémi Eismann, Jun 01 2006, May 04 2007

nonn

The prime numbers in this sequence are of the form (30i-7) with i=(level(n)+1)/2, level(n) defined in A117563.

			a(1)=743 because of 751=743+mod(743;15) and g(n)=751-743=8
30*((49+1)/2)-7=743
a(2)=1193 because of 1201=1193+mod(1193;15) and g(n)=1201-1193=8
30*((79+1)/2)-7=1193

Remi Eismann, Table of n, a(n) for n=1..10000

Cf. A117078, A001223, A117563, A001359, A118359, A074822, A118922, A118924, A119504.

A119596 Primes for which the weight as defined in A117078 is 11 and the gap as defined in A001223 is 10.

Original entry on oeis.org

241, 1627, 2089, 4201, 4663, 4861, 5323, 6247, 6379, 6709, 8821, 9283, 9679, 10141, 12253, 12517, 12781, 13441, 15091, 15289, 15619, 17599, 17929, 19249, 19447, 19843, 21757, 23539, 26839, 28687, 33703, 34429, 34693, 35089, 35353, 36343

Offset: 1

Rémi Eismann, Jun 01 2006, May 04 2007

nonn

The prime numbers in this sequence are of the form (22i-1) with i=(level(n)+1)/2, level(n) defined in A117563.

			a(1)=241 because of 251=241+mod(241;11) and 251-241=10.
22*((21+1)/2)-1=241, level=21
a(2)=1627 because of 1637=1627+mod(1627;11) and 1637-1627=10
22*((147+1)/2)-1=1627, level=147

Remi Eismann, Table of n, a(n) for n=1..10000

Cf. A117078, A001223, A117563, A001359, A118359, A074822, A118922, A118924, A119504.

A119597 Primes for which the weight as defined in A117078 is 11 and the gap as defined in A001223 is 6.

Original entry on oeis.org

61, 677, 941, 1117, 1601, 2063, 2371, 3691, 3911, 4021, 5297, 5407, 6067, 6353, 6991, 7541, 7717, 8311, 8641, 8663, 9103, 9851, 10973, 11897, 12491, 12953, 13591, 13613, 13723, 14537, 15131, 15263, 15307, 15461, 15901, 16363

Offset: 1

Rémi Eismann, Jun 01 2006, May 04 2007

nonn

The prime numbers in this sequence are of the form (22i-5) with i=(level(n)+1)/2, level(n) defined in A117563.

			a(1)=61 because of 67=61+mod(61;11) and 67-61=6.
22*((5+1)/2)-5=61, level=5
a(2)=677 because of 683=677+mod(677;11) and 683-677=6
22*((61+1)/2)-5=677, level=5

Remi Eismann, Table of n, a(n) for n=1..10000

Cf. A117078, A001223, A117563, A001359, A118359, A074822, A118922, A118924, A119504.

A184752 a(n) = largest k such that A014612(n+1) = A014612(n) + (A014612(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 16, 13, 26, 26, 18, 40, 43, 40, 48, 41, 60, 64, 66, 65, 74, 74, 64, 86, 97, 96, 99, 100, 106, 112, 115, 110, 123, 120, 122, 129, 146, 143, 152, 144, 163, 160, 169, 170, 170, 173, 168, 178, 184, 186, 185, 183, 202, 202, 214

Offset: 1

Rémi Eismann, Jan 21 2011

nonn

From the definition, a(n) = A014612(n) - A114403(n) if A014612(n) - A114403(n) > A114403(n), 0 otherwise where A014612 are the 3-almost primes and A114403 are the gaps between 3-almost primes.

			For n = 1 we have A014612(1) = 8, A014612(2) = 12; there is no k such that 12 - 8 = 4 = (8 mod k), hence a(1) = 0.
For n = 3 we have A014612(3) = 18, A014612(4) = 20; 16 is the largest k such that 20 - 18 = 2 = (18 mod k), hence a(3) = 16.
For n = 21 we have A014612(21) = 98, A014612(22) = 99; 97 is the largest k such that 99 - 98 = 1 = (97 mod k), hence a(21) = 97.

Rémi Eismann, Table of n, a(n) for n = 1..10000

Cf. A014612, A114403, A130650, A184753, A117078, A117563, A001223, A118534.

cp's OEIS Frontend

A130882 a(n) = smallest k such that A002808(n+1) = A002808(n) + (A002808(n) mod k), or 0 if no such k exists.

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A130889 a(n) = smallest k such that A000959(n+1) = A000959(n) + (A000959(n) mod k), or 0 if no such k exists.

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A125565 Primes p=prime(i) of level (1,12), i.e., such that A118534(i)=prime(i-12).

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A125572 Primes p=prime(i) of level (1,13), i.e., such that A118534(i)=prime(i-13).

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A125574 Primes p=prime(i) of level (1,14), i.e., such that A118534(i)=prime(i-14).

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A118123 a(n) = number of k's such that prime(n+1) = prime(n) + (prime(n) mod k).

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A119595 Primes for which the weight as defined in A117078 is 15 and the gap as defined in A001223 is 8.

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A119596 Primes for which the weight as defined in A117078 is 11 and the gap as defined in A001223 is 10.

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A119597 Primes for which the weight as defined in A117078 is 11 and the gap as defined in A001223 is 6.

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