A322977 -id:A322977 - cp's OEIS frontend

A322976 Number of divisors d of n such that d+2 is prime.

1, 1, 2, 1, 2, 2, 1, 1, 3, 2, 2, 2, 1, 1, 4, 1, 2, 3, 1, 2, 3, 2, 1, 2, 2, 1, 4, 1, 2, 4, 1, 1, 3, 2, 3, 3, 1, 1, 3, 2, 2, 3, 1, 2, 6, 1, 1, 2, 1, 2, 4, 1, 1, 4, 3, 1, 3, 2, 2, 4, 1, 1, 4, 1, 3, 3, 1, 2, 3, 3, 2, 3, 1, 1, 4, 1, 3, 3, 1, 2, 5, 2, 1, 3, 3, 1, 4, 2, 1, 6, 1, 1, 2, 1, 3, 2, 1, 1, 5, 2, 2, 4, 1, 1, 7

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

			10395 has 32 divisors: [1, 3, 5, 7, 9, 11, 15, 21, 27, 33, 35, 45, 55, 63, 77, 99, 105, 135, 165, 189, 231, 297, 315, 385, 495, 693, 945, 1155, 1485, 2079, 3465, 10395]. When 2 is added to each, as 1+2 = 3, 3+2 = 5, 5+2 = 7, etc, the only sums that are primes are: [3, 5, 7, 11, 13, 17, 23, 29, 37, 47, 79, 101, 107, 137, 167, 191, 233, 317, 947, 1487, 2081, 3467], thus (a10395) = 22.

Antti Karttunen, Table of n, a(n) for n = 1..10395
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A010051, A067513, A072627, A100821, A322358, A322975, A322977, A322978.

Array[DivisorSum[#, 1 &, PrimeQ[# + 2] &] &, 105] (* Michael De Vlieger, Jan 04 2019 *)

A322976(n) = sumdiv(n, d, isprime(d+2));

a(n) = Sum_{d|n} A010051(d+2).

a(A000040(n)) = 1 + A100821(n).

A322978 Number of even divisors d of 2n such that d-1 is prime.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..15120
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A010051, A059841.

Bisection of A322977.

Array[DivisorSum[2 #, 1 &, And[EvenQ@ #, PrimeQ[# - 1]] &] &, 105] (* Michael De Vlieger, Jan 04 2019 *)

A322978(n) = sumdiv(n+n, d, (!(d%2))*isprime(d-1));

A322977(n) = sumdiv(n, d, (!(d%2))*isprime(d-1));
A322978(n) = A322977(n+n);

a(n) = A322977(2*n).

a(n) = Sum_{d|(2*n), d>1} A059841(d)*A010051(d-1).

A323156 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A323155(n) for all n, except f(1) = 0.

Original entry on oeis.org

1, 2, 3, 4, 2, 5, 2, 6, 3, 2, 2, 7, 2, 8, 3, 6, 2, 9, 2, 10, 3, 2, 2, 11, 2, 2, 3, 12, 2, 13, 2, 14, 3, 2, 2, 15, 2, 16, 3, 17, 2, 18, 2, 19, 3, 2, 2, 20, 2, 2, 3, 4, 2, 21, 2, 22, 3, 2, 2, 23, 2, 24, 3, 14, 2, 5, 2, 25, 3, 8, 2, 26, 2, 27, 3, 28, 2, 5, 2, 29, 3, 2, 2, 30, 2, 2, 3, 31, 2, 32, 2, 4, 3, 2, 2, 33, 2, 34, 3, 10, 2, 35, 2, 36, 3

Offset: 1

Antti Karttunen, Jan 09 2019

nonn

For all i, j:
  a(i) = a(j) => A072627(i) = A072627(j),
  a(i) = a(j) => A323157(i) = A323157(j) => A322977(i) = A322977(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A323157, also A322315.

Cf. A072627, A322977.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A323155(n) = { my(m=1); fordiv(n, d, if(isprime(d-1), m *= (d-1)^(1+valuation(n,d-1)))); (m); };
v323156 = rgs_transform(vector(up_to, n, if(1==n,0,A323155(n))));
A323156(n) = v323156[n];

A322975 Number of divisors d of n such that d-2 is prime.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 2, 2, 0, 0, 1, 2, 1, 1, 2, 0, 2, 1, 1, 1, 0, 2, 2, 0, 1, 2, 2, 0, 2, 1, 1, 4, 0, 0, 1, 2, 2, 0, 2, 0, 1, 2, 2, 1, 0, 0, 3, 1, 1, 4, 1, 2, 1, 0, 1, 1, 2, 0, 2, 1, 0, 4, 2, 1, 2, 0, 2, 2, 0, 0, 3, 2, 1, 0, 1, 0, 4, 3, 1, 1, 0, 2, 1, 0, 2, 3, 3, 0, 0, 1, 2, 5

Offset: 1

Antti Karttunen, Jan 04 2019

nonn

			10395 has 32 divisors: [1, 3, 5, 7, 9, 11, 15, 21, 27, 33, 35, 45, 55, 63, 77, 99, 105, 135, 165, 189, 231, 297, 315, 385, 495, 693, 945, 1155, 1485, 2079, 3465, 10395]. When 2 is subtracted from each, as 1-2 = -1, 3-2 = 1, 5-2 = 3, etc, the only differences that are primes are: [3, 5, 7, 13, 19, 31, 43, 53, 61, 97, 103, 163, 229, 313, 383, 691, 1153, 1483, 3463], thus (a10395) = 19.

Antti Karttunen, Table of n, a(n) for n = 1..10395
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A010051, A062301, A067513, A072627, A322358, A322976, A322977, A322978.

Array[DivisorSum[#, 1 &, PrimeQ[# - 2] &] &, 105] (* Michael De Vlieger, Jan 04 2019 *)

A322975(n) = sumdiv(n, d, isprime(d-2));

a(n) = Sum_{d|n, d>2} A010051(d-2).

a(A000040(n)) = A062301(n).

A323157 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A000265(A323155(n)) for all n, except with f(1) = 0.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 2, 2, 2, 6, 2, 7, 2, 5, 2, 8, 2, 9, 2, 2, 2, 10, 2, 2, 2, 11, 2, 12, 2, 13, 2, 2, 2, 14, 2, 15, 2, 16, 2, 17, 2, 18, 2, 2, 2, 19, 2, 2, 2, 3, 2, 20, 2, 21, 2, 2, 2, 22, 2, 23, 2, 13, 2, 4, 2, 24, 2, 7, 2, 25, 2, 26, 2, 27, 2, 4, 2, 28, 2, 2, 2, 29, 2, 2, 2, 30, 2, 31, 2, 3, 2, 2, 2, 32, 2, 33, 2, 9, 2, 34, 2, 35, 2

Offset: 1

Antti Karttunen, Jan 09 2019

nonn

For all i, j:
  A305801(i) = A305801(j) => a(i) = a(j),
  A323156(i) = A323156(j) => a(i) = a(j),
  a(i) = a(j) => A322977(i) = A322977(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000265, A305801, A322977, A323155, A323156.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A000265(n) = (n/2^valuation(n, 2));
A323155(n) = { my(m=1); fordiv(n, d, if(isprime(d-1), m *= (d-1)^(1+valuation(n,d-1)))); (m); };
A323157aux(n) = if(1==n,0,A000265(A323155(n)));
v323157 = rgs_transform(vector(up_to, n, A323157aux(n)));
A323157(n) = v323157[n];

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A322976 Number of divisors d of n such that d+2 is prime.

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A322978 Number of even divisors d of 2n such that d-1 is prime.

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A323156 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A323155(n) for all n, except f(1) = 0.

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A322975 Number of divisors d of n such that d-2 is prime.

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A323157 Lexicographically earliest sequence such that for all i, j, a(i) = a(j) => f(i) = f(j), where f(n) = A000265(A323155(n)) for all n, except with f(1) = 0.

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