A031938 -id:A031938 - cp's OEIS frontend

A379239 Numbers k for which A003961(k)-sigma(k) is prime, where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

4, 6, 7, 10, 12, 13, 15, 19, 21, 22, 23, 28, 31, 33, 34, 35, 37, 39, 43, 45, 47, 48, 51, 53, 55, 58, 61, 67, 73, 76, 77, 79, 82, 83, 84, 89, 95, 97, 103, 105, 109, 111, 112, 113, 115, 118, 123, 124, 127, 129, 131, 141, 142, 143, 145, 148, 151, 153, 155, 156, 157, 159, 161, 163, 165, 167, 173, 185, 187, 192, 193, 199

Offset: 1

Antti Karttunen, Dec 23 2024

nonn

			10 is included as A003961(10)-sigma(10) = 21-18 = 3 which is prime.
13 is included as A003961(13)-sigma(13) = 17-14 = 3 which is prime.
23 is included as A003961(23)-sigma(23) = 29-24 = 5 which is prime.

Cf. A000203, A003961, A286385, A379238 (characteristic function).
Subsequences: A023200, A031924, A031926, A031930, A031932, A031936, A031938, etc, i.e., all primes for which the gap to the next prime is one more than some prime.
Cf. also A349165.

PARI
```
is_A379239 = A379238;
```

A031939 Upper prime of a difference of 20 between consecutive primes.

Original entry on oeis.org

907, 1657, 3109, 3433, 3967, 5737, 5923, 6007, 6823, 7669, 8263, 8563, 8803, 8887, 9277, 10243, 10453, 10687, 11047, 11113, 11197, 11467, 11617, 11677, 11887, 12007, 13147, 13441, 13669, 14107, 14197, 15493, 16963, 17539, 17827

Offset: 1

Jeff Burch

nonn

Index entries for primes, gaps between

Cf. A031937.

Prime[Select[Range[2, 5000], Prime[ # ] - Prime[ # - 1] == 20 &]] (* Stefan Steinerberger, May 14 2006 *)

a(n) = A031938(n)+20.

A079020 Suppose p and q = p+20 are primes. Define the difference pattern of (p,q) to be the successive differences of the primes in the range p to q. There are 56 possible difference patterns, shown in the Comments line. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

3, 11, 17, 23, 41, 47, 59, 83, 89, 107, 131, 137, 179, 191, 251, 293, 317, 347, 353, 359, 389, 401, 467, 503, 521, 593, 599, 653, 887, 947, 971, 1031, 1151, 1193, 1229, 1259, 1301, 1307, 1439, 1601, 1931, 1979, 1997, 2069, 2531, 3167, 3299, 4241, 5261, 5639, 5849, 8081, 10091, 17189, 18041, 19421

Offset: 1

Labos Elemer, Jan 24 2003

The 56 difference patterns are [20], [2,18], [6,14], [8,12], [12,8], [14,6], [18,2], [2,4,14], [2,6,12], [2,10,8], [2,12,6], [2,16,2], [6,2,12], [6,6,8], [6,8,6], [6,12,2], [8,4,8], [8,6,6], [8,10,2], [12,2,6], [12,6,2], [14,4,2], [2,4,2,12], [2,4,6,8], [2,4,8,6], [2,4,12,2], [2,6,4,8], [2,6,6,6], [2,6,10,2], [2,10,2,6], [2,10,6,2], [2,12,4,2], [6,2,4,8], [6,2,6,6], [6,2,10,2], [6,6,2,6], [6,6,6,2], [6,8,4,2], [8,4,2,6], [8,4,6,2], [8,6,4,2], [12,2,4,2], [2,4,2,4,8], [2,4,2,10,2], [2,4,6,2,6], [2,4,6,6,2], [2,6,4,2,6], [2,6,4,6,2], [2,6,6,4,2], [2,10,2,4,2], [6,2,4,6,2], [6,2,6,4,2], [8,4,2,4,2], [2,4,2,4,6,2], [2,6,4,2,4,2], [2,2,4,2,4,2,4].

Certain patterns are singular, i.e. occur only once like [2,2,4,2,4,2,4]. Impossible patterns are [2,14,4] or [10,10] etc.

			p=10091, q=10111 has difference pattern [2, 6, 4, 8] and {10091, 10093, 10099, 10103, 10111} is the corresponding consecutive prime 5-tuple.

A000230(10)=A031938(1)=887, A078951(1)=3299, A078965(1)=47, A078968(1)=251.

Cf. A079016-A079024.

Edited by Rick L. Shepherd, Sep 10 2003

A164513 Primes with gap to the next prime no less than 20.

Original entry on oeis.org

887, 1129, 1327, 1637, 1669, 1951, 2179, 2311, 2477, 2557, 2971, 3089, 3137, 3229, 3271, 3413, 3469, 3739, 3947, 3967, 4027, 4177, 4297, 4523, 4759, 4831, 5119, 5237, 5351, 5449, 5531, 5591, 5717, 5749, 5903, 5953, 5987, 6173, 6397, 6427, 6491, 6737

Offset: 1

Zak Seidov, Aug 14 2009

nonn

Includes all terms of A031938.

Harvey P. Dale, Table of n, a(n) for n = 1..5000

Cf. A031938 Lower prime of a difference of 20 between consecutive primes.

(*M6*)a=2;S={};Do[b=NextPrime[a];If[b-a>=20,AppendTo[S,a]];a=b,{2000}];S
Select[Partition[Prime[Range[1000]],2,1],#[[2]]-#[[1]]>19&][[All,1]] (* Harvey P. Dale, Jul 30 2020 *)

A231609 Table whose n-th row consists of primes p such that p + 2n is the next prime, read by antidiagonals.

Original entry on oeis.org

3, 7, 5, 23, 13, 11, 89, 31, 19, 17, 139, 359, 47, 37, 29, 199, 181, 389, 53, 43, 41, 113, 211, 241, 401, 61, 67, 59, 1831, 293, 467, 283, 449, 73, 79, 71, 523, 1933, 317, 509, 337, 479, 83, 97, 101, 887, 1069, 2113, 773, 619, 409, 491, 131, 103, 107

Offset: 1

T. D. Noe, Nov 26 2013

The plot has an unusual gap near 10^5. Why?

			The following sequences are read by antidiagonals
{   3,    5,   11,   17,   29,   41,   59,   71,  101,  107, ...}
{   7,   13,   19,   37,   43,   67,   79,   97,  103,  109, ...}
{  23,   31,   47,   53,   61,   73,   83,  131,  151,  157, ...}
{  89,  359,  389,  401,  449,  479,  491,  683,  701,  719, ...}
{ 139,  181,  241,  283,  337,  409,  421,  547,  577,  631, ...}
{ 199,  211,  467,  509,  619,  661,  797,  997, 1201, 1237, ...}
{ 113,  293,  317,  773,  839,  863,  953, 1409, 1583, 1847, ...}
{1831, 1933, 2113, 2221, 2251, 2593, 2803, 3121, 3373, 3391, ...}
{ 523, 1069, 1259, 1381, 1759, 1913, 2161, 2503, 2861, 3803, ...}
{ 887, 1637, 3089, 3413, 3947, 5717, 5903, 5987, 6803, 7649, ...}
...

T. D. Noe, Rows n = 1..100 of triangle, flattened

Cf. A001359, A029710, A031924, A031926, A031928.
Cf. A031930, A031932, A031934, A031936, A031938.
Cf. A000230 (numbers in first column).

nn = 10; t = Table[{}, {nn}]; complete = 0; lastP = 3; While[complete < nn, p = NextPrime[lastP]; diff = p - lastP; If[diff <= 2*nn && Length[t[[diff/2]]] < nn - diff/2 + 1, AppendTo[t[[diff/2]], lastP]; If[Length[t[[diff/2]]] == nn - diff/2 + 1, complete++]]; lastP = p]; t2 = PadRight[t, {nn, nn}, 0]; Table[t2[[n-j+1, j]], {n, nn}, {j, n}]

cp's OEIS Frontend

A379239 Numbers k for which A003961(k)-sigma(k) is prime, where A003961 is fully multiplicative with a(prime(i)) = prime(i+1), and sigma is the sum of divisors function.

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A031939 Upper prime of a difference of 20 between consecutive primes.

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A164513 Primes with gap to the next prime no less than 20.

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Mathematica

A231609 Table whose n-th row consists of primes p such that p + 2n is the next prime, read by antidiagonals.

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Mathematica