yes,definitely.

Of course, also because the samples have different (small) dimensions and also different correlations.

However, the p-value is only the tip of the iceberg in hypothesis testing, which cannot be based on the value of the p-value alone.

Perhaps it might be better to forget about p values especially when you are calculating correlations.

The expectation is that you can achieve a significant outcome if your sample size is large enough: increasing sample size results in decreasing p-value.

In your example, n=15 and n=30 there is no guarantee that this will be observed because you only have two data points (one at n=15 and one at n=30). Due to issues in sampling it is possible that either one or both of these particular samples is unusual.

In the way most people deal with statistical analyses the p-value is one immutable number. However, this viewpoint is mistaken. The p-value is an estimate of some unknowable true value. If you were to take other samples you would get other p-values. Do that enough and you can get a mean, standard deviation, 95% confidence interval, and so forth for a p-value.

Generally, p-value can hugely be dominated by sample sizes. Suppose you are running an obesity drug efficacy check. Suppose, you have 50 observations and the mean of 250 pounds has dropped to a mean of 249 pounds. Do you accept the drug to be effective? Of course, not. This will also be reflected by your p-value according to the null hypothesis that the drug has no effect. Now run a exercise. Say you have 1 million bivariate observations. Simulate a data with 1 million observations and the corresponding pre-drug and post-drug means (250 and 249), and just check the p-value! You shall reject the null and accept that the drug is effective.

The reason is that in popular tests (say, t-test), if you look at the expression of the p-values, it turns out to be a decreasing function in sample size and you tend to reject any null hypotheses. So, to say if a drug is effective on mice, just buy a million mice!

An alternate to p-value has been proposed; it is known as d-value. Instead of looking at the probability of difference of the means it actually calculates probability of individual differences (probability that X>Y). This is much more robust as long as sample size is concerned. You shall accept the null if the d-value is close to 0.5 and as it calculates the risk on application of drug, you will say a drug is effective if the d-value is low (here, cut-offs as such does not work, like level of a test). I should agree with Gioacchino de Candia that "p-value is only the tip of an iceberg". For more information, look at this article.

Article The P -value you can’t buy

The p-values is affected by the sample size. Larger the sample size, smaller is the p-values. However as already answered it is also effected by null hypothesis. Increasing the sample size will tend to result in a smaller P-value only if the null hypothesis is false.

There is one freely downloadable software named G*power 3.1.9.2. Where among the list of analysis, select correlation study and do input the sample size along with the Power value (i.e.,)(1- beta error probability).

We can observe in the output panel, that with the decrease in sample size, there is subsequent increase in the alpha error probability.

Likewise more discussion on this topic can be found in the manual

http://www.psychologie.hhu.de/arbeitsgruppen/allgemeine-psychologie-und-arbeitspsychologie/gpower.html