A031936 -id:A031936 - 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;
```

A079019 Suppose p and q = p+18 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 49 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

5, 11, 13, 19, 23, 29, 41, 43, 61, 71, 79, 83, 89, 109, 113, 131, 139, 149, 179, 181, 193, 211, 239, 251, 331, 401, 461, 491, 503, 523, 569, 601, 659, 691, 733, 739, 743, 821, 1303, 1531, 1601, 1861, 1931, 1933, 1993, 2069, 3313, 4201, 18043

Offset: 1

Labos Elemer, Jan 24 2003

Difference patterns are [18], [2,16], [4,14], [6,12], [8,10], [10,8], [12,6], [14,4], [16,2], [2,4,12], [2,6,10], [2,10,6], [2,12,4], [4,2,12], [4,6,8], [4,8,6], [4,12,2], [6,2,10], [6,4,8], [6,6,6], [6,8,4], [6,10,2], [8,4,6], [8,6,4], [10,2,6], [10,6,2], [12,2,4], [12,4,2], [2,4,2,10], [2,4,6,6], [2,6,4,6], [2,6,6,4], [2,10,2,4], [4,2,4,8], [4,2,10,2], [4,6,2,6], [4,6,6,2], [6,2,4,6], [6,2,6,4], [6,4,2,6], [6,4,6,2], [6,6,4,2], [8,4,2,4], [10,2,4,2], [2,4,2,4,6], [2,6,4,2,4], [4,2,4,6,2], [6,4,2,4,2], [2,4,2,4,2,4].

			p=18043, q=18061 has difference pattern [4,2,10,2] and {18043,18047,18049,18059,18061} is the corresponding consecutive prime 5-tuple.

A078947[1]=41, A078949[1]=71, A078950[1]=149, A078955[1]=19, A078956[1]=43, A078959[1]=23, A078962[1]=61, A078966[1]=601, A078958[1]=1601, A078963[1]=3313, A031936[1]=A000230[9]=523.

Cf. A079016-A079024.

Corrected by Rick L. Shepherd, Aug 30 2003

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}]

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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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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