A320701 -id:A320701 - cp's OEIS frontend

A320716 Indices of primes followed by a gap (distance to next larger prime) of 36.

1183, 1532, 1663, 1847, 2146, 2489, 2500, 2550, 2700, 2976, 3087, 3238, 3461, 4236, 4483, 4681, 4692, 4834, 4849, 4946, 5178, 5836, 6062, 6098, 6269, 6591, 6613, 6787, 6862, 6904, 7091, 7178, 7200, 7285, 7577, 7743, 8057, 8097, 8215, 8355, 8572, 8637, 8767, 8832, 8877, 9023, 9129, 9161

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A134117.

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 A134117.
Indices of 36's in A001223.
Row 18 of A174349.

A(N=100,g=36,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(A134117(n)).

A320716 = { i>0 | prime(i+1) = prime(i) + 36 }.

A343496 First point of the straight lines in A340649.

Original entry on oeis.org

5, 31, 194, 1061, 6456, 40080, 251721, 1617206, 10553419, 69709769, 465769825

Offset: 1

Simon Strandgaard and Jamie Morken, Apr 17 2021

prime(a(n)+1) - prime(a(n)) = n*2. E.g., for n=4: prime(a(4)+1) - prime(a(4)) = 4*2, prime(1062) - prime(1061) = 4*2, 8521 - 8513 = 8.

			For n=1, consider k's with prime gap 1*2 = 2, i.e., k's such that A001223(k)=2. k=5 is the first place where A001223(k)=2 and A141042(k)=A340649(k), so a(1)=5.
For n=2, consider k's with prime gap 2*2 = 4, i.e., k's such that A001223(k)=4. k=31 is the first place where A001223(k)=4 and A141042(k)=A340649(k), so a(2)=31.
For n=3, consider k's with prime gap 3*2 = 6, i.e., k's such that A001223(k)=6. k=194 is the first place where A001223(k)=6 and A141042(k)=A340649(k), so a(3)=194.

Simon Strandgaard, Visualization of the first 5 terms.

Cf. A000040, A001223, A141042, A340649, A029707, A029709, A320701, A320702, A320703.

n = 1
last_prime = 2
find_gap = 2
result = []
Prime.each(10_000) do |prime|
    next if prime == 2
    gap = prime - last_prime
    if gap == find_gap
        value = (n * prime) % last_prime
        if value == n * gap
            result << n
            find_gap += 2
        end
    end
    n += 1
    last_prime = prime
end
p result

a(n) = smallest k that satisfies A001223(k) = 2*n and A340649(k) = A141042(k).

A320717 Indices of primes followed by a gap (distance to next larger prime) of 38.

Original entry on oeis.org

3302, 4052, 4154, 4743, 5093, 5229, 5782, 5902, 6131, 6406, 6802, 7145, 7164, 7399, 7718, 7789, 8303, 8782, 9237, 9957, 10073, 10431, 10465, 10541, 10549, 10580, 10981, 11244, 11818, 11853, 12147, 12574, 13094, 13237, 13286, 13337, 13435, 13669, 13906, 14186, 14270, 14301, 14380, 14397

Offset: 1

M. F. Hasler, Oct 19 2018

nonn

Indices of the primes listed in A134118.

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 A134118.
Indices of 38's in A001223.
Row 19 of A174349.

A(N=100,g=38,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(A134118(n)).

cp's OEIS Frontend

A320716 Indices of primes followed by a gap (distance to next larger prime) of 36.

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A343496 First point of the straight lines in A340649.

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A320717 Indices of primes followed by a gap (distance to next larger prime) of 38.

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