A175144 -id:A175144 - cp's OEIS frontend

A189827 a(n) = d(n-1) + d(n+1), where d(k) is the number of divisors of k.

3, 5, 4, 7, 4, 8, 5, 8, 5, 10, 4, 10, 6, 9, 6, 11, 4, 12, 6, 10, 6, 12, 5, 12, 7, 10, 6, 14, 4, 14, 6, 10, 8, 13, 6, 13, 6, 12, 6, 16, 4, 14, 8, 10, 8, 14, 5, 16, 7, 12, 6, 14, 6, 16, 8, 12, 6, 16, 4, 16, 8, 11, 10, 15, 6, 14, 6, 14, 6, 20, 4, 16, 8, 10, 10

Offset: 2

T. D. Noe, Apr 28 2011

nonn

d(n-1) + d(n+1) is a measure of the compositeness of the numbers next to n. Sequence A189825 lists the first occurrence of each number.

It is conjectured that every number greater than 3 occurs an infinite number of times. Note that an infinite number of 4's is equivalent to there being an infinite number of twin primes (A001097). An infinite number of 5's is equivalent to there being an infinite number of primes of the form p^2-2 (A028871) or p^2+2 (A056899) for prime p. An infinite number of 6's is equivalent to there being an infinite number of primes of the form p^3-2 (A066878), p^3+2 (A048636), p*q-2 (A063637), or p*q+2 (A063638), where p and q are distinct primes.

			a(5) = d(4) + d(6) = 3 + 4 = 7.

T. D. Noe, Table of n, a(n) for n = 2..10000

Cf. A175144, A189825.

Table[DivisorSigma[0,n-1] + DivisorSigma[0,n+1], {n, 2, 100}]
First[#]+Last[#]&/@Partition[DivisorSigma[0,Range[80]],3,1] (* Harvey P. Dale, May 27 2013 *)

A090482 Smallest prime p such that tau(p-1) + tau(p+1) is n, or 0 if no such number exists.

Original entry on oeis.org

0, 0, 2, 0, 3, 0, 5, 7, 0, 11, 17, 19, 37, 29, 0, 41, 101, 79, 0, 71, 197, 179, 401, 199, 2917, 181, 577, 239, 3137, 883, 4357, 419, 1297, 701, 12101, 839, 62501, 881, 30977, 1429, 21317, 2351, 16901, 1259, 287297, 1871, 1008017, 2161, 7057, 4049, 215297, 3079

Offset: 1

Amarnath Murthy, Dec 02 2003

nonn

a(9)=0. Proof: Both p-1 and p+1 are even and composite hence 9=1+8 and 9=2+7 are ruled out, the only possibilities that remain are 9 = 3+6, or 9=4+5. 3+6 is ruled out as 4 is the only even number with 3 divisors. 4+5 is ruled out as 16 is the only even number with 5 divisors.

a(15) = a(19) = 0 is also provable. - David Wasserman, Nov 17 2005

			a(10) = 11, tau(10) = 4 and tau(12) = 6, 4+6=10.
a(16) = 41, a(17) = 101.

Cf. A090481, A090483, A175144.

nn = 60; t = Table[-1, {nn}]; t[[{1,2,4,6,9,15,19}]] = 0; cnt = 7; p = 1; While[cnt < nn, p = NextPrime[p]; s = DivisorSigma[0, p-1] + DivisorSigma[0, p+1]; If[s <= nn && t[[s]] == -1, t[[s]] = p; cnt++]]; t (* T. D. Noe, Apr 28 2011 *)

Least prime p such that A175144(p) = n.

More terms from David Wasserman, Nov 17 2005

A189825 Least number k such that d(k-1) + d(k+1) = n, where d(k) is the number of divisors of k.

Original entry on oeis.org

2, 4, 3, 14, 5, 7, 15, 11, 17, 19, 35, 29, 65, 41, 101, 79, 143, 71, 197, 161, 323, 169, 2917, 181, 577, 239, 575, 629, 899, 419, 1297, 701, 901, 721, 25599, 881, 5183, 1121, 9215, 2351, 4901, 1079, 107585, 1681, 36863, 2159, 3601, 2881, 11663, 2519

Offset: 3

T. D. Noe, Apr 28 2011

nonn

The function d(k-1) + d(k+1) is a measure of the compositeness of the numbers next to k. There is no k for n=1 and n=2. Some terms can be quite large; for example, a(99) = 6533135.

Donovan Johnson, Table of n, a(n) for n = 3..880

Cf. A175144.

nn = 100; t = Table[-1, {nn}]; t[[1]] = t[[2]] = 0; cnt = 2; n = 1; While[cnt < nn, n++; s = DivisorSigma[0,n-1] + DivisorSigma[0,n+1]; If[s <= nn && t[[s]] == -1, t[[s]] = n; cnt++]]; Drop[t,2]

Least k such that A189827(k) = n.

A175146 a(n) = d(A175145(n)-1)+d(A175145(n)+1), where d(m) is the number of divisors of m.

Original entry on oeis.org

3, 5, 7, 8, 10, 11, 12, 14, 16, 20, 22, 26, 28, 32, 34, 36, 38, 44, 46, 48, 56, 60, 68, 72, 76, 80, 84, 92, 96, 100, 106, 108, 128, 136, 148, 156, 160, 168, 174, 176, 196, 200, 208, 236, 244, 248, 260, 304, 328, 344, 392, 416, 436, 456, 484, 520, 528, 536, 592, 608

Offset: 1

Leroy Quet, Feb 24 2010

nonn

This sequence is strictly monotonically increasing.

Cf. A000005, A090481, A175144.

Extended by Ray Chandler, Mar 04 2010

A189536 The smallest prime p such that tau(p-1) + tau(p+1) = prime(n), or 0 if no such prime exists; where tau(k) is the number of divisors of k.

Original entry on oeis.org

0, 2, 3, 5, 17, 37, 101, 0, 401, 3137, 4357, 62501, 21317, 16901, 1008017, 15877, 1020101, 33857, 69697, 14401, 331777, 78401, 32401, 57601, 828101, 40195601, 32080897, 3326977, 876097, 476101, 199374401, 4326401, 14440001, 1299601, 33918977, 3459601, 2647719937, 145540097

Offset: 1

Juri-Stepan Gerasimov, Apr 23 2011

nonn

This is sequence A090482(n) for prime n.

Cf. A000005, A000668, A002496, A090482, A175144 (tau(p-1)+tau(p+1)).

nn = 25; t = Table[-1, {nn}]; t[[1]] = 0; t[[8]] = 0; cnt = 2; p = 1; While[cnt < nn, p = NextPrime[p]; s = DivisorSigma[0, p - 1] + DivisorSigma[0, p + 1]; If[PrimeQ[s], i = PrimePi[s]; If[i <= nn && t[[i]] == -1, t[[i]] = p; cnt++]]]; t (* T. D. Noe, Apr 28 2011 *)

a(26)-a(38) from Amiram Eldar, Jan 25 2025

cp's OEIS Frontend

A189827 a(n) = d(n-1) + d(n+1), where d(k) is the number of divisors of k.

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A090482 Smallest prime p such that tau(p-1) + tau(p+1) is n, or 0 if no such number exists.

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A189825 Least number k such that d(k-1) + d(k+1) = n, where d(k) is the number of divisors of k.

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A175146 a(n) = d(A175145(n)-1)+d(A175145(n)+1), where d(m) is the number of divisors of m.

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A189536 The smallest prime p such that tau(p-1) + tau(p+1) = prime(n), or 0 if no such prime exists; where tau(k) is the number of divisors of k.

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