A029709 -id:A029709 - cp's OEIS frontend

A076976 Product of the smallest prime divisors of composite numbers between successive primes.

1, 2, 2, 12, 2, 12, 2, 12, 120, 2, 120, 12, 2, 12, 168, 120, 2, 120, 12, 2, 168, 12, 120, 1680, 12, 2, 12, 2, 12, 2217600, 12, 168, 2, 15840, 2, 120, 168, 12, 312, 120, 2, 15840, 2, 12, 2, 221760, 262080, 12, 2, 12, 120, 2, 18720, 264, 168, 120, 2, 120, 12, 2, 34272

Offset: 1

Amarnath Murthy, Oct 23 2002

nonn

From Bernard Schott, Apr 09 2020: (Start)
a(n) = 2 iff prime(n) is in A001359 (prime gap=2).
a(n) = 12 iff prime(n) is in A029710 (prime gap=4).
a(n) = 24 * p with p prime >= 5 iff prime(n) is in A031924 (prime gap=6).
a(n) = 2^m * q with q odd >= 3 iff prime(n+1) - prime(n) = 2*m where m = A007814(a(n)). (End)

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A029707 (a(n)=2), A029709 (a(n)=12), A076977.

Cf. A001359, A029710, A031924.

p:= 2:
for i from 1 to 100 do
  q:= p; p:= nextprime(p);
  A[i]:= mul(min(numtheory:-factorset(i)),i=q+1..p-1);
od:
seq(A[i],i=1..100); # Robert Israel, Mar 30 2020

pspd[{p1_,p2_}]:=Times@@(FactorInteger[#][[1,1]]&/@Range[p1+1,p2-1]); pspd/@Partition[ Prime[Range[70]],2,1] (* Harvey P. Dale, Jan 12 2024 *)

a(n) = {my(p=1, pn=prime(n)); forcomposite(c=pn, nextprime(pn+1)-1, p *= vecmin(factor(c)[,1]);); p;} \\ Michel Marcus, Mar 31 2020

More terms from Sascha Kurz, Jan 22 2003

A320708 Indices of primes followed by a gap (distance to next larger prime) of 20.

Original entry on oeis.org

154, 259, 442, 480, 548, 753, 777, 783, 876, 971, 1035, 1066, 1095, 1106, 1147, 1254, 1277, 1302, 1337, 1345, 1355, 1381, 1396, 1400, 1423, 1438, 1562, 1592, 1613, 1662, 1669, 1808, 1955, 2016, 2043, 2081, 2116, 2129, 2147, 2226, 2302, 2307, 2387, 2517, 2547, 2563, 2694, 2724, 2745, 2755, 2766

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A031938.

Vincenzo Librandi, Table of n, a(n) for n = 1..10720
Index entries for primes, gaps between.

Equals A000720 o A031938.
Row 10 of A174349.
Subsequence of A107730 (prime(n+1) ends in same digit as prime(n)).
Indices of 20's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..3000] | NthPrime(n+1) - NthPrime(n) eq 20]; // Vincenzo Librandi, Mar 22 2019

Select[Range[3000], Prime[#] + 20 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 22 2019 *)

A(N=100,g=20,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A031938(n)).

A320708 = { i > 0 | prime(i+1) = prime(i) + 20 } = A001223^(-1)({20}).

A320704 Indices of primes followed by a gap (distance to next larger prime) of 12.

Original entry on oeis.org

46, 47, 91, 97, 114, 121, 139, 168, 197, 203, 214, 232, 239, 240, 242, 267, 278, 280, 290, 312, 317, 342, 357, 363, 376, 381, 404, 423, 437, 439, 449, 452, 461, 470, 472, 489, 499, 511, 546, 550, 562, 565, 599, 600, 617, 633, 634, 647, 653, 657, 675, 680, 692, 698, 716, 728

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes given in A031930.

Vincenzo Librandi, Table of n, a(n) for n = 1..10150
Index entries for primes, gaps between.

Equals A000720 o A031930.
Row 6 of A174349.
Indices of 12's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..1000] | NthPrime(n+1) - NthPrime(n) eq 12]; // Vincenzo Librandi, Mar 21 2019

Select[Range[1000], Prime[#] + 12 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 21 2019 *)

A320704_vec(N=100,g=12,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A031930(n)).

A320704 = { i > 0 | prime(i+1) = prime(i) + 12 }.

A320705 Indices of primes followed by a gap (distance to next larger prime) of 14.

Original entry on oeis.org

30, 62, 66, 137, 146, 150, 162, 223, 250, 283, 309, 350, 360, 382, 402, 410, 424, 434, 503, 514, 526, 532, 536, 570, 610, 649, 654, 666, 687, 704, 706, 747, 780, 790, 867, 906, 919, 929, 967, 978, 981, 992, 1011, 1023, 1038, 1042, 1057, 1072, 1133, 1154, 1160, 1177, 1184

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A031932.

Vincenzo Librandi, Table of n, a(n) for n = 1..8000
Index entries for primes, gaps between.

Equals A000720 o A031932.
Row 7 of A174349.
Indices of 14's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..1500] | NthPrime(n+1) - NthPrime(n) eq 14]; // Vincenzo Librandi, Mar 19 2019

Select[Range[1500], Prime[#] + 14 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 19 2019 *)
Position[Differences[Prime[Range[1200]]],14]//Flatten (* Harvey P. Dale, Nov 28 2024 *)

A(N=100,g=14,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A031932(n)).

A320705 = { i > 0 | prime(i+1) = prime(i) + 14 }.

A320706 Indices of primes followed by a gap (distance to next larger prime) of 16.

Original entry on oeis.org

282, 295, 319, 331, 335, 378, 409, 445, 476, 478, 481, 510, 560, 566, 619, 624, 674, 701, 739, 775, 856, 871, 881, 886, 935, 941, 1007, 1069, 1077, 1121, 1146, 1193, 1222, 1261, 1286, 1322, 1331, 1356, 1372, 1388, 1405, 1460, 1487, 1500, 1587, 1603, 1608, 1612, 1699, 1719, 1734, 1740, 1811, 1876, 1924, 1956, 1969, 1977, 2002, 2022, 2034, 2042, 2071

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A031934.

Vincenzo Librandi, Table of n, a(n) for n = 1..9790
Index entries for primes, gaps between.

Equals A000720 o A031934.
Row 8 of A174349.
Indices of 16's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..2100] | NthPrime(n+1) - NthPrime(n) eq 16]; // Vincenzo Librandi, Mar 19 2019

Select[Range[2500], Prime[#] + 16 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 19 2019 *)
Position[Partition[Prime[Range[2100]],2,1],?(#[[2]]-#[[1]]== 16&),1,Heads-> False]//Flatten (* _Harvey P. Dale, Nov 01 2020 *)

A(N=100,g=16,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A031934(n)).

A320706 = { i > 0 | prime(i+1) = prime(i) + 16 }.

A320707 Indices of primes followed by a gap (distance to next larger prime) of 18.

Original entry on oeis.org

99, 180, 205, 221, 274, 293, 326, 368, 416, 529, 539, 573, 597, 602, 607, 623, 635, 639, 677, 693, 725, 785, 811, 838, 844, 852, 855, 916, 937, 939, 942, 945, 968, 997, 1028, 1093, 1130, 1151, 1203, 1227, 1252, 1304, 1311, 1349, 1508, 1514, 1519, 1523, 1540, 1547, 1629, 1636, 1641, 1654, 1656

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A031936.

Vincenzo Librandi, Table of n, a(n) for n = 1..10300
Index entries for primes, gaps between.

Equals A000720 o A031936.
Row 9 of A174349.
Indices of 18's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

[n: n in [1..1700] | NthPrime(n+1) - NthPrime(n) eq 18]; // Vincenzo Librandi, Mar 22 2019

Select[Range[1700], Prime[#] + 18 == Prime[# + 1] &] (* Vincenzo Librandi, Mar 22 2019 *)
Flatten[Position[Differences[Prime[Range[2000]]],18]] (* Harvey P. Dale, May 12 2022 *)

A(N=100,g=18,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A031936(n)).

A320707 = { i > 0 | prime(i+1) = prime(i) + 18 } = A001223^(-1)({18}).

A320713 Indices of primes followed by a gap (distance to next larger prime) of 30.

Original entry on oeis.org

590, 650, 708, 757, 842, 890, 928, 985, 1006, 1051, 1108, 1556, 1570, 1648, 1650, 1675, 1754, 1900, 1919, 2027, 2125, 2149, 2321, 2391, 2397, 2429, 2631, 2637, 2699, 2781, 2866, 2918, 2989, 2993, 3010, 3085, 3153, 3207, 3315, 3340, 3350, 3373, 3420, 3511, 3551, 3580, 3637, 3751, 3777, 3948

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A124596.

Index entries for primes, gaps between.

Equals A000720 o A124596.
Indices of 30's in A001223.
Row 15 of A174349.
Subsequence of A107730 (prime(n+1) ends in same digit as prime(n)).
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

A(N=100,g=30,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A124596(n)).

A320713 = { i>0 | prime(i+1) = prime(i) + 30 } = A001223^(-1)({30}).

A320718 Indices of primes followed by a gap (distance to next larger prime) of 40.

Original entry on oeis.org

2191, 2344, 2524, 2788, 3562, 4058, 4677, 5030, 5349, 6076, 6145, 6256, 6320, 6442, 6454, 6902, 7232, 7488, 8119, 8152, 8245, 8366, 8553, 8567, 8591, 8746, 9260, 9361, 10536, 10735, 11095, 11407, 11534, 11781, 12227, 12312, 12663, 12815, 12940, 13015, 13333, 13676, 13873, 14065, 14123

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A126721.

Index entries for primes, gaps between.

Equals A000720 o A126721.
Row 20 of A174349.
Subsequence of A107730 (prime(n+1) ends in same digit as prime(n)).
Indices of 40's in A001223.
Cf. A029707, A029709, A320701, A320702, ..., A320720 (analog for gaps 2, 4, 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).

A(N=100,g=40,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A126721(n)).

A320718 = { i > 0 | prime(i+1) = prime(i) + 40 } = A001223^(-1)({40}).

A320719 Indices of primes followed by a gap (distance to next larger prime) of 42.

Original entry on oeis.org

1879, 3732, 4059, 4135, 4714, 5355, 5948, 6160, 6841, 7434, 7724, 7746, 7952, 7980, 8081, 8269, 8580, 9303, 9395, 9971, 10045, 10305, 10968, 11023, 11135, 11251, 11338, 11399, 11515, 11807, 11888, 11901, 12089, 12374, 12488, 13277, 13447, 14497, 14802, 15086, 15089, 15350, 15612, 15785

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A134120.

Harvey P. Dale, Table of n, a(n) for n = 1..3000
Index entries for primes, gaps between.

Cf. A029707, A029709 (analog for gaps 2 & 4), A320701, A320702, ... A320720 (analog for gaps 6, 8, ..., 44), A116493 (gap 70), A116496 (gap 100), A116497 (gap 200), A116495 (gap 210).
Equals A000720 o A134120.
Indices of 42's in A001223.
Row 21 of A174349.

Position[Differences[Prime[Range[16000]]],42]//Flatten (* Harvey P. Dale, Feb 22 2020 *)

A(N=100,g=42,p=2,i=primepi(p)-1,L=List())={forprime(q=1+p,,i++; if(p+g==p=q, listput(L,i); N--||break));Vec(L)} \\ returns the list of first N terms of the sequence

a(n) = A000720(A134120(n)).

A356221 Position of second appearance of 2n in the sequence of prime gaps A001223; if 2n does not appear at least twice, a(n) = -1.

Original entry on oeis.org

3, 6, 11, 72, 42, 47, 62, 295, 180, 259, 297, 327, 446, 462, 650, 1315, 1059, 1532, 4052, 2344, 3732, 3861, 8805, 7234, 4754, 2810, 4231, 14124, 5949, 9834, 17200, 10229, 19724, 25248, 15927, 30765, 42673, 28593, 24554, 50523, 44227, 44390, 29040, 89715, 47350

Offset: 1

Gus Wiseman, Aug 02 2022

nonn

Prime gaps (A001223) are the differences between consecutive prime numbers. They begin: 1, 2, 2, 4, 2, 4, 2, 4, 6, ...

The position of the first (instead of second) appearance of 2n is A038664.
Column k = 2 of A356222.
The position of the n-th appearance of 2n is A356223.
A001223 lists the prime gaps, reduced A028334.
A073491 lists numbers with gapless prime indices.
A274121 counts appearances of the n-th prime gap in those prior.
A356226 gives the lengths of maximal gapless intervals of prime indices.
Cf. A029709, A066205, A137921, A193829, A287170, A328335, A328457, A356224, A356225.

nn=1000;
gaps=Differences[Array[Prime,nn]];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[gaps,2*n][[2,1]],{n,mnrm[Select[Range[nn],Length[Position[gaps,2*#]]>=2&]]}]

cp's OEIS Frontend

A076976 Product of the smallest prime divisors of composite numbers between successive primes.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A320708 Indices of primes followed by a gap (distance to next larger prime) of 20.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A320704 Indices of primes followed by a gap (distance to next larger prime) of 12.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A320705 Indices of primes followed by a gap (distance to next larger prime) of 14.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A320706 Indices of primes followed by a gap (distance to next larger prime) of 16.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A320707 Indices of primes followed by a gap (distance to next larger prime) of 18.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A320713 Indices of primes followed by a gap (distance to next larger prime) of 30.

Views

Author

Keywords

Comments

Links

Crossrefs